# On the classification of two-qubit group orbits and the use of coarse-
grained ‘shape’ as a superselection property

Thomas Hebdige  Controlled Quantum Dynamics Theory Group, Imperial College
London, Prince Consort Road, London SW7 2BW, UK  David Jennings  Controlled
Quantum Dynamics Theory Group, Imperial College London, Prince Consort Road,
London SW7 2BW, UK  Department of Physics, University of Oxford, Oxford, OX1
3PU, UK  School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT,
UK.

(August 27, 2024)

###### Abstract

Recently a complete set of entropic conditions has been derived for the
interconversion structure of states under quantum operations that respect a
specified symmetry action, however the core structure of these conditions is
still only partially understood. Here we develop a coarse-grained description
with the aim of shedding light on both the structure and the complexity of
this general problem. Specifically, we consider the degree to which one can
associate a basic ‘shape’ property to a quantum state or channel that captures
coarse-grained data either for state interconversion or for the use of a state
within a simulation protocol. We provide a complete solution for the two-qubit
case under the rotation group, give analysis for the more general case and
discuss possible extensions of the approach.

Figure 1: Residual symmetry-types in two-qubit systems. We associate to each
state a ‘shape’ that describes the residual symmetries of that state. As an
example, in the above figure we show the symmetry properties of the set of all
mixtures of Bell states under $SU(2)$. This is determined solely by the
triplet component of the state. The only possibilities are for the state to
have a residual symmetry of (a) a cylinder, (b) a double-sided rectangle, or
(c) a sphere. This ‘shape’ can be used to provide superselection rules when
using the two-qubit state to simulate quantum operations. Details are given in
section 4.

##  1 Introduction

Symmetry principles are ubiquitous in both the classical and quantum realms.
They constrain dynamics, connect with conservation laws and simplify
computations of physical properties. In crystallography the crystal structure
is described by a symmetry group and largely determines the properties of the
material [1].

Symmetry considerations have been extremely successful in the field of quantum
information theory [2, 3, 4]. Moreover a substantial component of this work
has addressed the regime in which symmetry principles disconnect from
conservation laws [5], and which has found application in the study of quantum
features of thermodynamics [6, 7, 8]. Quantum states that break the symmetry
can also be used to circumvent limitations on the precision of measurements
imposed by conservation laws, as described by the WAY theorem [9, 10, 11, 12,
13, 14], while metrology can be viewed as using broken symmetries to
distinguish different group transformations [15]. Symmetry principles have
also provided new insights into quantum speed limits and how quickly quantum
operations can be performed [16]. Finally, symmetry groups provide an
invaluable toolkit in the context of quantum computing [17, 18, 19, 20].

In this work we look at how general quantum states $\rho$ can be classified by
groups that break a given symmetry constraint to an equal degree. A concrete
example makes this clearer: both a chair and an arrow break rotational
symmetry, however an arrow has a residual (axial) symmetry group to it, and so
breaks rotations to a weaker degree than the chair. For a fixed quantum system
of dimension $d$, and symmetry action $G$, we can ask what residual symmetries
does the system admit, and how do these residual symmetries transform under
symmetric dynamics. We also address the degree to which this aspect of a
quantum system constitutes a property in the ‘resource-theoretic’ sense.

The structure of the paper is as follows. In the next section we provide
notation and background motivation. In section 3 we provide the coarse-
graining scheme and some basic relations involved. This demonstrates the
complexity of the topic, and highlights some subtleties if one wishes to
associate the scheme to a quantum resource. In section 4 we provide a complete
classification for two-qubit systems (partly pictured in Fig. 1 and more fully
in Fig. 5), and which illustrates how quantum state spaces admit a coarse-
grained partial order.

##  2 Background and Motivation

In this paper we shall use the following notation. For any quantum system $A$
we denote by ${\mathcal{H}}_{A}$ its associated Hilbert space, and
${\mathcal{B}}({\mathcal{H}}_{A})$ the set of linear operators on
${\mathcal{H}}_{A}$. We assume that a symmetry action of $G$ is defined on
${\mathcal{H}}_{A}$ through the unitary representation $g\mapsto
U(g)\in{\mathcal{B}}({\mathcal{H}}_{A})$. The group action on a quantum state
$\rho\in{\mathcal{B}}({\mathcal{H}}_{A})$ is given by the adjoint action
$\mathcal{U}_{g}(\rho):=U(g)\ \rho\ U^{\dagger}(g)$.

For any quantum channel
${\mathcal{E}}:{\mathcal{B}}({\mathcal{H}}_{A})\rightarrow{\mathcal{B}}({\mathcal{H}}_{A^{\prime}})$,
the action of the group on $\mathcal{E}$ is
$\mathfrak{U}_{g}({\mathcal{E}}):=\mathcal{U}_{g}\circ\mathcal{E}\circ\mathcal{U}_{g^{-1}}$
(where $\circ$ denotes the concatenation of channels). This can be interpreted
as moving to a different frame, performing the channel and returning to the
original frame.

If $\mathfrak{U}_{g}({\mathcal{E}})=\mathcal{E}$ for all $g\in G$ then
${\mathcal{E}}$ is called a _symmetric operation_. This includes certain
preparation procedures, and the resulting states are known as _symmetric
states_ because they are invariant under the group action:
$\mathcal{U}_{g}(\rho)=\rho$ for all $g\in G$.

###  2.1 General state transformations under a symmetry constraint

A primary concern here is the study of general quantum states or quantum
operations under a symmetry action. In particular we would like to study the
degree to which any given quantum state breaks the symmetry (a ‘resource
state’ [21, 22]). One way of comparing states in this manner is to say that
$\rho\succ\sigma$ if and only if $\rho\rightarrow\sigma={\mathcal{E}}(\rho)$
for some _symmetric_ quantum operation ${\mathcal{E}}$, which defines a
partial order on quantum states [23, 24]. A _measure_ of how much a state
breaks the symmetry is then a function $M$ that respects the partial order –
namely, if $\rho\succ\sigma$ then $M(\rho)\geq M(\sigma)$ [21, 22]. A central
question is whether a _complete_ set of measures exist that fully capture this
ordering of states, and thus asks when two quantum states can or cannot be
interconverted under the symmetry. Very recently [7] an answer was given for
this general problem in terms of single-shot entropies of the form
$H_{\mbox{\tiny min}}(R|A)$ that quantify the amount of information in a
system $A$ about an external quantum reference frame $R$. Thus it is the case
that $\rho\rightarrow\sigma$ under the symmetry constraint if and only if
$H_{\rm min}(R|A)$ decreases with respect to all reference frames $R$.

However this set of measures for the resource theory is over-complete and
highly complex to use. Applying it to general systems requires sophisticated
techniques in single-shot information theory. Here we wish to tackle a more
modest goal, and instead of describing the complete symmetry properties of a
quantum state, wish to coarse-grain states into groups that break the symmetry
in the same way. Notably, this itself turns out to be a highly non-trivial
problem and so sheds light on the structure of the more general set of
measures given in [7].

###  2.2 Efficient simulation of quantum channels

One very grounded way of probing the degree to which a quantum state breaks a
symmetry is to see how useful it is when we wish to perform tasks, such as
induce a channel on another system. More generally, if we have some target
quantum operation ${\mathcal{E}}$ on a system, we would like to know what kind
of resource states and interactions are required to realise that operation.
The degree to which a resource $\sigma$ can do this under symmetric dynamics
therefore constitutes a measure of its properties.

In [25], the symmetry properties of quantum channels were analysed, with a
focus on the orbit of a channel ${\mathcal{E}}$, defined as

$\mathcal{M}(\mathcal{E}):=\\{\mathfrak{U}_{g}({\mathcal{E}}):\ g\in G\\}.$
(1)

A similar definition applies to the orbit of states under the group, so
${\mathcal{M}}(\rho)=\\{{\mathcal{U}}_{g}(\rho):\ g\in G\\}$.

Now the type of orbit one obtains provides a natural context in which to
understand both how the channel breaks a symmetry constraint, and moreover how
one might simulate such a channel using an auxiliary quantum system $B$
prepared in a non-symmetric state $\sigma_{B}$.

It is clear that an auxiliary resource state may be used with varying
efficiency in the simulation of a channel on a system $S$, however in [25] the
question of when a state $\sigma_{B}$ can be used to induce a channel
${\mathcal{E}}$ such that its ability to simulate the very same channel on
subsequent systems is undiminished. This was motivated by the discovery of a
“catalytic coherence” protocol [26] in which a quantum state $\sigma_{B}$ is
used as a phase reference. The state undergoes non-trivial changes
$\sigma_{B}\rightarrow\sigma_{B}^{(1)}\rightarrow\sigma_{b}^{(2)}\rightarrow\cdots\sigma_{B}^{(n)}$
yet for all $n\geq 1$ its abilities as a phase reference never diminish. In
[25] the structure of such _arbitrarily repeatable_ protocols was spelled out
using harmonic analysis. It was shown that if a channel ${\mathcal{E}}$ on $S$
is simulated using a quantum state $\sigma_{B}$ under an arbitrarily
repeatable protocol then there exists a POVM measurement $\\{M_{k}\\}$ on $B$
such that

${\mathcal{E}}(\rho)=\sum_{k}\tr(M_{k}\sigma)\Phi_{k}(\rho)$ (2)

where $\\{\Phi_{k}\\}$ are completely positive maps on the primary system $S$.
The POVM $\\{M_{k}\\}$ on the auxiliary system depends explicitly on the
target channel ${\mathcal{E}}$. Crucially however, it can be understood as
resolving the location $\boldsymbol{x}$ of ${\mathcal{E}}$ on its unitary
orbit ${\mathcal{M}}({\mathcal{E}})$ to some scale that depends on the
dimension $d$ of $S$. For example, in the case of catalytic coherence, the
POVM measurement on $B$ corresponds simply to resolving a point
$\boldsymbol{x}$ on a circle (the orbit of the channel ${\mathcal{E}}$ under
phase rotations) down to an angular scale $\sim\frac{2\pi}{d}$.

The degree to which $\boldsymbol{x}$ is resolved on
${\mathcal{M}}({\mathcal{E}})$ clearly depends on the resource state
$\sigma_{B}$, and corresponds to the fact that $\sigma_{B}$ is a _quantum_
reference frame. This in turn can be viewed as inducing non-classical geometry
on ${\mathcal{M}}({\mathcal{E}})$ – for example, for $G=SU(2)$ a channel can
have orbit ${\mathcal{M}}({\mathcal{E}})$ that is a 2-sphere. The use of a
bounded reference system in this case means that only finite resolution of
points on ${\mathcal{M}}({\mathcal{E}}$) is possible, however since the
2-sphere carries a phase space structure the bounded reference case
corresponds to the so-called “fuzzy sphere” model from non-commutative
geometry [27]. This differs from catalytic coherence in that we now have
complementarity in resolving the coordinates on the sphere, and which
ultimately arises because the two orbits have quite different structure.

Thus the general efficient use of a quantum state $\sigma_{B}$ to simulate a
channel ${\mathcal{E}}$ on $S$ can be analysed in terms of two aspects:

  1. 1.

(‘Shape’) The type of orbit ${\mathcal{M}}(\sigma)$ the state has under the
group action compared to ${\mathcal{M}}({\mathcal{E}})$, independent of
metrical aspects.

  2. 2.

(‘Geometry’) The ability of $\sigma$ to encode classical coordinate data for
$\boldsymbol{x}$ in ${\mathcal{M}}({\mathcal{E}})$.

What is a necessary relation between ${\mathcal{M}}({\mathcal{E}})$ and
${\mathcal{M}}(\sigma)$? It is clear that we need $\sigma$ to break the
symmetry to a ‘larger’ degree than ${\mathcal{E}}$, however in this paper we
would like to make this statement more precise. Given the complexity of the
general problem, one motivation in this work is to start with the above
separation into ‘shape’ and ‘geometry’, and develop an organisational setting
in which we provide a coarse-graining over states of the same ‘shape’, with
the remaining task being to analyse the ‘geometry’ aspect of the state.

###  2.3 Related topics in quantum state tomography and quantum computation

Beyond the abstract problem of simulating an arbitrary quantum channel using a
resource state, there are specific contexts of importance where such a coarse-
grained division naturally arises. In [28], quantum state tomography was
studied in the context of prior information that restricts the state to a
lower-dimensional submanifold of the state space. The analysis shows that the
topological genus of the manifold can be used to bound the number of
measurements needed to discriminate states on the submanifold. The orbit of a
state (or channel) is one such submanifold, and so the study of what kinds of
orbits can exist and how they transform among themselves is of potential
relevance to quantum tomography under symmetry constraints.

A wholly practical direction where such analysis might be of relevance is in
the computational power of gate-sets in quantum computation, and how such
gate-sets interact with states that increase the computational power of the
gate-set (such as magic states for the stabilizers [29]). For example, a
recent work provides a classification of all Clifford gate-sets [17] in a
lattice hierarchy, and so aspects of the present work might be applicable in
studying how noisy quantum states increase the computational power of easily
realisable gate-sets.

###  2.4 Core questions of the present work

In subsection 2.1 we highlighted that state interconversion
$\rho\rightarrow\sigma$ under a symmetry constraint is highly non-trivial, and
while progress has been made on this fundamental problem, it is not clear how
complex it is in general. Also, as discussed in subsection 2.2 recent results
on optimal protocols show that the structure of this theory naturally
decomposes into a geometric component and the particular kind of group orbit
the quantum channel has. In light of these considerations one expects that the
group orbit level of description might provide a form of coarse-grained
description that sheds light on the more complex parent resource theory. By
studying this coarse-grained theory we can shed light on the parent theory,
and moreover extend the notion of a resource theory from being about pre-
orders on quantum states to considering pre-orders on _sub-sets_ of states.
However, in order for this to work and be meaningful, it is crucial that the
coarse-grained theory be (a) physically sensible and (b) relate naturally to
the parent theory. Therefore the core questions we address in this work are
the following:

  1. 1.

Does a coarse-grained resource-theory framework exist?

  2. 2.

Is this theory consistent with its parent resource theory? Is it physically
sensible?

  3. 3.

How complex is the general structure of symmetry-based resource theories?

In the next section we tackle the first two of these questions, by showing how
a natural partition of the state space arises that is consistent with the
finer-grained resource theory, and describe its limitations and physical
robustness. In the process, and through exhaustive analysis of the $G=SU(2)$
case for two qubits, we shed light on the crucial third question posed.

##  3 Coarse-graining states and channels under a Symmetry Action

###  3.1 Simulating Operations Under Symmetry Constraints

If one wishes to realise a quantum channel ${\mathcal{E}}$ via a symmetric
interaction with a state $\sigma$, the most natural way to model this is as

${\mathcal{E}}(\rho)=\tr_{B}V(\rho_{A}\otimes\sigma_{B})V^{\dagger}$ (3)

where $V$ is a covariant unitary, namely $[V,\ U_{A}(g)\otimes U_{B}(g)]=0$
for all $g\in G$. If such a relation holds, we say that ${\mathcal{E}}$ can be
_simulated_ using the state $\sigma_{B}$.

A basic classification of states and channels under a symmetry can be done by
studying the subgroup of residual symmetries for the state or channel, called
the isotropy subgroup or stabilizer of ${\mathcal{E}}$ and defined as

$\rm{Iso}({\mathcal{E}}):=\\{g\in G\ :\
\mathfrak{U}_{g}({\mathcal{E}})={\mathcal{E}}\\}.$ (4)

It specifies the residual symmetry that the channel has under the group
action. Symmetric channels are invariant under all group transformations, and
simply have $\rm{Iso}({\mathcal{E}})=G$. This notion also applies to states,
where $\rm{Iso}(\rho):=\\{g\in G\ :\ \mathcal{U}_{g}(\rho)=\rho\\}$.

In many cases the quantum operation ${\mathcal{E}}$ has some residual
symmetry, which is characterised an isotropy subgroup $H$ of $G$. The possible
isotropy subgroups form an abstract structure called a _lattice_ [30]. The
partial ordering of the subgroups is defined by subset inclusion, namely
$H_{1}\prec H_{2}$ if $H_{1}\subseteq H_{2}$ for $H_{1},H_{2}\subseteq G$. In
addition, every pair of elements have a unique supremum and unique infimum,
which define binary operations called the meet and join [31]. For two
subgroups $H_{1}$ and $H_{2}$, their meet is denoted $H_{1}\wedge H_{2}$ and
defined as $H_{1}\cap H_{2}$, while their join is denoted $H_{1}\vee H_{2}$,
defined as the subgroup _generated_ by $H_{1}\cup H_{2}$ [30]. This structure
is illustrated by a Hasse diagram of the subgroups, as shown in Fig. 2.

Figure 2: Coarse-grained classification of quantum states under a symmetry.
The set of all density operators is partitioned into sets $C(H)$ with definite
symmetries (seen on the left). The $C(H)$ correspond to the subgroups shown in
the Hasse diagram in the middle. The Hasse diagram represents the subgroup
lattice, with lines indicating subgroup inclusion, i.e. $H_{5}\prec H_{4}\prec
H_{3}$ etc. Note that for finite-dimensional systems many of the $C(H)$ will
be empty. The subsets $C(H)$ can be associated, up to diffeomorphisms, with
group orbits (indicated on the right). The left figure shows the action of
group-averaging (over $H_{2}$), going from $\sigma$ to
$\mathcal{P}_{H_{2}}(\sigma)$ as described in section 3.4.

Since quantum operations can be combined in a number of ways, we present some
basic statements that constrain the residual symmetry of the resultant quantum
operation. The proofs for this section are provided in Appendix A.

######  Lemma 1.

Given any two quantum channels
${\mathcal{E}}:{\mathcal{B}}({\mathcal{H}}_{A})\rightarrow{\mathcal{B}}({\mathcal{H}}_{A^{\prime}})$
and
${\mathcal{F}}:{\mathcal{B}}({\mathcal{H}}_{B})\rightarrow{\mathcal{B}}({\mathcal{H}}_{B^{\prime}})$,
with unitary representations of a group $G$ defined on all input and output
spaces. Then we have the following:

  1. 1.

$\rm{Iso}({\mathcal{E}}\otimes{\mathcal{F}})\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$
under the tensor product group action
$\mathfrak{U}_{g}({\mathcal{E}}\otimes{\mathcal{F}})=\mathfrak{U}_{g}({\mathcal{E}})\otimes\mathfrak{U}_{g}({\mathcal{F}})$.

  2. 2.

If $B^{\prime}=A$ then
$\rm{Iso}({\mathcal{E}}\circ{\mathcal{F}})\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$.

  3. 3.

If $A=B$ and $A^{\prime}=B^{\prime}$, and $p$ some probability $0\leq p\leq
1$, then
$\mathrm{Iso}(p{\mathcal{E}}+(1-p){\mathcal{F}})\succ\mathrm{Iso}({\mathcal{E}})\wedge\mathrm{Iso}({\mathcal{F}})$.

  4. 4.

$\mathrm{Iso}({\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g}^{\dagger})=g[\mathrm{Iso}({\mathcal{E}})]g^{-1}$.

These results also apply for quantum states, once we view the state as the
state preparation map $1\rightarrow\rho$. We can now make precise a coarse-
grained (necessary, but far from sufficient) requirement that a state $\sigma$
allow the simulation of a channel ${\mathcal{E}}$ under symmetric dynamics and
show that this bound is tight in general. The proof is provided in Appendix A.

######  Theorem 1.

If a system $B$ in a state $\sigma_{B}$ can be used to simulate a CPTP map
${\mathcal{E}}$ under symmetric dynamics, then
$\text{Iso}(\sigma_{B})\prec\text{Iso}({\mathcal{E}})$. Moreover, if
${\mathcal{E}}$ has isotropy group $Iso({\mathcal{E}})$ then there exists a
quantum system $B$ and quantum state $\sigma_{B}$ with
$\text{Iso}(\sigma_{B})=\text{Iso}({\mathcal{E}})$ that allows the simulation
of ${\mathcal{E}}$.

This result gives the basic relation, in the coarse-grained picture, for when
a quantum state $\sigma_{B}$ can be used to simulate a quantum channel and
also shows that this relation is a tight one. This is necessary in order for
the isotropy subgroup approach to be consistent with the parent resource
theory, where one is concerned with the minimal resources needed to realise a
particular quantum channel.

A similar result can be given in the case of approximate simulation. For any
target operation ${\mathcal{E}}$ we can define the noisy version given by

${\mathcal{E}}_{\epsilon}=(1-\epsilon)\ {\mathcal{E}}+\epsilon\ {\mathcal{D}}$
(5)

where ${\mathcal{D}}$ is the complete depolarization map
$\rho\mapsto\frac{1}{d}\mathbbm{1}$ for all $\rho$. From Lemma 1 we see that
$\mbox{Iso}({\mathcal{E}}_{\epsilon})=\mbox{Iso}({\mathcal{E}})\wedge
G=\mbox{Iso}({\mathcal{E}})$. If $\sigma$ allows the simulation of
${\mathcal{E}}_{\epsilon}$ for some $0<\epsilon<1$ then this corresponds to an
approximate, ‘isotropic’ simulation of the original map with noise parameter
$\epsilon$. Theorem 1 extends to this case in an obvious way.

###  3.2 Is ‘shape’ a resource-theoretic property?

While we normally associate measurable properties with either projective
measurements or more generally POVMs, there is an alternative way that is more
general again. Specifically, a property is associated with a _pre-order_
defined on the set of all quantum states. This recent approach is called the
resource-theory method, and has found success in areas such as entanglement,
coherence, thermodynamics, and many other scenarios [32, 4, 33, 34]. Quantum
maps that respect the pre-order are called ‘free operations’ and any real-
valued function on quantum states that respects the pre-order is a _measure_
of the property.

Given an ability to order the isotropy subgroups of quantum states and
channels under the group action, we can ask if a meaningful notion of ‘shape’
can be defined for quantum systems, along such a resource-theoretic line. This
would essentially characterise the asymmetry of states and channels without
reference to a measure.

For continuous Lie groups, we find that the orbit of a state or channel is
always a homogeneous space [35, 36], specified by both the group $G$ and the
particular isotropy subgroup $H$ of the channel. More precisely the orbit of
the channel is ${\mathcal{M}}({\mathcal{E}})$ and coincides with the quotient
$G/H$ up to diffeomorphisms, which we write ${\mathcal{M}}({\mathcal{E}})\cong
G/H$ (and likewise for states).111Technically we define the ‘shape’ of the
group orbit with respect to the conjugacy class of the isotropy subgroup $H$.
This ‘shape’ is also called the Orbit-Type in the literature. See [37] for
further details. This is what we call the ‘shape’ of a state or channel.

A simple example can be given for a single qubit state under an $SU(2)$
symmetry, where the possible types of group orbit are

$\mathcal{M}\left(\frac{1}{2}[\mathds{1}+\mathbf{r}\cdot\sigma]\right)=\begin{cases}SU(2)/U(1)\cong
S^{2}&\mathbf{r}\neq 0\\\ SU(2)/SU(2)\cong\\{e\\}&\mathbf{r}=0,\end{cases}$
(6)

and these are illustrated in Fig. 3.

Figure 3: Basic orbits of states. Group orbits for a single qubit under the
action of $SU(2)$ shown in the Bloch sphere. There are only two possibilities
in this case: a sphere for non-zero Bloch vectors (left), or a point for the
symmetric maximally mixed state (right).

The lattice of isotropy subgroups gives us a way of comparing the ‘shapes’ of
group orbits under a group action $G$, with
${\mathcal{M}}({\mathcal{E}})\prec{\mathcal{M}}({\mathcal{F}})$ iff
$\rm{Iso}({\mathcal{E}})\succ\rm{Iso}({\mathcal{F}})$ (up to isomorphism of
the isotropy subgroups). Likewise for states,
${\mathcal{M}}(\rho)\prec{\mathcal{M}}(\sigma)$ iff
$\rm{Iso}(\rho)\succ\rm{Iso}(\sigma)$ (up to isomorphism). The convention here
is to match with resource-theoretic measures of asymmetry [23, 21], so that
one ‘shape’ is ‘bigger’ than another if it has a smaller isotropy group. For
example, a single-qubit state with non-zero Bloch vector
(${\mathcal{M}}(\rho)\cong S^{2}$) has a ‘bigger’ group orbit than the
maximally mixed state (${\mathcal{M}}(\mathds{1}/2)\cong\\{e\\}$). This gives
valuable information regarding the resources required to perform a quantum
channel in the presence of symmetry constraints.

The isotropy subgroup gives us a natural equivalence relation $\sim$ between
states that behave in the same way under the group action. We can then say
that $\rho_{1}$ and $\rho_{2}$ are related $\rho_{1}\sim\rho_{2}$ if we have
that $\rm{Iso}(\rho_{1})=\rm{Iso}(\rho_{2})$. This equivalence relation
partitions the state space into sets $C(H):=\\{\rho:\ \rm{Iso}(\rho)=H\\}$,
and collects together all states that break the symmetry in the same manner.
Taking the union of $C(H)$ corresponding to isomorphic $H$ partitions the
states and channels according to their ‘shape’.

Viewed as a mapping, $C$ maps from the subgroup lattice onto a partition of
the state space, and so we can simply allow the partition to inherit the
lattice structure, and order subsets of states as $C(H_{1})\succ C(H_{2})$
whenever $H_{1}\prec H_{2}$.

Note however that the sets $\\{C(H)\\}$ are not convex in general. For example
a qubit system under $G=SU(2)$, both $\ket{0}\bra{0}$ and $\ket{1}\bra{1}$
have the same $U(1)$ isotropy subgroup, but
$\frac{1}{2}(\ket{0}\bra{0}+\ket{1}\bra{1})=\frac{1}{2}\mathds{1}$, which is
symmetric. However, $C(G)$ is always a convex set. This follows because if
$\rho_{1},\rho_{2}\in C(G)$ then Lemma 1 implies that
$G\succ\rm{Iso}(p_{1}\rho_{1}+p_{2}\rho_{2})\succ\rm{Iso}(\rho_{1})\wedge\rm{Iso}(\rho_{2})=G$,
and so any mixture has the same isotropy subgroup.

The $SU(2)$ qubit example also highlights that if $C(H)\neq\emptyset$ it does
not imply that $C(H^{\prime})\neq\emptyset$ for all $H^{\prime}\succ H$, since
there are subgroups of $SU(2)$ containing $U(1)$ which do not appear as the
isotropy subgroup of a single qubit state. Not all possible isotropy subgroups
will be seen in a given system, and for a finite dimensional system many of
the sets $C(H)$ will be empty.

However in order to have a meaningful resource-theory interpretation, the
coarse-grained ordering must be compatible with the set of free operations –
namely the symmetric operations. It is easy to show that this is in fact the
case at the level individual quantum states.

######  Lemma 2.

Under a symmetric operation ${\mathcal{E}}$,
$\mathrm{Iso}({\mathcal{E}}(\rho))\succ\rm{Iso}(\rho)$.

This shows that the coarse-grained ordering of states is consistent with the
set of free (symmetric) operations, as expected. However it does not imply
that a functional mapping is defined on the sets $C(H)$; it is possible to
have $\rho_{1}$ and $\rho_{2}$ in the same set $C(H)$, but each get sent to
different sets $C(H^{\prime}_{1})$ and $C(H^{\prime}_{2})$. This aspect means
that interpreting the coarse-graining as defining a quantum resource needs
care.

###  3.3 The speck of dust argument – tiny perturbations to symmetric states &
channels

Having outlined how the set of quantum states or the set of quantum channels
of a quantum system is partitioned according to the symmetry action one might
raise the following concern: what happens if one perturbs a channel via some
small perturbation? How does this affect its isotropy subgroup?

It is readily seen that one can make an arbitrarily small perturbation to any
quantum channel ${\mathcal{E}}$ so as to break the symmetry in a “maximal”
way. Informally this amounts to the “speck of dust” argument where, for
example, a perfectly rotationally symmetric ball can be made completely
asymmetric by sticking arbitrarily small pieces of dust onto it (Fig. 4).

Figure 4: The speck of dust argument. If we start with a perfectly
rotationally symmetric object (left image) we can always add arbitrarily small
perturbations that completely break the rotational symmetry (right image).
However physically the fact that we have finite resolution means that quantum
states and channels should always be understood as being defined up to some
$\epsilon$–smoothing scale. This does not affect the resource-theoretic
account and therefore the ‘shape’ is a robust feature in the presence of such
tiny symmetry-breaking perturbations.

In more abstract terms, one can use the principal orbit-type theorem [37] for
the space of quantum channels of a system to see that there is always a
_principal isotropy_ subset that forms a dense subset in the set of all
quantum channels. Thus “most” quantum channels on a system will break a
symmetry to a maximal degree (the principal isotropy).

This argument of course applies equally well to the partitioning of the
quantum state space into different subsets with fixed isotropies, however in
both cases this does not pose a problem for our formalism for the following
reasons. Firstly, in resource theories we are fundamentally interested in the
most efficient use of resources and therefore our task is to look for those
quantum resource states that break the symmetry not in a maximal way, but in a
_minimal_ way.

Secondly, at the level of channels, the most interesting and commonly
addressed channels in quantum information science are _not_ of principal
isotropy type — namely they have non-trivial isotropies. For example, consider
quantum channels on a single qubit, and the irreducible group action of
$G=SU(2)$. A typical quantum channel on the qubit will only have a small
residual isotropy group $\mathbb{Z}_{2}=\\{\mathds{1},-\mathds{1}\\}$ (the
principal isotropy subgroup of the set of all qubit channels). However
consider the following standard single qubit channels:

  1. 1.

The depolarizing channel:
${\mathcal{E}}(\rho)=p\rho+(1-p)(\frac{1}{2}\mathbbm{1})$, with $0\leq p\leq
1$.

  2. 2.

A rotation about the $\hat{n}$ axis:
$U_{\hat{n}}(\theta)=\exp[-i\hat{n}\cdot\boldsymbol{\sigma}]$.

  3. 3.

A projective measurement along the $\hat{n}$ axis:
${\mathcal{E}}(\rho)=\sum_{k}\tr(\Pi_{k}\rho)\Pi_{k}$, where
$\Pi_{0}=\ket{\hat{n}}\bra{\hat{n}}$ and
$\Pi_{1}=\mathds{1}-\Pi_{0}=\ket{-\hat{n}}\bra{-\hat{n}}$.

  4. 4.

Partial dephasing along the axis $\hat{n}$:
${\mathcal{E}}(\rho)=p\rho+(1-p)\sum_{k}\tr(\Pi_{k}\rho)\Pi_{k}$, where
$\Pi_{0}=\ket{\hat{n}}\bra{\hat{n}}$ and
$\Pi_{1}=\mathds{1}-\Pi_{0}=\ket{-\hat{n}}\bra{-\hat{n}}$.

  5. 5.

A qubit state preparation channel:
${\mathcal{E}}(\rho)=\frac{1}{2}(\mathds{1}+p\ \hat{n}\cdot\sigma)$.

It is readily seen that the isotropy subgroup for channel (1) is $SU(2)$,
while for channels (2–5) it is a $U(1)$ subgroup of $SU(2)$. The orbits will
have ‘shapes’ $SU(2)/SU(2)\cong\\{e\\}$ and $SU(2)/U(1)\cong S^{2}$
respectively. None of these channels have principal isotropy type
$\\{\mathbbm{1},-\mathbbm{1}\\}$, however they are clearly key quantum
operations in quantum information science for which one might wish to
determine the minimal quantum resources necessary to realise them.

Thirdly, the fact that something is “measure zero” in some space does not
imply that it is physically irrelevant. For example, a 2-d plane is a measure
zero set in three spatial dimensions, however this does not mean that anyonic
physics or the quantum hall effect is irrelevant. Or of closer bearing on our
present work, we can consider the case of Noether’s theorem: clearly the set
of dynamics with rotational symmetry is again “measure zero”, however this
does not mean that Noether’s theorem is irrelevant for physics. In both cases
while the restrictions are strictly unstable under small perturbations, the
important thing is that the features of interest are operationally robust
under small perturbations.

The speck of dust subtlety makes itself apparent in our setting if one tries
to naively exploit some metric on quantum states to quantify how far apart the
sets $\\{C(H)\\}$ are from each other. It is readily seen that the distance
been any two such sets is in fact zero.

######  Lemma 3.

Let $d(\cdot,\cdot)$ be any metric on the space of quantum states. In terms of
this metric we define

$\mathbf{d}(C(H_{1}),C(H_{2})):=\inf_{\begin{subarray}{c}\sigma_{1}\in
C(H_{1})\\\ \sigma_{2}\in C(H_{2})\end{subarray}}d(\sigma_{1},\sigma_{2}).$
(7)

Then $\mathbf{d}(C(H_{1}),C(H_{2}))=0$ for all $H_{1},H_{2}\prec G$.

This also shows that any set $C(H)$ is arbitrarily close to the set $C(G)$.
However, this simply means that any symmetry properties must always be
considered up to some _finite resolution scale_ – arbitrarily small
perturbations can always eliminate residual symmetries, even though this state
is practically indistinguishable from the unperturbed one.

Therefore any membership of a quantum state $\rho$ to a subset should only be
considered up to some smoothing scale $\epsilon$ based on a distance measure
$d$ (such as from the $L_{1}$ norm). For each state there is an
$\epsilon$-ball of nearby states,
$\mathcal{B}_{\epsilon}(\rho)=\\{\sigma:d(\rho,\sigma)\leq\epsilon\\}$. Rather
than $\rm{Iso}(\rho)$, we should consider
$\rm{Iso}_{\epsilon}(\rho):=\max_{\sigma\in\mathcal{B}_{\epsilon}(\rho)}\rm{Iso}(\sigma)$.
Intuitively, if a state $\rho$ is within a distance $\epsilon$ of another
state with more residual symmetries, we associate those additional residual
symmetries with $\rho$. The smoothing scale selects the size of symmetry-
breaking perturbations that we wish to consider, and will depend on the
particular physical context.

This reasoning applies equally to quantum channels. For example, the
depolarisation channel on a qubit is fully invariant under the $G=SU(2)$ and
so no non-trivial resource state is needed to simulate it. If one perturbs
this channel to ${\mathcal{E}}\rightarrow{\mathcal{E}}_{\epsilon}$ by some
$\epsilon>0$ perturbation in the diamond norm one can clearly break this
isotropy to the principal isotropy $\\{\mathbbm{1},-\mathbbm{1}\\}$, however
it is also clear (e.g. from the continuity of the Stinespring dilation [38])
that ${\mathcal{E}}_{\epsilon}$ can either be simulated exactly with a
resource state on $B$ that $\epsilon$-close to be being symmetric, or
simulated approximately to the same threshold with a perfectly symmetric state
on $B$. Thus the use of isotropy groups within the resource theory context is
robust under such small perturbations and the speck of dust argument is moot.
We illustrate this point explicitly in Section 4, where we demonstrate that
for the case of two qubits under $SU(2)$ the notion of different ‘shapes’ is
perfectly robust under the particular case of perturbations that allow a $4\%$
error rate.

###  3.4 Projecting out residual symmetries via quantum operations

Given the structure of states under the above partition, we can ask how easy
it is to move from one set $C(H)$ to another. As discussed one does not in
general have quantum operations that map any $C(H)$ neatly into some other
subset. Instead it makes sense to consider the sets
$\hat{C}(H):=\bigcup_{W\succ H}C(W)$. These sets are quite natural to consider
because Lemma 2 implies that each $\hat{C}(H)$ is closed under symmetric
operations, and any symmetric operation provides a well-defined mapping of
$\hat{C}(H)\rightarrow\hat{C}(f(H))$.

The quantum operation

$\mathcal{P}_{H}(\rho):=\int_{H}dh\ \mathcal{U}_{h}(\rho)$ (8)

is the average of the state $\rho$ over a fixed subgroup $H$ weighted by the
invariant Haar measure $dh$. In the case of a finite subgroup of size $|H|$,
the integral $\int_{H}dh$ is replaced by the sum $\frac{1}{|H|}\sum_{h\in H}$.

######  Lemma 4.

The map $\mathcal{P}_{H}$ has the following properties:

  1. 1.

$\mathcal{P}_{H}$ is the (orthogonal) projector onto $\hat{C}(H)$.

  2. 2.

$\mathcal{P}_{H}(\rho)=\arg\min_{\sigma\in\hat{C}(H)}S(\rho||\sigma)$, where
$S(\rho||\sigma)$ is the relative entropy.

Therefore $\mathcal{P}_{H}$ projects onto $\hat{C}(H)$, as illustrated in Fig
2. Dephasing [20], $\mathcal{E}(\rho)=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\
e^{i\theta Z}\rho e^{-i\theta Z}$, can be viewed as ${\mathcal{P}}_{U(1)}$,
sending $\rho$ to the nearest incoherent state while preserving the diagonal
terms of a density matrix.

The mapping $\mathcal{P}_{H}$ moves down chains of the subgroup Hasse diagram,
however this is not always a symmetric operation. The following tells us when
such a transformation can be performed freely in the resource theory.

######  Lemma 5.

Given a group action for $G$, $\rm{Iso}(\mathcal{P}_{H})=N_{G}(H)$, where
$N_{G}(H)=\\{g\in G\ :\ gHg^{-1}=H\\}$ is the normalizer [39] of $H$ in $G$,
and therefore if $H$ is a normal subgroup of $G$ ($H\triangleleft G$) then
${\mathcal{P}}_{H}$ is a symmetric operation.

This constrains the kind of resources needed to move from one $C(H)$ to
another using $\mathcal{P}_{H}$, since this projects onto a given
$\hat{C}(H)$, although less resource-hungry ways may exist to perform the same
transformation.

###  3.5 Discussion of the section results

In this section we have given an analysis of the what happens when one
classifies quantum states or channels in terms of their residual symmetries.
We discussed how composition and mixing affect the ordering, and how one can
relate these features to the issue of simulation (exact or approximate) of a
target quantum operation with some resource state. We also saw that the
ordering has subtleties if one wishes to interpret it in a resource theoretic
sense. The statements one makes are also quite blunt, and a good example of
this is the fact that the sets $\\{C(H)\\}$ are all arbitrarily close to one
another. However the relations still carry non-trivial content, and for
example Theorem 1 can be viewed as a form of superselection rule on quantum
operations that tells us which states are ruled out and which are not.

While this is conceptually neat and provides high-level insight into the
complexities of the full classification of states (as described in [7]), it is
less clear how computationally useful or simple these structures are in
practice. To address this point, in the next section we consider the important
case of a two-qubit system under an $SU(2)$ action. We find that already in
just this simple scenario the hierarchy of states is quite complex, however
the example does provide insight into what to expect in the more general case.

##  4 Illustrative Example: Two Qubits Under $SU(2)$ Symmetry Constraints

The goal of this section is to illustrate the classification of quantum states
in a simple quantum system that has sufficient structure, yet is tractable to
the point of being fully solvable. The state space of a two-qubit system is 15
dimensional and is sufficiently non-trivial, moreover there is a very natural
group action to consider, namely the tensor product representation of $SU(2)$.
The orbits of 2-qubit states have been studied in relation to thermodynamics
and correlations within these states [40, 41].

Any two qubit state $\rho_{AB}$ can be written

$\displaystyle\rho_{AB}$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma$
$\displaystyle\qquad\qquad\left.+\sum_{i,j=1}^{3}T_{ij}\
\sigma_{i}\otimes\sigma_{j}\right),$ (9)

with local Bloch vectors $\mathbf{a}$ and $\mathbf{b}$ and correlation terms
determined by the correlation matrix $T_{ij}$ [42]. In general
$|\mathbf{a}|\leq 1$ and $|\mathbf{b}|\leq 1$. Moreover, we can put all
2-qubit states into diagonal form,

$\displaystyle\rho_{AB}$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma$
$\displaystyle\qquad\qquad\left.+\sum_{i=1}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right)$ (10)

where $\\{\mathbf{c}_{i}\\}$ and $\\{\mathbf{d}_{i}\\}$ are orthonormal bases
of $\mathbb{R}^{3}$. The coefficients $\tau_{i}$ specify a position
$(\tau_{1},\tau_{2},\tau_{3})$ in a tetrahedron with vertices
$(-1,-1,-1),(1,1,-1),(1,-1,1)$ and $(-1,1,1)$. When $\mathbf{a}=\mathbf{b}=0$,
then $\rho_{AB}$ is a quantum state with maximally mixed marginals represented
by a point in the tetrahedron. In the case $\mathbf{a}$ or $\mathbf{b}$ being
non-zero, every quantum state $\rho_{AB}$ corresponds to a point in the
tetrahedron, however the converse is not true: not all triples
$(\mathbf{a},\mathbf{b},\mathbf{t})$ correspond to a valid quantum states.

Via local unitaries $U_{1}\otimes U_{2}$, these can be transformed into the
canonical form

$\displaystyle\tilde{\rho}_{AB}$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+\mathbf{a}^{\prime}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}^{\prime}\cdot\sigma$
$\displaystyle\qquad\qquad\left.+\sum_{i=1}^{3}\tau_{i}\
\sigma_{i}\otimes\sigma_{i}\right)$ (11)

where
$\mathbf{a}^{\prime}\cdot\sigma=U_{1}(\mathbf{a}\cdot\sigma)U_{1}^{\dagger}$
and
$\mathbf{b}^{\prime}\cdot\sigma=U_{2}(\mathbf{b}\cdot\sigma)U_{2}^{\dagger}$.
The states in this canonical form with maximally mixed marginals are called
T-states [42],

$\rho_{T}=\frac{1}{4}\left(\mathds{1}\otimes\mathds{1}+\sum_{j}\tau_{j}\
\sigma_{j}\otimes\sigma_{j}\right).$ (12)

The set of T-states is the convex hull of the four Bell states. The corners of
the tetrahedron defined by the coefficients $\\{\tau_{i}\\}$ correspond to the
Bell states.

We now consider the symmetry properties of 2-qubit states under $SU(2)$
symmetry constraints, assuming an $SU(2)$ tensor product representation for
the group action, i.e. $\mathcal{U}_{g}(\rho)=U(g)\otimes U(g)\ \rho\
U^{\dagger}(g)\otimes U^{\dagger}(g)$, where $U(g)$ is a two dimensional irrep
of $SU(2)$.

Our description of the group orbit as the quotient space $G/\rm{Iso}(\rho)$
starts with the group manifold of $G$ and eliminates the redundancy caused by
the symmetries of $\rho$. The group transformations in $\rm{Iso}(\rho)$ have
trivial group action on $\rho$, so we quotient out the equivalence classes $g\
\text{Iso}(\rho):=\\{gh_{i}:\ h_{i}\in\text{Iso}(\rho)\\}$.

Certain groups have structure which permits us to further simplify our
description of the group orbit manifold. Here we consider $SU(2)$ symmetry
constraints, however this $SU(2)$ action has a subgroup
$\mathbb{Z}_{2}=\\{\pm\mathds{1}\\}$ that is always a symmetry of a quantum
state, and which is associated with $SU(2)$ being the double cover of $SO(3)$.
Moreover this sub-group has a key property which allows us to simplify our
description of the $SU(2)$ isotropy groups to those of $SO(3)$. We can
therefore think in terms of $SO(3)$ subgroups, which are the familiar chiral
_point groups_ one encounters in for example crystallography [1].

A subgroup $N$ of $G$ is a normal subgroup, denoted $N\triangleleft G$, if it
satisfies $gNg^{-1}=N$ for all $g\in G$ [1]. These partition $G$ into
equivalence classes $gN:=\\{gn_{i}:\ n_{i}\in N\\}$, which themselves
constitute elements of the quotient group $G/N$. Every element of $G$ is
specified by the pairing of the equivalence class $gN$ and an element of the
normal subgroup $N$. Likewise, elements of $H$ are specified by the pairing of
an element of $H/N$ and an element of $N$ because $N\triangleleft H$. This
means that the group orbit manifold $G/H$ can be described by the quotient
groups $G/N$ and $H/N$, and we can write ${\mathcal{M}}(\rho)\cong
G/H\cong(G/N)/(H/N)$. In our present analysis $\mathbb{Z}_{2}$ is a normal
sub-group of $SU(2)$ common to all quantum states and so
${\mathcal{M}}(\rho)\cong SO(3)/(H/\mathbb{Z}_{2})$ where $H$ is the isotropy
group of $\rho$ in $SU(2)$. We can therefore use the more intuitive $SO(3)$
subgroups, listed in Table 1.

Group | Description
---|---
$C_{n}$ | Cyclic group of order $n$
$C_{\infty}$ | Symmetries of Cone
$D_{n}$ | Symmetries of a regular $n$-sided Polygon
$D_{\infty}$ | Symmetries of Cylinder
$T$ | Symmetries of Tetrahedron
$O$ | Symmetries of Cube/Octahedron
$I$ | Symmetries of Dodecahedron/Icosahedron
Table 1: Chiral point groups: subgroups of $SO(3)$, as detailed in [1]. All
quantum states have isotropy subgroups under $SU(2)$ that can be related to
particular chiral point group.

In the next section we look at the isotropy subgroups of the Bell states, and
then go on to classify the symmetry properties of the T-states. Then we
continue onto the wider class of 2-qubit states with maximally mixed marginals
and finally we complete the classification for general 2-qubit states by
considering non-zero local Bloch vectors.

###  4.1 Bell States

The Bell states correspond to the extremal points of the tetrahedron of
T-states. Their isotropy subgroups under the $SU(2)$ group action can be
calculated directly, as described in Appendix B.

The singlet Bell state $\psi^{-}$ is symmetric under $SU(2)$, therefore
$\rm{Iso}(\psi^{-})=SU(2)$ and so ${\mathcal{M}}(\psi^{-})=\\{e\\}$ as
expected. The triplet Bell states do break the $SU(2)$ symmetry, with

$\displaystyle\rm{Iso}(\phi^{+})$ $\displaystyle=\\{(iZ)^{\alpha}e^{i\theta
Y}:\ 0\leq\theta<2\pi,\ \alpha=0,1,2,3\\}.$ (13)

We will call this subgroup $K_{\infty}\cong U(1)\rtimes\mathbb{Z}_{4}$, and it
provides an example of a semi-direct product group222 The semi-direct product
[1] is constructed from two groups, in this case $\mathbb{Z}_{4}$ and $U(1)$.
Each element of the semidirect product group can be thought of as a pairing
$(h,n)$ where $h\in H=\mathbb{Z}_{4}$ and $n\in N=U(1)$. However the
multiplication law of the semidirect product group does not treat the elements
of this pairing independently, with
$(h_{1},n_{1})(h_{2},n_{2})=(h_{1}h_{2},f_{h_{2}}(n_{1})n_{2})$ where
$f_{h}:N\rightarrow N$ is a automorphism on the second group specified by an
element of the first group. The semi-direct product $\rtimes$ is defined by
the choice of automorphism $f_{h}$. Note that $N\triangleleft(N\rtimes H)$. If
a trivial (identity) automorphism is chosen, where $f_{h}(n)=n$ for any $h$
and $n$, we have a direct product group [1]. In the
$U(1)\rtimes\mathbb{Z}_{4}$ example, the group multiplication law is
$\left[(iZ)^{\alpha_{1}}e^{i\theta_{1}Y}\right]\
\left[(iZ)^{\alpha_{2}}e^{i\theta_{2}Y}\right]=(iZ)^{\alpha_{1}+\alpha_{2}}\
e^{i[(-1)^{\alpha_{2}}\theta_{1}+\theta_{2}]Y}.$ . Similarly,

$\displaystyle\rm{Iso}(\phi^{-})=\\{(iY)^{\alpha}e^{i\theta X}:\
0\leq\theta<2\pi,\ \alpha=0,1,2,3\\}$ (14)

and

$\displaystyle\rm{Iso}(\psi^{+})=\\{(iX)^{\alpha}e^{i\theta Z}:\
0\leq\theta<2\pi,\ \alpha=0,1,2,3\\},$ (15)

both of which are isomorphic to $K_{\infty}$.

Given the triplet Bell states have isomorphic isotropy subgroups, their group
orbits will have the same shape,
${\mathcal{M}}(\phi^{+})\cong{\mathcal{M}}(\phi^{-})\cong{\mathcal{M}}(\psi^{+})\cong
SU(2)/K_{\infty}$. In the previous section we described how normal subgroups
simplify our description of the group orbit, and we can use this here to put
the description in terms of $SO(3)$ subgroups. There is a 2-to-1 homomorphism
from the quotient $G/\\{\mathds{1},-\mathds{1}\\}$. Elimination of the
$\mathbb{Z}_{2}$ normal subgroup from the group orbit description gives
$SU(2)/K_{\infty}\cong SO(3)/D_{\infty}$, where $D_{\infty}$ is the isotropy
subgroup of the cylinder.

###  4.2 T-States

From the Bell states, we can begin to map out the possible shapes of group
orbit in the tetrahedron of T-states. Consider the convex hull of triplet Bell
states to be the base of the tetrahedron, and $\psi^{-}$ to be the peak. The
vertical height above the base of this tetrahedron indicates the proportion of
the singlet state in the convex mixture
$p_{1}\phi^{+}+p_{2}\phi^{-}+p_{3}\psi^{+}+p_{4}\psi^{-}$, where
$p_{1}+p_{2}+p_{3}+p_{4}=1$.

Since $\psi^{-}$ is symmetric under the $SU(2)$ group action,

$\displaystyle{\mathcal{M}}(p_{1}\phi^{+}+p_{2}\phi^{-}+p_{3}\psi^{+}+p_{4}\psi^{-})$
$\displaystyle\quad={\mathcal{M}}(p_{1}\phi^{+}+p_{2}\phi^{-}+p_{3}\psi^{+}),$
(16)

therefore the possible types of group orbit for T-states will all be exhibited
in the convex hull of the triplet states. These can be visualized as a
triangle of states, as shown in Fig. 5.

Figure 5: The partitioning of the $2$-qubit state space. a) Tetrahedron of
T-states coloured to indicate different group orbit shapes: grey –
$SU(2)/K_{2}\cong SO(3)/D_{2}$, light blue – $SO(3)/D_{\infty}$, black –
$\\{e\\}$. Slices through the tetrahedron that are parallel to the plane
containing the triplet Bell states exhibit the same structure, exhibited in
the triangle to the right. b) Tetrahedron mapping out the smoothed isotropy
subgroups with smoothing scale $\epsilon=0.04$. The bands along the diagonals
indicate $\rm{Iso}_{\epsilon}(\rho)\cong K_{\infty}$, while states in the
central hexagon are assigned $\rm{Iso}_{\epsilon}(\rho)\cong SU(2)$. The
remaining states retain their $K_{2}$ isotropy subgroup. c) Hasse diagram of
the observed isotropy subgroups (up to isomorphism) for two qubits. Arrows
indicate subset inclusion. This is a sublattice of the full $SU(2)$ subgroup
lattice.

The vertices of this triangle are the triplet Bell states, and each have a
$K_{\infty}$ isotropy subgroup. Appendix B also shows that the midpoints of
the edges of this triangle, corresponding to states
$\frac{1}{4}(\mathds{1}\otimes\mathds{1}+\sigma_{i}\otimes\sigma_{i})$, also
have $K_{\infty}$ isotropy subgroups. Therefore the diagonals of this
triangle, corresponding to states

$\displaystyle\rho$
$\displaystyle=p\phi^{+}+(1-p)\frac{1}{4}(\mathds{1}\otimes\mathds{1}+Y\otimes
Y)$ (17) $\displaystyle\rho$
$\displaystyle=p\phi^{-}+(1-p)\frac{1}{4}(\mathds{1}\otimes\mathds{1}+X\otimes
X)$ (18) $\displaystyle\rho$
$\displaystyle=p\psi^{+}+(1-p)\frac{1}{4}(\mathds{1}\otimes\mathds{1}+Z\otimes
Z)$ (19)

for $0\leq p<1$, have $K_{\infty}\prec\rm{Iso}(\rho)$. The states not on these
diagonals remain invariant under the subgroup $K_{2}=\\{\pm\mathds{1},\pm
iX,\pm iY,\pm iZ\\}$.

The possible isotropy subgroups exhibited by the T-states can be seen with the
projectors ${\mathcal{P}}_{H}$, as detailed in Appendix C. These are:

  * •

$\rm{Iso}(\rho_{T})=SU(2)\Rightarrow{\mathcal{M}}(\rho)\cong\\{e\\}$

Werner states along the central axis of the T-state tetrahedron, where
$\tau_{1}=\tau_{2}=\tau_{3}$. These are the states

$\rho=(1-p)\ \psi^{-}+\frac{p}{3}\ (\psi^{+}+\phi^{+}+\phi^{-})$ (20)

for $0\leq p\leq 1$.

  * •

$\rm{Iso}(\rho_{T})=K_{\infty}\Rightarrow{\mathcal{M}}(\rho)\cong
SO(3)/D_{\infty}$

T-states with $\tau_{i}=\tau_{j}\neq\tau_{k}$. These states lie on or above
the diagonals of the triangle of triplet Bell states, as seen in Fig.5,
excluding the Werner states on the central axis of the tetrahedron.

  * •

$\rm{Iso}(\rho_{T})=K_{2}\Rightarrow{\mathcal{M}}(\rho)\cong SO(3)/D_{2}$

T-states with $\tau_{1}\neq\tau_{2}\neq\tau_{3}$. These are states that do not
lie on or above the diagonals of the triangle formed by the triplet Bell
states.

The smoothed isotropy subgroups are illustrated in the lower part of Fig. 5,
with a smoothing scale of $\epsilon=0.04$ under the trace distance. Under this
smoothing, the horizontal slices through the tetrahedron are no longer
equivalent; near $\psi^{-}$ all states are assigned $SU(2)$ as their isotropy
subgroup, while at the base there remain many states with
$\rm{Iso}_{\epsilon}(\rho)=K_{2}$.

###  4.3 Maximally Mixed Marginals

The set of 2-qubit states with maximally mixed marginals is a wider class of
states reached from the T-states through local $SU(2)$ unitaries
$U(g_{1})\otimes U(g_{2})$. Appendix C details how the isotropy subgroups are
enumerated.

If these local unitaries are the same for each qubit, we reach states of the
form

$\displaystyle\tilde{\rho}_{T}$ $\displaystyle=U(g)\otimes U(g)\ \rho_{T}\
U^{\dagger}(g)\otimes U^{\dagger}(g)={\mathcal{U}}_{g}(\rho_{T})$
$\displaystyle=\frac{1}{4}\left(\mathds{1}\otimes\mathds{1}+\sum_{i=1}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma\right),$ (21)

where the vectors $\\{\mathbf{c}_{i}\\}$ form an orthonormal basis of
$\mathbb{R}^{3}$.

These states behave similarly to the T-states, but with isomorphic isotropy
subgroups. This can be seen from the final part of Lemma 1, where
$\text{Iso}({\mathcal{U}}_{g}\circ\rho_{T}\circ{\mathcal{U}}_{g^{-1}})=g[\text{Iso}(\rho_{T})]g^{-1}$.
Explicitly, instead of $\\{\pm\mathds{1},\pm iX,\pm iY,\pm iZ\\}$, these
rotated T-states will be invariant under $\\{\pm\mathds{1},\ \pm
i(\mathbf{c}_{1}\cdot\sigma),\ \pm i(\mathbf{c}_{2}\cdot\sigma),\ \pm
i(\mathbf{c}_{3}\cdot\sigma)\\}\cong K_{2}$.

The possible types of group orbit for these states are the same as for the
T-states:

  * •

$\rm{Iso}(\tilde{\rho}_{T})=SU(2)\Rightarrow{\mathcal{M}}(\tilde{\rho}_{T})\cong\\{e\\}$

The same symmetric states as the T-states, unchanged by the unitary
transformation $U(g)\otimes U(g)$. These are states $\tilde{\rho}_{T}$ with
$\tau_{1}=\tau_{2}=\tau_{3}$.

  * •

$\rm{Iso}(\tilde{\rho}_{T})=K_{\infty}\Rightarrow{\mathcal{M}}(\tilde{\rho}_{T})\cong
SO(3)/D_{\infty}$

States $\tilde{\rho}_{T}$ with $\tau_{i}=\tau_{j}\neq\tau_{k}$. These are
reached via unitary transformations $U(g)\otimes U(g)$ from the T-states with
$K_{\infty}$ isotropy subgroups.

  * •

$\rm{Iso}(\tilde{\rho}_{T})=K_{2}\Rightarrow{\mathcal{M}}(\tilde{\rho}_{T})\cong
SO(3)/D_{2}$

States $\tilde{\rho}_{T}$ with $\tau_{1}\neq\tau_{2}\neq\tau_{3}$. These are
reached via unitary transformations $U(g)\otimes U(g)$ from the T-states with
$K_{2}$ isotropy subgroups.

Further states with maximally mixed marginals are reached from the T-states by
different local unitaries on each qubit. Suppose that these two local
unitaries have the same generator $\mathbf{r}\cdot\sigma$, such that
$U(g_{1})\otimes U(g_{2})=e^{i\theta_{1}\mathbf{r}\cdot\sigma}\otimes
e^{i\theta_{2}\mathbf{r}\cdot\sigma}$. This gives states of the form

$\displaystyle\rho_{M}=e^{i\theta_{1}\mathbf{r}\cdot\sigma}\otimes
e^{i\theta_{2}\mathbf{r}\cdot\sigma}\ \rho_{T}\
e^{-i\theta_{1}\mathbf{r}\cdot\sigma}\otimes
e^{-i\theta_{2}\mathbf{r}\cdot\sigma}$
$\displaystyle=\frac{1}{4}\left(\mathds{1}\otimes\mathds{1}+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i=2}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right)$ (22)

where $\\{\mathbf{r},\mathbf{c}_{2},\mathbf{c}_{3}\\}$ and
$\\{\mathbf{r},\mathbf{d}_{2},\mathbf{d}_{3}\\}$ are two sets of orthonormal
bases for $\mathbb{R}^{3}$. If $U(g_{1})=U(g_{2})$ in the local unitaries,
these sets of bases will be the same and we retrieve $\tilde{\rho}_{T}$.
However, in the case that $\theta_{1}\neq\theta_{2}$ in the local unitaries,
new possibilities for the isotropy subgroup are introduced:

  * •

$\rm{Iso}(\rho_{M})=U(1)\Rightarrow{\mathcal{M}}(\rho_{M})\cong S^{2}$

States $\rho_{M}$ with $\tau_{2}=\tau_{3}$. These are reached from T-states
with $K_{\infty}$ isotropy subgroups via local unitaries
$e^{i\theta_{1}\mathbf{r}\cdot\sigma}\otimes
e^{i\theta_{2}\mathbf{r}\cdot\sigma}$ where $\theta_{1}\neq\theta_{2}$.

  * •

$\rm{Iso}(\rho_{M})=\mathbb{Z}_{4}\Rightarrow{\mathcal{M}}(\rho_{M})\cong
SO(3)/C_{2}$

States $\rho_{M}$ with $\tau_{2}\neq\tau_{3}$. These are reached from T-states
with $K_{2}$ isotropy subgroups via local unitaries
$e^{i\theta_{1}\mathbf{r}\cdot\sigma}\otimes
e^{i\theta_{2}\mathbf{r}\cdot\sigma}$ where $\theta_{1}\neq\theta_{2}$.

The states $\tilde{\rho}_{T}$ are only invariant under a $K_{2}$ subgroup
because the correlation terms are put into diagonal form by the same
orthonormal bases $\\{\mathbf{c}_{i}\\}$ and $\\{\mathbf{d}_{i}\\}$ on each
qubit. For example, the correlation terms of the T-states are put into the
canonical form by the standard cartesian orthonormal basis
$\\{\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\\}$ on each qubit. When
the local unitaries are applied, this can break the $K_{2}$ symmetry, and the
local bases become ‘misaligned’. However, when
$\mathbf{c}_{1}=\mathbf{d}_{1}=\mathbf{r}$ there is still a subgroup
$\mathbb{Z}_{4}=\\{\pm\mathds{1},\pm i(\mathbf{r}\cdot\sigma)\\}$ that leaves
these states invariant.

The remaining 2-qubit states with maximally mixed marginals are reached from
the T-states through local $SU(2)$ unitaries that do not share a generator,
such that $U(g_{1})\otimes
U(g_{2})=e^{i\theta_{1}\mathbf{r}_{1}\cdot\sigma}\otimes
e^{i\theta_{2}\mathbf{r}_{2}\cdot\sigma}$ where
$|\mathbf{r}_{1}|=|\mathbf{r}_{2}|=1$ and $\mathbf{r}_{1}\neq\mathbf{r}_{2}$.
These states can be expressed in the form

$\displaystyle\tilde{\rho}_{M}$
$\displaystyle=e^{i\theta_{1}\mathbf{r}_{1}\cdot\sigma}\otimes
e^{i\theta_{2}\mathbf{r}_{2}\cdot\sigma}\ \rho_{T}\
e^{-i\theta_{1}\mathbf{r}_{1}\cdot\sigma}\otimes
e^{-i\theta_{2}\mathbf{r}_{2}\cdot\sigma}$
$\displaystyle=\frac{1}{4}\left(\mathds{1}\otimes\mathds{1}+\sum_{i=1}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right),$ (23)

where $\\{\mathbf{c}_{i}\\}$ and $\\{\mathbf{d}_{i}\\}$ are orthonormal bases
of $\mathbb{R}^{3}$ that do not share any elements, i.e.
$\mathbf{c}_{i}\neq\mathbf{d}_{j}$ for all $i$ and $j$. These states exhibit
another type of group orbit:

  * •

$\rm{Iso}(\rho)=\mathbb{Z}_{2}\Rightarrow{\mathcal{M}}(\rho)\cong SO(3)$

States $\tilde{\rho}_{M}$. The orthonormal bases $\\{\mathbf{c}_{i}\\}$ and
$\\{\mathbf{d}_{i}\\}$ do not share any elements, therefore the only subgroup
that leaves these states invariant is $\mathbb{Z}_{2}=\\{\pm\mathds{1}\\}$.
These are in a sense maximally asymmetric states, as they have no non-trivial
residual symmetries.

###  4.4 Local Bloch Vectors

Finally, the introduction of local Bloch vectors completes the set of 2-qubit
states. In the states with maximally mixed marginals, the alignment of the
orthonormal bases describing the correlation terms played a key role in
determining the symmetry properties. The effect of the local Bloch vectors
depends on the how they relate to each other, as well as how they relate to
the bases for the correlation terms.

There are three possible isotropy subgroups for these states:

  * •

$\rm{Iso}(\rho)=\mathbb{Z}_{2}\Rightarrow{\mathcal{M}}(\rho)\cong SO(3)$

This will occur for any states

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma$
$\displaystyle\qquad\left.+\sum_{i=1}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right)$ (24)

whose local Bloch vectors $\mathbf{a}\cdot\sigma$ and $\mathbf{b}\cdot\sigma$
are not parallel or anti-parallel, such that $\mathbf{a}\neq\gamma\mathbf{b}$
for any $\gamma\in\mathbb{R}$.

This isotropy subgroup will also occur in states where the basis in which the
correlation terms are diagonalised does not align with the local Bloch
vectors. These are states of the form

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma$
$\displaystyle\qquad\left.+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i=2}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right)$ (25)

where $\\{\mathbf{r},\mathbf{c}_{2},\mathbf{c}_{3}\\}$ and
$\\{\mathbf{r},\mathbf{d}_{2},\mathbf{d}_{3}\\}$ are two sets of orthonormal
bases for $\mathbb{R}^{3}$ and either $\mathbf{a}$ or
$\mathbf{b}\neq\gamma\mathbf{r}$ for any $\gamma\in\mathbb{R}$. In this case,
the correlation terms would be invariant under
$\mathbb{Z}_{4}=\\{\pm\mathds{1},\pm i\mathbf{r}\cdot\sigma\\}$, however the
local Bloch vectors prevent this. Similarly if $\tau_{2}=\tau_{3}$ in states
of this form, then the local Bloch vectors break a $U(1)$ symmetry, while if
$\\{\mathbf{c}_{i}\\}=\\{\mathbf{d}_{i}\\}$ then the local Bloch vectors break
a $K_{2}$ symmetry. Likewise if both
$\\{\mathbf{c}_{i}\\}=\\{\mathbf{d}_{i}\\}$ and $\tau_{2}=\tau_{3}$, the
correlation terms are invariant under a $K_{\infty}$ subgroup that is again
broken by the local Bloch vectors.

  * •

$\rm{Iso}(\rho)=\mathbb{Z}_{4}\Rightarrow{\mathcal{M}}(\rho)\cong SO(3)/C_{2}$

States of the form

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+a\mathbf{r}\cdot\sigma\otimes\mathds{1}+b\mathds{1}\otimes\mathbf{r}\cdot\sigma$
$\displaystyle\qquad\left.+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i=2}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right)$ (26)

where $\\{\mathbf{r},\mathbf{c}_{2},\mathbf{c}_{3}\\}$ and
$\\{\mathbf{r},\mathbf{d}_{2},\mathbf{d}_{3}\\}$ are two sets of orthonormal
bases for $\mathbb{R}^{3}$, $a,b\in\mathbb{R}$ and $\tau_{2}\neq\tau_{3}$. The
local Bloch vectors are invariant under the same $\mathbb{Z}_{4}$ subgroup
that leaves the correlation terms invariant. This also includes the case where
$\mathbf{c}_{i}=\mathbf{d}_{i}$ for $i=2,3$. The correlation terms are
invariant under a $K_{2}$ subgroup, but the local Bloch vectors break this to
a $\mathbb{Z}_{4}$ subgroup.

Similarly, states of the form

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+a\mathbf{c}_{1}\cdot\sigma\otimes\mathds{1}+b\mathds{1}\otimes\mathbf{c}_{1}\cdot\sigma$
$\displaystyle\qquad\left.+\sum_{i=1}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma\right)$ (27)

where $\\{\mathbf{c}_{1},\mathbf{c}_{2},\mathbf{c}_{3}\\}$ is an orthonormal
basis for $\mathbb{R}^{3}$ and $\tau_{1}=\tau_{2}\neq\tau_{3}$. The
correlation terms are invariant under a $K_{\infty}$ subgroup, but the local
Bloch vectors break the residual symmetry down to a $\mathbb{Z}_{4}$ subgroup
as the local Bloch vectors are orthogonal to $\mathbf{c}_{3}\cdot\sigma$, the
generator of the relevant $U(1)$ subgroup.

  * •

$\rm{Iso}(\rho)=U(1)\Rightarrow{\mathcal{M}}(\rho)\cong S^{2}$

These are states

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+a\mathbf{r}\cdot\sigma\otimes\mathds{1}+b\mathds{1}\otimes\mathbf{r}\cdot\sigma$
$\displaystyle\qquad\left.\tau_{1}\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\tau\sum_{i=2}^{3}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right)$ (28)

where $\\{\mathbf{r},\mathbf{c}_{2},\mathbf{c}_{3}\\}$ and
$\\{\mathbf{r},\mathbf{d}_{2},\mathbf{d}_{3}\\}$ are two sets of orthonormal
bases for $\mathbb{R}^{3}$, $a,b\in\mathbb{R}$. The local Bloch vectors are
invariant under the same $U(1)$ subgroup that leaves the correlation terms
invariant. This also includes the cases when
$\\{\mathbf{c}_{i}\\}=\\{\mathbf{d}_{i}\\}$ and when $\tau_{1}=\tau$, where
the correlation terms are invariant under $K_{\infty}$ and $SU(2)$
respectively.

This completes the classification of the isotropy subgroups of 2-qubit states.

###  4.5 Some simple consequences of the classification

This classification allows us to infer constraints on the resources required
to simulate asymmetric channels under symmetry constraints. If we wish to
simulate an axial channel (i.e. ${\mathcal{M}}({\mathcal{E}})=S^{2}$) under
$SU(2)$ symmetry constraints using the T-states, the resource state must have
${\mathcal{M}}(\sigma)\succ S^{2}$. T-states on the diagonals of the
tetrahedron, with ${\mathcal{M}}(\rho)\prec SO(3)/D_{\infty}$, will not be
able to perform such simulations; only the T-states with
$\tau_{1}\neq\tau_{2}\neq\tau_{3}$ will achieve this task up to some
approximation as discussed earlier.

However, convex combinations of different T-states can create more states that
break the symmetry to a larger degree. For example,
$\rm{Iso}(p\phi^{+}+(1-p)\phi^{-})=K_{2}$ when $p\neq 1/2$, despite
$\rm{Iso}(\phi^{+})\cong\rm{Iso}(\phi^{-})\cong K_{\infty}$.

Group orbits also constrain the dynamics of the T-states. For instance, Lemma
2 implies that symmetric operations move states situated on a diagonal of the
tetrahedron only along that diagonal. The T-states are closed under symmetric
dynamics – leaving would entail further symmetries being broken.

Also note that $H_{1}\subseteq H_{2}$ does not imply there exists a symmetric
channel from $C(H_{1})$ into $C(H_{2})$. T-states on the diagonals of the base
of the tetrahedron cannot be reached from those that lie off these diagonals
under symmetric operations [25], despite $K_{2}\prec K_{\infty}$. Group orbits
impose the highest level constraints; further techniques such as those of [25]
then impose more detailed constraints.

##  5 Outlook

Our analysis has shown that the basic problem of state interconversion under a
symmetry constraint has a rich and non-trivial structure. We have established
a high-level description of the problem that is consistent with the resource-
theoretic picture, however the question whether the framework can find
practical use in the same way that superselection rules simplify computation
remains to be explored.

We have classified the set of all two-qubit states under the $SU(2)$ tensor
product representation, which may be of use in its own right for the study of
quantum correlations. However, beyond two-qubits, the general problem can be
extremely difficult – for example even the finite subgroups of $SU(5)$ remain
unclassified [17]. Despite this complexity, we believe that the tools we have
developed here provide useful insight into the structure of information
processing under symmetry constraints.

Other techniques for studying the effect of symmetry constraints on quantum
operations connect with the present analysis. As already mentioned, recent
work on the harmonic analysis of quantum channels suggests that the geometry
of group orbits provides an insight into the use of finite quantum resources
[25]. The present paper has focused purely on the ‘shape’ of this orbit, but
the role of the induced geometry on the orbits is only partially understood.
To extend such an analysis would require adapting techniques in metrology [15]
and differential geometry [43]. It would also be of interest to draw
connections with recent work [44] that extends classical coarse-graining into
the quantum regime with symmetries present.

Finally, these ideas may inform us about other resource theories, using the
symmetry constraints directly (e.g. in thermodynamics). Along similar lines,
the techniques we have described may find application in the context of
realising universal quantum computation via the combination of resource states
with simple gate-sets, for example Clifford operations and magic states.

##  6 Acknowledgements

We would like to thank Cristina Cirstoiu, Erick Hinds Mingo, Zoe Holmes and
Markus Frembs for many useful discussions. TH is funded by the EPSRC Centre
for Doctoral Training in Controlled Quantum Dynamics. DJ is supported by the
Royal Society.

## References

  * Tinkham [1964] M. Tinkham.  _Group theory and quantum mechanics_.  International series in pure and applied physics. McGraw-Hill, 1964.
  * Caves et al. [2002] Carlton M. Caves, Christopher A. Fuchs, and Rüdiger Schack.  Unknown quantum states: The quantum de finetti representation.  _Journal of Mathematical Physics_ , 43(9):4537–4559, 2002.  doi: 10.1063/1.1494475.
  * Jozsa [2001] R. Jozsa.  Quantum factoring, discrete logarithms, and the hidden subgroup problem.  _Computing in Science Engineering_ , 3(2):34–43, Mar 2001.  ISSN 1521-9615.  doi: 10.1109/5992.909000.
  * Horodecki et al. [2009] Ryszard Horodecki, Pawel Horodecki, Michal Horodecki, and Karol Horodecki.  Quantum entanglement.  _Rev. Mod. Phys._ , 81:865–942, Jun 2009.  doi: 10.1103/RevModPhys.81.865.
  * Marvian and Spekkens [2014] I. Marvian and R. W. Spekkens.  Extending Noether’s theorem by quantifying the asymmetry of quantum states.  _Nature Communications_ , 5:3821, May 2014.  doi: 10.1038/ncomms4821.
  * Lostaglio et al. [2015] Matteo Lostaglio, Kamil Korzekwa, David Jennings, and Terry Rudolph.  Quantum coherence, time-translation symmetry, and thermodynamics.  _Phys. Rev. X_ , 5:021001, Apr 2015.  doi: 10.1103/PhysRevX.5.021001.
  * Gour et al. [2018] Gilad Gour, David Jennings, Francesco Buscemi, Runyao Duan, and Iman Marvian.  Quantum majorization and a complete set of entropic conditions for quantum thermodynamics.  _Nature Communications_ , 9(1):5352, 2018.  ISSN 2041-1723.  doi: 10.1038/s41467-018-06261-7.
  * Lostaglio et al. [2015] M. Lostaglio, D. Jennings, and T. Rudolph.  Description of quantum coherence in thermodynamic processes requires constraints beyond free energy.  _Nature Communications_ , 6:6383, March 2015.  doi: 10.1038/ncomms7383.
  * Wigner [1952] E. P. Wigner.  Die messung quantenmechanischer operatoren.  _Zeitschrift für Physik A Hadrons and nuclei_ , 133(1):101–108, Sep 1952.  ISSN 0939-7922.  doi: 10.1007/BF01948686.
  * Araki and Yanase [1960] Huzihiro Araki and Mutsuo M. Yanase.  Measurement of quantum mechanical operators.  _Phys. Rev._ , 120:622–626, Oct 1960.  doi: 10.1103/PhysRev.120.622.
  * Loveridge and Busch [2011] L. Loveridge and P. Busch.  ‘measurement of quantum mechanical operators’ revisited.  _The European Physical Journal D_ , 62(2):297–307, Apr 2011.  ISSN 1434-6079.  doi: 10.1140/epjd/e2011-10714-3.
  * Ahmadi et al. [2013] Mehdi Ahmadi, David Jennings, and Terry Rudolph.  The wigner–araki–yanase theorem and the quantum resource theory of asymmetry.  _New Journal of Physics_ , 15(1):013057, 2013\.  doi: 10.1088/1367-2630/15/1/013057.
  * Marvian and Spekkens [2012] I. Marvian and R. W. Spekkens.  An information-theoretic account of the Wigner-Araki-Yanase theorem.  _ArXiv e-prints_ , December 2012.  URL https://arxiv.org/abs/1212.3378.
  * Navascués and Popescu [2014] Miguel Navascués and Sandu Popescu.  How energy conservation limits our measurements.  _Phys. Rev. Lett._ , 112:140502, Apr 2014.  doi: 10.1103/PhysRevLett.112.140502.
  * Šafránek et al. [2015] Dominik Šafránek, Mehdi Ahmadi, and Ivette Fuentes.  Quantum parameter estimation with imperfect reference frames.  _New Journal of Physics_ , 17(3):033012, 2015\.  doi: 10.1088/1367-2630/17/3/033012.
  * Marvian et al. [2016] Iman Marvian, Robert W. Spekkens, and Paolo Zanardi.  Quantum speed limits, coherence, and asymmetry.  _Phys. Rev. A_ , 93:052331, May 2016.  doi: 10.1103/PhysRevA.93.052331.
  * Grier and Schaeffer [2016] D. Grier and L. Schaeffer.  The Classification of Stabilizer Operations over Qubits.  _ArXiv e-prints_ , March 2016.  URL https://arxiv.org/abs/1603.03999.
  * Gottesman [1997] D. Gottesman.  _Stabilizer codes and quantum error correction_.  PhD thesis, California Institute of Technology, 1997.  URL https://arxiv.org/abs/quant-ph/9705052.
  * França and Hashagen [2018] D S França and A K Hashagen.  Approximate randomized benchmarking for finite groups.  _Journal of Physics A: Mathematical and Theoretical_ , 51(39):395302, aug 2018.  doi: 10.1088/1751-8121/aad6fa.
  * Nielsen and Chuang [2010] M.A. Nielsen and I.L. Chuang.  _Quantum Computation and Quantum Information: 10th Anniversary Edition_.  Cambridge University Press, 2010.  ISBN 9781139495486.  doi: 10.1017/CBO9780511976667.
  * Vaccaro et al. [2008] J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs.  Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules.  _Phys. Rev. A_ , 77:032114, Mar 2008.  doi: 10.1103/PhysRevA.77.032114.
  * Gour et al. [2009] Gilad Gour, Iman Marvian, and Robert W. Spekkens.  Measuring the quality of a quantum reference frame: The relative entropy of frameness.  _Phys. Rev. A_ , 80:012307, Jul 2009.  doi: 10.1103/PhysRevA.80.012307.
  * Gour and Spekkens [2008] Gilad Gour and Robert W Spekkens.  The resource theory of quantum reference frames: manipulations and monotones.  _New Journal of Physics_ , 10(3):033023, 2008\.  doi: 10.1088/1367-2630/10/3/033023.
  * Marvian and Spekkens [2013] Iman Marvian and Robert W Spekkens.  The theory of manipulations of pure state asymmetry: I. basic tools, equivalence classes and single copy transformations.  _New Journal of Physics_ , 15(3):033001, 2013\.  doi: 10.1088/1367-2630/15/3/033001.
  * Cirstoiu and Jennings [2017] C. Cirstoiu and D. Jennings.  Irreversibility and quantum information flow under global and local gauge symmetries.  _ArXiv e-prints_ , Dec 2017.  URL https://arxiv.org/abs/1707.09826.
  * Åberg [2014] Johan Åberg.  Catalytic coherence.  _Phys. Rev. Lett._ , 113:150402, Oct 2014.  doi: 10.1103/PhysRevLett.113.150402.
  * Madore [1995] J. Madore.  _An Introduction to Noncommutative Differential Geometry and its Physical Applications_.  London Mathematical Society Lecture Note Series. Cambridge University Press, 1995.  ISBN 9780521467919.  doi: 10.1017/CBO9780511569357.
  * Kech et al. [2015] Michael Kech, Peter Vrana, and Michael M Wolf.  The role of topology in quantum tomography.  _Journal of Physics A: Mathematical and Theoretical_ , 48(26):265303, 2015.  doi: 10.1088/1751-8113/48/26/265303.
  * Veitch et al. [2014] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson.  The resource theory of stabilizer quantum computation.  _New Journal of Physics_ , 16(1):013009, 2014\.  doi: 10.1088/1367-2630/16/1/013009.
  * Schmidt [1994] R. Schmidt.  _Subgroup Lattices of Groups_.  De Gruyter Expositions in Mathematics. De Gruyter, 1994.  ISBN 9783110868647.  URL https://www.degruyter.com/view/product/172483.
  * Burris and Sankappanavar [1981] S. Burris and H.P. Sankappanavar.  _A course in universal algebra_.  Graduate texts in mathematics. Springer-Verlag, 1981.  ISBN 9780387905785.  URL http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html.
  * Coecke et al. [2016] Bob Coecke, Tobias Fritz, and Robert W. Spekkens.  A mathematical theory of resources.  _Information and Computation_ , 250(Supplement C):59 – 86, 2016.  ISSN 0890-5401.  doi: 10.1016/j.ic.2016.02.008.
  * Streltsov et al. [2017] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio.  Colloquium: Quantum coherence as a resource.  _Rev. Mod. Phys._ , 89:041003, Oct 2017.  doi: 10.1103/RevModPhys.89.041003.
  * Goold et al. [2016] John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk.  The role of quantum information in thermodynamics—a topical review.  _Journal of Physics A: Mathematical and Theoretical_ , 49(14):143001, 2016.  doi: 10.1088/1751-8113/49/14/143001.
  * Bengtsson and Życzkowski [2017] I. Bengtsson and K. Życzkowski.  _Geometry of Quantum States: An Introduction to Quantum Entanglement_.  Cambridge University Press, 2017.  ISBN 9781107026254.  doi: 10.1017/CBO9780511535048.
  * Arvanitogeōrgos [2003] A. Arvanitogeōrgos.  _An Introduction to Lie Groups and the Geometry of Homogeneous Spaces_.  Student mathematical library. American Mathematical Society, 2003.  ISBN 9780821827789.  doi: 10.1090/stml/022.
  * Bredon [1972] G.E. Bredon.  _Introduction to Compact Transformation Groups_.  Pure and Applied Mathematics. Elsevier Science, 1972.  ISBN 9780080873596.
  * Kretschmann et al. [2008] Dennis Kretschmann, Dirk Schlingemann, and Reinhard F. Werner.  A continuity theorem for stinespring’s dilation.  _Journal of Functional Analysis_ , 255(8):1889 – 1904, 2008.  ISSN 0022-1236.  doi: 10.1016/j.jfa.2008.07.023.
  * Isaacs [1994] I.M. Isaacs.  _Algebra: A Graduate Course_.  Graduate studies in mathematics. American Mathematical Society, 1994.  ISBN 9780821847992.
  * Jevtic et al. [2012a] Sania Jevtic, David Jennings, and Terry Rudolph.  Quantum mutual information along unitary orbits.  _Physical Review A_ , 85(5):052121, 2012a.  doi: 10.1103/PhysRevA.85.052121.
  * Jevtic et al. [2012b] Sania Jevtic, David Jennings, and Terry Rudolph.  Maximally and minimally correlated states attainable within a closed evolving system.  _Physical review letters_ , 108(11):110403, 2012b.  doi: 10.1103/PhysRevLett.108.110403.
  * Horodecki and Horodecki [1996] Ryszard Horodecki and Michal Horodecki.  Information-theoretic aspects of inseparability of mixed states.  _Phys. Rev. A_ , 54:1838–1843, Sep 1996.  doi: 10.1103/PhysRevA.54.1838.
  * Amari and Nagaoka [2007] S. Amari and H. Nagaoka.  _Methods of Information Geometry_.  Translations of mathematical monographs. American Mathematical Society, 2007.  ISBN 9780821843024.
  * Kabernik [2018] Oleg Kabernik.  Quantum coarse graining, symmetries, and reducibility of dynamics.  _Phys. Rev. A_ , 97:052130, May 2018.  doi: 10.1103/PhysRevA.97.052130.
  * Keyl and Werner [1999] M. Keyl and R. F. Werner.  Optimal cloning of pure states, testing single clones.  _Journal of Mathematical Physics_ , 40(7):3283–3299, 1999.  doi: 10.1063/1.532887.
  * Marvian Mashhad, Iman [2012] Marvian Mashhad, Iman.  _Symmetry, Asymmetry and Quantum Information_.  PhD thesis, 2012.  URL http://hdl.handle.net/10012/7088.
  * Gilchrist et al. [2009] A. Gilchrist, D. R. Terno, and C. J. Wood.  Vectorization of quantum operations and its use.  _ArXiv e-prints_ , November 2009.  URL https://arxiv.org/abs/0911.2539.

##  Appendix A Proofs

######  Lemma 1.

Given any two quantum channels
${\mathcal{E}}:{\mathcal{B}}({\mathcal{H}}_{A})\rightarrow{\mathcal{B}}({\mathcal{H}}_{A^{\prime}})$
and
${\mathcal{F}}:{\mathcal{B}}({\mathcal{H}}_{B})\rightarrow{\mathcal{B}}({\mathcal{H}}_{B^{\prime}})$,
with unitary representations of a group $G$ defined on all input and output
spaces. Then we have the following

  1. 1.

$\rm{Iso}({\mathcal{E}}\otimes{\mathcal{F}})\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$
under the tensor product group action
$\mathfrak{U}_{g}({\mathcal{E}}\otimes{\mathcal{F}})=\mathfrak{U}_{g}({\mathcal{E}})\otimes\mathfrak{U}_{g}({\mathcal{F}})$.

  2. 2.

If $B^{\prime}=A$ then
$\rm{Iso}({\mathcal{E}}\circ{\mathcal{F}})\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$.

  3. 3.

If $A=B$ and $A^{\prime}=B^{\prime}$, and $p$ some some probability $0\leq
p\leq 1$, then
$\rm{Iso}(p{\mathcal{E}}+(1-p){\mathcal{F}})\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$.

  4. 4.

$\rm{Iso}({\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g}^{\dagger})=g[\rm{Iso}({\mathcal{E}})]g^{-1}$.

###### Proof.

For $g\in\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$, the global
group action gives
$\mathfrak{U}_{g}({\mathcal{E}}\otimes{\mathcal{F}})=\mathfrak{U}_{g}({\mathcal{E}})\otimes\mathfrak{U}_{g}({\mathcal{F}})={\mathcal{E}}\otimes{\mathcal{F}}$,
therefore $g\in\rm{Iso}({\mathcal{E}}\otimes{\mathcal{F}})$. However
${\mathcal{E}}\otimes{\mathcal{F}}$ may have further symmetries from taking
the tensor product (for example
$\mathrm{Iso}(\ket{0}\bra{0}\otimes\ket{0}\bra{0})\succ\mathrm{Iso}(\ket{0}\bra{0})$),
therefore
$\rm{Iso}({\mathcal{E}}\otimes{\mathcal{F}})\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$.

The group action on ${\mathcal{E}}\circ{\mathcal{F}}$ is given by
$\mathfrak{U}_{g}({\mathcal{E}}\circ{\mathcal{F}})={\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{F}}\circ{\mathcal{U}}_{g^{-1}}={\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}}\circ{\mathcal{U}}_{g}\circ{\mathcal{F}}\circ{\mathcal{U}}_{g^{-1}}=\mathfrak{U}_{g}({\mathcal{E}})\circ\mathfrak{U}_{g}({\mathcal{F}})$.
Therefore if $g\in\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$, then
$g\in\rm{Iso}({\mathcal{E}}\circ{\mathcal{F}})$, implying
$\rm{Iso}({\mathcal{E}}\circ{\mathcal{F}})\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$.

If $g\in\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})$, then
$\mathfrak{U}_{g}(p{\mathcal{E}}+(1-p){\mathcal{F}})=p\
\mathfrak{U}_{g}({\mathcal{E}})+(1-p)\
\mathfrak{U}_{g}({\mathcal{F}})=p{\mathcal{E}}+(1-p){\mathcal{F}}$, implying
$\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}({\mathcal{F}})\prec\rm{Iso}(p{\mathcal{E}}+(1-p){\mathcal{F}})$.

For
$h\in\text{Iso}({\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}})$,
we have
${\mathcal{U}}_{h}\circ{\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}}\circ{\mathcal{U}}_{h^{-1}}={\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}}$
which simplifies to
${\mathcal{U}}_{(g^{-1}hg)}\circ{\mathcal{E}}\circ{\mathcal{U}}_{(g^{-1}hg)^{-1}}$.
Therefore $g^{-1}hg\in\text{Iso}({\mathcal{E}})$ or $h\in
g[\text{Iso}({\mathcal{E}})]g^{-1}$. When $gh^{\prime}g^{-1}\in
g[\text{Iso}({\mathcal{E}})]g^{-1}$ acts upon
${\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}}$, we have
$({\mathcal{U}}_{g}\circ{\mathcal{U}}_{h^{\prime}}\circ{\mathcal{U}}_{g^{-1}})\circ({\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}})\circ({\mathcal{U}}_{g}\circ{\mathcal{U}}_{h^{\prime-1}}\circ{\mathcal{U}}_{g^{-1}})={\mathcal{U}}_{g}\circ{\mathcal{U}}_{h^{\prime}}\circ{\mathcal{E}}\circ{\mathcal{U}}_{h^{\prime-1}}\circ{\mathcal{U}}_{g^{-1}}={\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}}$,
therefore
$gh^{\prime}g^{-1}\in\text{Iso}({\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}})$.
Therefore
$\text{Iso}({\mathcal{U}}_{g}\circ{\mathcal{E}}\circ{\mathcal{U}}_{g^{-1}})=g[\text{Iso}({\mathcal{E}})]g^{-1}$.
∎

######  Theorem 1.

If a system $B$ in a state $\sigma_{B}$ can be used to simulate a CPTP map
${\mathcal{E}}$ under symmetric dynamics, then
$\text{Iso}(\sigma_{B})\prec\text{Iso}({\mathcal{E}})$. Conversely, if
${\mathcal{E}}$ has isotropy group $Iso({\mathcal{E}})$ then there exists a
quantum system $B$ and quantum state $\sigma_{B}$ that can be used to simulate
${\mathcal{E}}$ with $Iso(\sigma_{B})=Iso({\mathcal{E}})$.

###### Proof.

Viewing the state $\sigma_{B}$ as a CPTP map, we can view the simulation as a
composite channel ${\mathcal{E}}={\mathcal{F}}\circ\sigma_{B}$. Therefore
$\rm{Iso}({\mathcal{E}})=\rm{Iso}({\mathcal{F}}\circ\sigma_{B})\succ\rm{Iso}({\mathcal{F}})\wedge\rm{Iso}(\sigma_{B})=\rm{Iso}(\sigma_{B})$
since ${\mathcal{F}}$ is symmetric. Therefore
$\rm{Iso}({\mathcal{E}})\succ\rm{Iso}(\sigma_{B})$, so
${\mathcal{M}}(\sigma_{B})\succ{\mathcal{M}}({\mathcal{E}})$.

Alternatively, from the Covariant Stinespring theorem [45, 46], we can write
${\mathcal{E}}(\rho)={\mathcal{F}}(\rho\otimes\sigma_{B})=\rm{tr}_{BC}[V(\rho\otimes\sigma_{B}\otimes\gamma_{C})V^{\dagger}]$,
where $\gamma_{C}=|\eta\rangle\langle\eta|$ is some pure state,
$[V,U_{A}(g)\otimes U_{B}(g)\otimes U_{C}(g)]=0$ and $\rm{Iso}(\gamma_{C})=G$.
Supposing $g\in\rm{Iso}(\sigma_{B})$,

$\mathcal{E}(\rho)=\mathrm{tr}_{BC}[V(\rho\otimes{\mathcal{U}}_{g}(\sigma_{B})\otimes{\mathcal{U}}_{g}(\gamma_{C}))V^{\dagger}]=\mathrm{tr}_{BC}[V(\mathds{1}\otimes{\mathcal{U}}_{g}\otimes{\mathcal{U}}_{g})(\rho\otimes\sigma_{B}\otimes\gamma_{C})V^{\dagger}].$
(29)

$V$ is a symmetric unitary, so
$V(\mathds{1}\otimes{\mathcal{U}}_{g}\otimes{\mathcal{U}}_{g})=V({\mathcal{U}}_{g}\otimes{\mathcal{U}}_{g}\otimes{\mathcal{U}}_{g})({\mathcal{U}}_{g}^{\dagger}\otimes\mathds{1}\otimes\mathds{1})=({\mathcal{U}}_{g}\otimes{\mathcal{U}}_{g}\otimes{\mathcal{U}}_{g})V({\mathcal{U}}_{g}^{\dagger}\otimes\mathds{1}\otimes\mathds{1})$,
giving

$\displaystyle\mathcal{E}(\rho)$
$\displaystyle=\mathrm{tr}_{BC}[\mathcal{U}_{g}(V(\mathcal{U}_{g^{-1}}(\rho)\otimes\sigma_{B}\otimes\gamma_{C})V^{\dagger})]=\mathcal{U}_{g}(\mathrm{tr}_{BC}[V(\mathcal{U}_{g^{-1}}(\rho)\otimes\sigma_{B}\otimes\gamma_{C})V^{\dagger}])$
$\displaystyle=\mathcal{U}_{g}\circ\mathcal{E}\circ\mathcal{U}_{g^{-1}}(\rho).$
(30)

Therefore $\mathrm{Iso}(\sigma_{B})\prec\mathrm{Iso}(\mathcal{E})$, and
${\mathcal{M}}(\sigma_{B})\succ{\mathcal{M}}({\mathcal{E}})$.

We now establish the second part of the theorem, and construct the protocol
that realises ${\mathcal{E}}$ with a quantum state that has the same isotropy
subgroup as ${\mathcal{E}}$. Firstly, we note that we can decompose any
${\mathcal{E}}$ into irreducible process modes [25] as

${\mathcal{E}}=\sum_{\lambda,k}\alpha_{\lambda^{*},k}(\boldsymbol{x})\Phi_{\lambda,k}$
(31)

where $\lambda$ labels irreps, including multiplicities and $k$ labels the
basis vector of the irrep. As shown in [25] the coefficients
$\alpha_{\lambda,k}(\boldsymbol{x})$ are non-normalised harmonic wavefunctions
on the space ${\mathcal{M}}:=G/\text{Iso}({\mathcal{E}})$, which is
diffeomorphic to the orbit of ${\mathcal{E}}$ under the group $G$, and where
the point $\boldsymbol{x}$ corresponds to the reference data needed to
simulate ${\mathcal{E}}$ under the symmetry constraint.

For our reference system $B$, we take its Hilbert space to be
$\mathcal{L}^{2}({\mathcal{M}},\mathbb{C})$, where
$\sigma_{B}=|\psi\rangle\langle\psi|$ is a reference state with wavefunction
$\psi(\boldsymbol{z})=\delta(\boldsymbol{z}-\boldsymbol{x})$, understood in
the distributional sense. The protocol to realise ${\mathcal{E}}$ via
$\sigma_{B}$ involves a measurement of the position $\boldsymbol{z}$ of $B$,
which provides the necessary reference frame data to realise ${\mathcal{E}}$
on the system $A$. Explicitly, we have

$\displaystyle{\mathcal{E}}$
$\displaystyle=\sum_{\lambda,k}\alpha_{\lambda^{*},k}(\boldsymbol{x})\
\Phi_{\lambda,k}$
$\displaystyle=\tr_{B}\sum_{\lambda,k}|\boldsymbol{x}\rangle_{B}\langle\boldsymbol{x}|\otimes\alpha_{\lambda^{*},k}(\boldsymbol{x})\
\Phi_{\lambda,k}$ $\displaystyle=\sum_{\lambda,k}\int dz\
\tr[M(\boldsymbol{z})\ \sigma_{B}]\ \alpha_{\lambda^{*},k}(\boldsymbol{z})\
\Phi_{\lambda,k}$ (32)

where $\\{dzM(\boldsymbol{z})\\}$ is the position measurement on
${\mathcal{M}}$, which is covariant since $G$ acts transitively on
${\mathcal{M}}$. Finally, the action of $G$ on $\sigma_{B}$ is given by the
action of $G$ on the quotient space $G/\text{Iso}({\mathcal{E}})$ where at
each point we have an isotropy group $\text{Iso}({\mathcal{E}})$, and thus
$\text{Iso}(\sigma_{B})=\text{Iso}({\mathcal{E}})$ as required. ∎

######  Lemma 2.

Under a symmetric operation ${\mathcal{E}}$,
$\rm{Iso}({\mathcal{E}}(\rho))\succ\rm{Iso}(\rho)$.

###### Proof.

We can view the state $\rho$ as a CPTP map $1\rightarrow\rho$, therefore from
Lemma 1,
$\rm{Iso}({\mathcal{E}}(\rho))=\rm{Iso}({\mathcal{E}}\circ\rho)\succ\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}(\rho)$.
Since $G\wedge H=H$ for any subgroup $H\prec G$, then
$\rm{Iso}({\mathcal{E}})\wedge\rm{Iso}(\rho)=\rm{Iso}(\rho)$, therefore
$\rm{Iso}({\mathcal{E}}(\rho))\succ\rm{Iso}(\rho)$, and thus the group orbits
also obey ${\mathcal{M}}({\mathcal{E}}(\rho))\prec{\mathcal{M}}(\rho)$. ∎

######  Lemma 3.

Let $d(\cdot,\cdot)$ be any metric on the space of quantum states. In terms of
this metric we define

$\mathbf{d}(C(H_{1}),C(H_{2})):=\inf_{\begin{subarray}{c}\sigma_{1}\in
C(H_{1})\\\ \sigma_{2}\in C(H_{2})\end{subarray}}d(\sigma_{1},\sigma_{2}).$
(33)

Then $\mathbf{d}(C(H_{1}),C(H_{2}))=0$ for all $H_{1},H_{2}\prec G$.

###### Proof.

For a state $\rho_{1}\in C(H_{1})$,
${\mathcal{U}}_{g}(\epsilon\rho_{1}+(1-\epsilon)\ \sigma)=\epsilon\
{\mathcal{U}}_{g}(\rho_{1})+(1-\epsilon)\ {\mathcal{U}}_{g}(\sigma)=\epsilon\
{\mathcal{U}}_{g}(\rho_{1})+(1-\epsilon)\ \sigma$ when $\sigma\in C(G)$. Hence
$\text{Iso}(\epsilon\rho_{1}+(1-\epsilon)\sigma)=\text{Iso}(\rho_{1})$, and
$\epsilon\rho_{1}+(1-\epsilon)\sigma\in C(H_{1})$. Likewise, for $\rho_{2}\in
C(H_{2})$, the state $\epsilon\rho_{2}+(1-\epsilon)\sigma$ (with the same
$\sigma\in C(G)$) is also in $C(H_{2})$. As $\epsilon\rightarrow 0$,
$d(\epsilon\rho_{1}+(1-\epsilon)\sigma,\epsilon\rho_{2}+(1-\epsilon)\sigma)\rightarrow
0$, therefore $\mathbf{d}(C(H_{1}),C(H_{2}))=0$ for all $H_{1},H_{2}\prec G$.
∎

######  Lemma 4.

The map $\mathcal{P}_{H}$ has the following properties:

  1. 1.

$\mathcal{P}_{H}$ is the (orthogonal) projector onto $\hat{C}(H)$.

  2. 2.

$\mathcal{P}_{H}(\rho)=\arg\min_{\sigma\in\hat{C}(H)}S(\rho||\sigma)$, where
$S(\rho||\sigma)$ is the relative entropy.

###### Proof.

For states $\rho\in C(W)$ for $W\succ H$, the unitary $U(h)$ for $h\in H$ will
leave the state unchanged. Therefore $\mathcal{P}_{H}(\rho)=\int_{H}dh\
{\mathcal{U}}_{h}(\rho)=\int_{H}dh\ \rho=\rho$. Moreover, ${\mathcal{P}}_{H}$
is a projector because ${\mathcal{P}}_{H}\circ{\mathcal{P}}_{H}=\int_{H}dh\
\int_{H}dh^{\prime}\
{\mathcal{U}}_{h}\circ{\mathcal{U}}_{h^{\prime}}=\int_{H}dh\
\int_{H}dh^{\prime}\ {\mathcal{U}}_{hh^{\prime}}=\int_{H}dh^{\prime\prime}\
{\mathcal{U}}_{h^{\prime\prime}}={\mathcal{P}}_{H}$. Furthermore,
${\mathcal{P}}_{H}^{\dagger}(\rho)=\int_{H}dh\
{\mathcal{U}}_{h^{-1}}(\rho)=\int_{H}dh\
{\mathcal{U}}_{h}(\rho)={\mathcal{P}}_{H}(\rho)$, meaning ${\mathcal{P}}_{H}$
is an orthogonal projector.

We follow the proof in [22]. By Klein’s Inequality, we have
$S(\mathcal{P}_{H}(\rho)||\sigma)\geq 0$, with equality iff
$\sigma=\mathcal{P}_{H}(\rho)$, therefore
$\min_{\sigma\in\hat{C}(H)}S(\mathcal{P}_{H}(\rho)||\sigma)=0$. From the
definition of relative entropy,
$S(\mathcal{P}_{H}(\rho))=\min_{\sigma\in\hat{C}(H)}[-\rm{Tr}(\rho\ \log\
\sigma)]$ because $\log$ is analytic and $\sigma\in\hat{C}(H)$. Using the
idempotency of $\mathcal{P}_{H}$,
$S(\rho||\mathcal{P}_{H}(\rho))=-S(\rho)-\rm{Tr}(\rho\ \log\
\mathcal{P}_{H}(\rho))=-S(\rho)+S(\mathcal{P}_{H}(\rho))=-S(\rho)+\min_{\sigma\in\hat{C}(H)}[-\rm{Tr}(\rho\
\log\ \sigma)]=\min_{\sigma\in\hat{C}(H)}S(\rho||\sigma)$. ∎

######  Lemma 5.

Given a group action for $G$, $\rm{Iso}(\mathcal{P}_{H})=N_{G}(H)$, where
$N_{G}(H)=\\{g\in G\ :\ gHg^{-1}=H\\}$ is the normalizer [39] of $H$ in $G$,
and therefore if $H$ is a normal subgroup of $G$ ($H\triangleleft G$) then
${\mathcal{P}}_{H}$ is a symmetric operation.

###### Proof.

The group action on ${\mathcal{P}}_{H}$ is
$\mathfrak{U}_{g}({\mathcal{P}}_{H})=\int_{H}dh\
\mathfrak{U}_{g}({\mathcal{U}}_{h})=\int_{H}dh\
{\mathcal{U}}_{g}\circ{\mathcal{U}}_{h}\circ{\mathcal{U}}_{g^{-1}}=\int_{H}dh\
{\mathcal{U}}_{ghg^{-1}}$. An element $g\in G$ will therefore be a member of
$\text{Iso}({\mathcal{P}}_{H})$ iff $gHg^{-1}=H$. This is the definition of
the normalizer, so we have that $\rm{Iso}(\mathcal{P}_{H})=N_{G}(H)$.
Averaging over a normal subgroup $H\triangleleft G$ is therefore symmetric,
because $N_{G}(H)=G$ by the definition of a normal subgroup. ∎

##  Appendix B Bell States and Single Component T-States

The extremal points of the T-states are the Bell states, which can be
expressed as the vectorisation [47] of Pauli matrices:

$\displaystyle\ket{\phi^{+}}$ $\displaystyle\propto\ket{\rm{vec}(\mathds{1})}$
(34) $\displaystyle\ket{\phi^{-}}$ $\displaystyle\propto\ket{\rm{vec}(Z)}$
(35) $\displaystyle\ket{\psi^{+}}$ $\displaystyle\propto\ket{\rm{vec}(X)}$
(36) $\displaystyle\ket{\psi^{-}}$ $\displaystyle\propto\ket{\rm{vec}(Y)}.$
(37)

Elements of the isotropy subgroup of a Bell state therefore satisfy
${\mathcal{U}}_{g}(\ket{\rm{vec}(\sigma_{i})}\bra{\rm{vec}(\sigma_{i})})=\ket{\rm{vec}(\sigma_{i})}\bra{\rm{vec}(\sigma_{i})}$,
hence

$U(g)\otimes
U(g)\ket{\rm{vec}(\sigma_{i})}=e^{i\phi}\ket{\rm{vec}(\sigma_{i})}$ (38)

where $\phi\in\mathbb{R}$ is some arbitrary phase. From the identity $A\otimes
B\ket{\rm{vec}(M)}=\ket{\rm{vec}(AMB^{T})}$ [47], the residual symmetries
satisfy

$U(g)\sigma_{i}U^{T}(g)=e^{i\phi}\sigma_{i}.$ (39)

We now consider each of the Bell states in turn.

###  B.1 $\rm{Iso}(\phi^{+})\cong K_{\infty}$

$\ket{\phi^{+}}\propto\ket{\rm{vec}(\mathds{1})}$ is invariant under group
transformations satisfying $U(g)U^{T}(g)=e^{i\phi}\mathds{1}$. $SU(2)$ group
elements take the form $U(g)=\mathrm{exp}(i\varphi[aX+bY+cZ])$ with
$a^{2}+b^{2}+c^{2}=1$ ($a,b,c\in\mathbb{R}$) and $0\leq\varphi<2\pi$,
therefore

$e^{i\varphi(aX+bY+cZ)}e^{i\varphi(aX-bY+cZ)}=e^{i\phi}\mathds{1}\implies
e^{i\varphi(aX+bY+cZ)}=e^{i\phi}e^{i\varphi(-aX+bY-cZ)}.$ (40)

Expansion of the matrix exponentials gives

$\cos{\varphi}\mathds{1}+i\sin{\varphi}(aX+bY+cZ)=e^{i\phi}[\cos{\varphi}\mathds{1}+i\sin{\varphi}(-aX+bY-
cZ)]$ (41)

which provides the conditions

$\displaystyle\cos{\varphi}(1-e^{i\phi})$ $\displaystyle=0$ (42)
$\displaystyle ia\sin{\varphi}(1+e^{i\phi})$ $\displaystyle=0$ (43)
$\displaystyle ib\sin{\varphi}(1-e^{i\phi})$ $\displaystyle=0$ (44)
$\displaystyle ic\sin{\varphi}(1+e^{i\phi})$ $\displaystyle=0.$ (45)

The problem simplifies because $\phi\in\mathbb{R}$ is restricted in the values
it may take. Suppose $\cos{\varphi}\neq 0$, then
$\cos{\varphi}(1-e^{i\phi})=0$ implies $\phi=0$ (modulo $2\pi$). Now suppose
$\cos{\varphi}=0$, ensuring that $\sin{\varphi}\neq 0$. Since
$a^{2}+b^{2}+c^{2}=1$, at least one of $a$, $b$ or $c$ is non-zero, therefore
either $1+e^{i\phi}=0$ or $1-e^{i\phi}=0$ in order for the conditions to hold.
Together, these imply that $\phi=0$ or $\pi$ (modulo $2\pi$).

When $\phi=0$, the conditions reduce to

$\displaystyle 2ia\sin{\varphi}$ $\displaystyle=0$ (46) $\displaystyle
2ic\sin{\varphi}$ $\displaystyle=0.$ (47)

One possible solution is $a=c=0$, leaving $\varphi$ unconstrained and
therefore $b=\pm 1$ (without loss of generality choose $b=1$, since $b=-1$
gives ‘rotations’ of the opposite handedness). These solutions take the form
$U(g)=e^{i\varphi Y}$. The equations are also satisfied when
$\sin{\varphi}=0$, so when $\varphi=0,\pi$, which give group transformations
$U(g)=\pm\mathds{1}$.

In the $\phi=\pi$ case, the conditions become

$\displaystyle 2\cos{\varphi}$ $\displaystyle=0$ (48) $\displaystyle
2ib\sin{\varphi}$ $\displaystyle=0.$ (49)

If $b\neq 0$, then $\cos{\varphi}=\sin{\varphi}=0$, which is not possible.
Therefore the only solutions for these conditions is $b=0$ and
$\cos{\varphi}=0$, which gives $\varphi=\pi/2$ or $3\pi/2$. In this case $a$
and $c$ are only constrained by $a^{2}+c^{2}=1$. The $\varphi=\pi/2$ solution
corresponds to the $SU(2)$ transformations

$U(g)=i(aX+cZ)=(iZ)(c\mathds{1}+iaY)=(iZ)(\cos{\theta}\mathds{1}+i\sin{\theta}Y)=(iZ)e^{i\theta
Y},$ (50)

since $a^{2}+c^{2}=1$, so we parametrise in terms of $\cos{\theta}$ and
$\sin{\theta}$. Similarly $U(g)=(-iZ)e^{i\theta Y}$ for the $\varphi=3\pi/2$
case. This shows that

$\rm{Iso}(\phi^{+})=\\{(iZ)^{\alpha}e^{i\theta Y}:\ 0\leq\theta<2\pi,\
\alpha=0,1,2,3\\}\cong K_{\infty}.$ (51)

###  B.2 $\rm{Iso}(\phi^{-})=K_{\infty}$

Analogous arguments hold for $\ket{\phi^{-}}\propto\ket{\rm{vec}(Z)}$. The
condition for membership of $\rm{Iso}(\phi^{-})$ is

$e^{i\varphi(aX+bY+cZ)}Ze^{i\varphi(aX-bY+cZ)}=e^{i\phi}Z$ (52)

which simplifies to

$e^{i\varphi(aX+bY+cZ)}=e^{i\phi}e^{i\varphi(aX-bY-cZ)}.$ (53)

This gives the conditions

$\displaystyle\cos{\varphi}(1-e^{i\phi})$ $\displaystyle=0$ (54)
$\displaystyle ia\sin{\varphi}(1-e^{i\phi})$ $\displaystyle=0$ (55)
$\displaystyle ib\sin{\varphi}(1+e^{i\phi})$ $\displaystyle=0$ (56)
$\displaystyle ic\sin{\varphi}(1+e^{i\phi})$ $\displaystyle=0,$ (57)

solved in the same way as the previous example. The isotropy subgroup is

$\rm{Iso}(\phi^{-})=\\{(iY)^{\alpha}e^{i\theta X}:\ 0\leq\theta<2\pi,\
\alpha=0,1,2,3\\}\cong K_{\infty}.$ (58)

###  B.3 $\rm{Iso}(\psi^{+})=K_{\infty}$

$\ket{\psi^{+}}\propto\ket{\rm{vec}(X)}$ gives the condition

$e^{i\varphi(aX+bY+cZ)}=e^{i\phi}e^{i\varphi(-aX-bY+cZ)}.$ (59)

This is satisfied when

$\displaystyle\cos{\varphi}(1-e^{i\phi})$ $\displaystyle=0$ (60)
$\displaystyle ia\sin{\varphi}(1+e^{i\phi})$ $\displaystyle=0$ (61)
$\displaystyle ib\sin{\varphi}(1+e^{i\phi})$ $\displaystyle=0$ (62)
$\displaystyle ic\sin{\varphi}(1-e^{i\phi})$ $\displaystyle=0,$ (63)

which gives the isotropy subgroup as

$\rm{Iso}(\psi^{+})=\\{(iX)^{\alpha}e^{i\theta Z}:\ 0\leq\theta<2\pi,\
\alpha=0,1,2,3\\}\cong K_{\infty}.$ (64)

###  B.4 $\rm{Iso}(\psi^{-})=SU(2)$

The singlet Bell state $\ket{\psi^{-}}\propto\ket{\rm{vec}(Y)}$ behaves
differently to the other Bell states. The isotropy subgroup condition is

$e^{i\varphi(aX+bY+cZ)}=e^{i\phi}e^{i\varphi(aX+bY+cZ)},$ (65)

which is satisfied for all elements of $SU(2)$, therefore
$\rm{Iso}(\psi^{-})=SU(2)$.

###  B.5 $\rm{Iso}(\frac{1}{4}(\mathds{1}\otimes\mathds{1}+\tau_{i}\
\sigma_{i}\otimes\sigma_{i})=K_{\infty}$

For states,
$\rho=\frac{1}{4}(\mathds{1}\otimes\mathds{1}+\tau_{i}\sigma_{i}\otimes\sigma_{i})$,
the $\mathds{1}\otimes\mathds{1}$ term can be neglected because
$\rm{Iso}(\mathds{1}\otimes\mathds{1}+\rho^{\prime})=\rm{Iso}(\rho^{\prime})$.
For $\rho=\frac{1}{4}(\mathds{1}\otimes\mathds{1}+\tau_{1}X\otimes X)$, an
$SU(2)$ group element belongs to $\rm{Iso}(\rho)$ when

${\mathcal{U}}_{g}(X\otimes
X)={\mathcal{U}}_{g}(X)\otimes{\mathcal{U}}_{g}(X)=X\otimes X,$ (66)

satisfied when

$e^{i\varphi(aX+bY+cZ)}Xe^{-i\varphi(aX+bY+cZ)}=\pm X,$ (67)

or equivalently

$e^{i\varphi(aX+bY+cZ)}=\pm e^{i\varphi(aX-bY-cZ)}.$ (68)

This is the same condition as for $\psi^{+}$, therefore

$\rm{Iso}(X\otimes X)=\rm{Iso}(\psi^{+})=\\{(iY)^{\alpha}e^{i\theta X}:\
0\leq\theta<2\pi,\ \alpha=0,1,2,3\\}\cong K_{\infty}.$ (69)

Likewise, elements of $\rm{Iso}(Y\otimes Y)$ satisfy

$e^{i\varphi(aX+bY+cZ)}=\pm e^{i\varphi(-aX+bY-cZ)},$ (70)

the same as for $\phi^{+}$, therefore

$\rm{Iso}(Y\otimes Y)=\\{(iZ)^{\alpha}e^{i\theta Y}:\ 0\leq\theta<2\pi,\
\alpha=0,1,2,3\\}\cong K_{\infty},$ (71)

while the condition for membership of $\rm{Iso}(Z\otimes Z)$ is

$e^{i\varphi(aX+bY+cZ)}=\pm e^{i\varphi(-aX+bY-cZ)},$ (72)

showing that

$\rm{Iso}(Z\otimes Z)=\\{(iX)^{\alpha}e^{i\theta Z}:\ 0\leq\theta<2\pi,\
\alpha=0,1,2,3\\}\cong K_{\infty}.$ (73)

##  Appendix C 2-Qubit Isotropy Subgroups

The isotropy subgroups of 2-qubit states under a tensor product $SU(2)$ group
action can be found with projection operators ${\mathcal{P}}_{H}$. These do
not indicate which subgroups of $SU(2)$ will appear as isotropy subgroups,
however the following lemma restricts the possibilities.

######  Lemma 6.

For a 2-qubit state $\rho$ transforming under an $SU(2)$ tensor product
representation, any cyclic subgroup of the isotropy subgroup $\rm{Iso}(\rho)$
must be either $\mathbb{Z}_{2}$, $\mathbb{Z}_{4}$ or $U(1)$.

###### Proof.

A group element belongs to $\rm{Iso}(\rho)$ when
${\mathcal{U}}_{g}(\rho)=\rho$, or in vectorised form [47],

$U(g)\otimes U(g)\otimes U^{*}(g)\otimes
U^{*}(g)\ket{\rm{vec}(\rho)}=\ket{\rm{vec}(\rho)}.$ (74)

For $\rho$ to be invariant under the group transformation ${\mathcal{U}}_{g}$,
it requires $\ket{\rm{vec}(\rho)}$ to be in $\rm{ker}[U(g)\otimes U(g)\otimes
U^{*}(g)\otimes U^{*}(g)-\mathds{1}]$.

All groups contain cyclic subgroups from repeated application of a single
generator, because $g\in G$ implies that $g^{n}\in G$. For any cyclic subgroup
of $\rm{Iso}(\rho)$, there is some basis in which its representation is
diagonal, i.e. $U(g)=e^{i\theta}\ket{0}\bra{0}+e^{-i\theta}\ket{1}\bra{1}$.
Therefore

$\displaystyle U(g)\otimes U(g)\otimes U^{*}(g)\otimes U^{*}(g)-\mathds{1}$
$\displaystyle\qquad=(e^{2i\theta}-1)(\ket{1}\bra{1}+\ket{2}\bra{2}+\ket{7}\bra{7}+\ket{11}\bra{11})+(e^{4i\theta}-1)\ket{3}\bra{3}+(e^{-4i\theta}-1)\ket{12}\bra{12}$
$\displaystyle\hskip
56.9055pt+(e^{-2i\theta}-1)(\ket{4}\bra{4}+\ket{8}\bra{8}+\ket{13}\bra{13}+\ket{14}\bra{14}),$
(75)

where we have assumed for simplicity that $U(g)$ is diagonal in the
computational basis (with for example $\ket{2}=\ket{0010}$). Vectors in
$\rm{span}(\ket{0},\ket{5},\ket{6},\ket{9},\ket{10},\ket{15})$ are invariant
under this $U(1)$ subgroup. Vectors in
$\rm{span}(\ket{1},\ket{2},\ket{4},\ket{7},\ket{8},\ket{11},\ket{13},\ket{14})$
are also in $\rm{ker}[U(g)\otimes U(g)\otimes U^{*}(g)\otimes
U^{*}(g)-\mathds{1}]$ for $\theta=0,\pi$, and are therefore invariant under a
$\mathbb{Z}_{2}$ subgroup. Finally, the vectors of
$\rm{span}(\ket{3},\ket{12})$ are in $\rm{ker}[U(g)\otimes U(g)\otimes
U^{*}(g)\otimes U^{*}(g)-\mathds{1}]$ when $\theta=0,\pi/2,\pi,3\pi/2$, and
hence invariant under a $\mathbb{Z}_{4}$ subgroup.

This holds for any cyclic subgroup of $SU(2)$, therefore any cyclic subgroup
of $\rm{Iso}(\rho)$ must be $\mathbb{Z}_{2}$, $\mathbb{Z}_{2}$ or $U(1)$. ∎

This plays a similar role to the crystallographic restriction theorem in
crystallography [1], and reduces the number of subgroups to check. For
example, any 2-qubit state invariant under a $\mathbb{Z}_{6}$ must be
invariant under the $U(1)$ subgroup with the same generator. This allows us to
enumerate the possible isotropy subgroups for 2-qubit states. It also suggests
that an $N$-qubit state will have only cyclic subgroups
$\mathbb{Z}_{2},\mathbb{Z}_{4},\dots\mathbb{Z}_{2N},U(1)$ within
$\rm{Iso}(\rho)$.

The channels ${\mathcal{P}}_{H}$ provide a way to find the sets $C(H)$.
Consider two subgroups $H_{1}$ and $H_{2}$, with $H_{1}\prec H_{2}$ and there
are no subgroups $H$ that could be the isotropy subgroup of some state such
that $H_{1}\prec H\prec H_{2}$. If a state $\rho$ satisfies
${\mathcal{P}}_{H_{1}}(\rho)=\rho$ but ${\mathcal{P}}_{H_{2}}(\rho)\neq\rho$,
then $\rm{Iso}(\rho)=H_{1}$, and $\rho\in C(H_{1})$. We now use this technique
to identify the isotropy subgroups of all 2-qubit states.

###  C.1 $\mathbb{Z}_{2}\prec\rm{Iso}(\rho)$

All 2-qubit states are symmetric under $\mathbb{Z}_{2}=\\{\pm\mathds{1}\\}$,
confirmed by

${\mathcal{P}}_{\mathbb{Z}_{2}}(\rho)=\frac{1}{2}[\
{\mathcal{U}}_{\mathds{1}}(\rho)+{\mathcal{U}}_{-\mathds{1}}(\rho)\ ]=\rho$
(76)

for any 2-qubit state $\rho$.

###  C.2 $\mathbb{Z}_{4}\prec\rm{Iso}(\rho)$

From Lemma 6, if a 2-qubit state has $\mathbb{Z}_{3}\prec\rm{Iso}(\rho)$, then
it also has $U(1)\prec\rm{Iso}(\rho)$. The next subgroup to check is
$\mathbb{Z}_{4}=\\{\pm\mathds{1},\pm i\mathbf{r}\cdot\sigma\\}$. For
illustrative purposes, let us consider the particular $\mathbb{Z}_{4}$
subgroup $\\{\pm\mathds{1},\pm iZ\\}$. For a general 2-qubit state $\rho$,

$\displaystyle{\mathcal{P}}_{\mathbb{Z}_{4}}(\rho)$
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+{\mathcal{P}}_{\mathbb{Z}_{4}}(\mathbf{a}\cdot\sigma\otimes\mathds{1})+{\mathcal{P}}_{\mathbb{Z}_{4}}(\mathds{1}\otimes\mathbf{b}\cdot\sigma)+\sum_{i,j=1}^{3}T_{ij}\
{\mathcal{P}}_{\mathbb{Z}_{4}}(\sigma_{i}\otimes\sigma_{j})\ \right]$ (77)
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+a_{z}\
Z\otimes\mathds{1}+b_{z}\ \mathds{1}\otimes Z+T_{33}\ Z\otimes
Z+\sum_{i,j=1}^{2}T_{ij}\ \sigma_{i}\otimes\sigma_{j}\right].$ (78)

Now consider the more general $\mathbb{Z}_{4}=\\{\pm\mathds{1},\pm
i\mathbf{r}\cdot\sigma\\}$ subgroup. The useful identities

$\displaystyle(\mathbf{r}\cdot\sigma)(\mathbf{v}\cdot\sigma)$
$\displaystyle=(\mathbf{r}\cdot\mathbf{v})\mathds{1}+i(\mathbf{r}\times\mathbf{v})\cdot\sigma$
(79)
$\displaystyle(\mathbf{r}\cdot\sigma)(\mathbf{v}\cdot\sigma)(\mathbf{r}\cdot\sigma)$
$\displaystyle=2(\mathbf{r}\cdot\mathbf{v})(\mathbf{r}\cdot\sigma)-\mathbf{v}\cdot\sigma$
(80)

allow us to calculate ${\mathcal{P}}_{\mathbb{Z}_{4}}(\rho)$. The local Bloch
vectors give

$\displaystyle{\mathcal{P}}_{\mathbb{Z}_{4}}(\mathbf{a}\cdot\sigma\otimes\mathds{1})$
$\displaystyle=\frac{1}{2}[\mathbf{a}\cdot\sigma\otimes\mathds{1}+(\mathbf{r}\cdot\sigma)(\mathbf{a}\cdot\sigma)(\mathbf{r}\cdot\sigma)\otimes\mathds{1}]$
(81)
$\displaystyle=\frac{1}{2}[\mathbf{a}\cdot\sigma\otimes\mathds{1}+(2(\mathbf{r}\cdot\mathbf{a})(\mathbf{r}\cdot\sigma)-\mathbf{a}\cdot\sigma)\otimes\mathds{1}]$
(82)
$\displaystyle=(\mathbf{r}\cdot\mathbf{a})(\mathbf{r}\cdot\sigma)\otimes\mathds{1}$
(83)

and similarly
${\mathcal{P}}_{\mathbb{Z}_{4}}(\mathds{1}\otimes\mathbf{b}\cdot\sigma)=(\mathbf{r}\cdot\mathbf{b})(\mathds{1}\otimes\mathbf{r}\cdot\sigma)$.
The correlation terms may be written

$\sum_{i,j=1}^{3}T_{ij}\
\sigma_{i}\otimes\sigma_{j}=\sum_{i,j=1}^{3}T^{\prime}_{ij}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma,$ (84)

where $\\{\mathbf{c}_{i}\\}$ is an orthonormal basis for $\mathbb{R}^{3}$ and
we choose $\mathbf{c}_{1}=\mathbf{r}$. It is clear that
${\mathcal{P}}_{\mathbb{Z}_{4}}(\mathbf{c}_{1}\cdot\sigma\otimes\mathbf{c}_{1}\cdot\sigma)={\mathcal{P}}_{\mathbb{Z}_{4}}(\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma)=\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma$.
The identity

$\displaystyle(\mathbf{r}\cdot\sigma)(\mathbf{u}\cdot\sigma)(\mathbf{r}\cdot\sigma)\otimes(\mathbf{r}\cdot\sigma)(\mathbf{v}\cdot\sigma)(\mathbf{r}\cdot\sigma)$
$\displaystyle=4(\mathbf{r}\cdot\mathbf{u})(\mathbf{r}\cdot\mathbf{v})\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma-2(\mathbf{r}\cdot\mathbf{u})\mathbf{r}\cdot\sigma\otimes\mathbf{v}\cdot\sigma$
$\displaystyle\qquad-2(\mathbf{r}\cdot\mathbf{v})\mathbf{u}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\mathbf{u}\cdot\sigma\otimes\mathbf{v}\cdot\sigma.$
(85)

allows us to show that when $j\neq 1$,

$\displaystyle{\mathcal{P}}_{\mathbb{Z}_{4}}(\mathbf{c}_{1}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma)$
$\displaystyle=\mathbf{c}_{1}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma+2(\mathbf{r}\cdot\mathbf{c}_{1})(\mathbf{r}\cdot\mathbf{c}_{j})\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma-(\mathbf{r}\cdot\mathbf{c}_{1})\mathbf{r}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma-(\mathbf{r}\cdot\mathbf{c}_{j})\mathbf{c}_{1}\cdot\sigma\otimes\mathbf{r}\cdot\sigma$
$\displaystyle=\mathbf{r}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma-\mathbf{r}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma=0$
(86)

because $\mathbf{r}\cdot\mathbf{c}_{j}=\mathbf{c}_{1}\cdot\mathbf{c}_{j}=0$.
The same holds for the
$\mathbf{c}_{j}\cdot\sigma\otimes\mathbf{c}_{1}\cdot\sigma$ terms with $j\neq
1$, which also vanish when ${\mathcal{P}}_{\mathbb{Z}_{4}}$ is applied.
Finally, when $i,j\neq 1$,

$\displaystyle{\mathcal{P}}_{\mathbb{Z}_{4}}(\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma)$
$\displaystyle=\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma+2(\mathbf{r}\cdot\mathbf{c}_{i})(\mathbf{r}\cdot\mathbf{c}_{j})\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma-(\mathbf{r}\cdot\mathbf{c}_{i})\mathbf{r}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma-(\mathbf{r}\cdot\mathbf{c}_{j})\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{r}\cdot\sigma$
$\displaystyle=\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma.$
(87)

Therefore

$\displaystyle{\mathcal{P}}_{\mathbb{Z}_{4}}(\rho)$
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{a})(\mathbf{r}\cdot\sigma)\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{b})(\mathds{1}\otimes\mathbf{r}\cdot\sigma)+T^{\prime}_{11}\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i,j=2}^{3}T^{\prime}_{ij}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma\ \right]$
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{a})(\mathbf{r}\cdot\sigma)\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{b})(\mathds{1}\otimes\mathbf{r}\cdot\sigma)+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i=2}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\ \right],$ (88)

where $\\{\mathbf{r},\mathbf{c}_{i}\\}$ and $\\{\mathbf{r},\mathbf{d}_{i}\\}$
are orthonormal bases of $\mathbb{R}^{3}$.

###  C.3 $\rm{Iso}(\rho)=\mathbb{Z}_{2}$

The image of ${\mathcal{P}}_{\mathbb{Z}_{4}}$ forms a proper subset of the
image of ${\mathcal{P}}_{\mathbb{Z}_{2}}$, therefore there exist 2-qubit
states with $\rm{Iso}(\rho)=\mathbb{Z}_{2}$, and these arise in several ways:

  * •

States of the form

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma+\sum_{i=1}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\ \right],$ (89)

where $\\{\mathbf{c}_{i}\\}$ and $\\{\mathbf{d}_{i}\\}$ are orthonormal bases
of $\mathbb{R}^{3}$ and $\mathbf{c}_{i}\neq\mathbf{d}_{i}$ for $i=1,2,3$.

  * •

States of the form

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i=2}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\ \right],$ (90)

where $\\{\mathbf{r},\mathbf{c}_{i}\\}$ and $\\{\mathbf{r},\mathbf{d}_{i}\\}$
are orthonormal bases of $\mathbb{R}^{3}$ but $\mathbf{a}$ and/or $\mathbf{b}$
are not parallel or anti-parallel to $\mathbf{r}$, i.e. there are no
$\gamma_{1},\gamma_{2}\in\mathbb{R}$ such that both
$\mathbf{a}=\gamma_{1}\mathbf{r}$ and $\mathbf{b}=\gamma_{2}\mathbf{r}$.

  * •

States of the form

$\displaystyle\rho$
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma+\sum_{i,j=1}^{3}T_{ij}\
\sigma_{i}\otimes\sigma_{j}\ \right],$ (91)

where $\mathbf{a}\neq\gamma\mathbf{b}$ for any $\gamma\in\mathbb{R}$, so the
local Bloch vectors are neither parallel nor anti-parallel to each other.

###  C.4 $U(1)\prec\rm{Iso}(\rho)$

The next subgroup to consider is $U(1)$. We apply ${\mathcal{P}}_{U(1)}$ to
states ${\mathcal{P}}_{\mathbb{Z}_{4}}(\rho)$, because if
$\mathbb{Z}_{4}\nprec\rm{Iso}(\rho)$ then $U(1)\nprec\rm{Iso}(\rho)$. For
illustrative purposes, consider the $U(1)$ subgroup $\\{e^{i\theta Z}:\
0\leq\theta<2\pi\\}$, which contains $\mathbb{Z}_{4}=\\{\pm\mathds{1},\pm
iZ\\}$. The states invariant under this $U(1)$ subgroup are

$\displaystyle{\mathcal{P}}_{U(1)}(\rho)$
$\displaystyle={\mathcal{P}}_{U(1)}\circ{\mathcal{P}}_{\mathbb{Z}_{4}}(\rho)$
(92) $\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+a_{z}\
Z\otimes\mathds{1}+b_{z}\ \mathds{1}\otimes Z+T_{33}\ Z\otimes
Z+\sum_{i,j=1}^{2}T_{ij}\ {\mathcal{P}}_{U(1)}(\sigma_{i}\otimes\sigma_{j})\
\right]$ (93)

because the terms $Z\otimes\mathds{1}$, $\mathds{1}\otimes Z$ and $Z\otimes Z$
are certainly invariant under the $U(1)$ subgroup generated by $Z$. For the
remaining correlation terms,

$\displaystyle{\mathcal{P}}_{U(1)}(X\otimes X)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\ e^{i\theta Z}Xe^{-i\theta
Z}\otimes e^{i\theta Z}Xe^{-i\theta Z}$ (94)
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\ \cos^{2}{2\theta}\
X\otimes X+\sin^{2}{2\theta}\ Y\otimes Y-\cos{2\theta}\sin{2\theta}\ (X\otimes
Y+Y\otimes X)$ (95) $\displaystyle=\frac{1}{2}(X\otimes X+Y\otimes
Y)={\mathcal{P}}_{U(1)}(Y\otimes Y)$ (96)

and similarly ${\mathcal{P}}_{U(1)}(X\otimes Y)=-{\mathcal{P}}_{U(1)}(Y\otimes
X)=\frac{1}{2}(X\otimes Y-Y\otimes X)$. Together these give

$\displaystyle{\mathcal{P}}_{U(1)}(\rho)$
$\displaystyle=\frac{1}{4}\bigg{[}\mathds{1}\otimes\mathds{1}+a_{z}\
Z\otimes\mathds{1}+b_{z}\ \mathds{1}\otimes Z+T_{33}\ Z\otimes
Z+\frac{1}{2}(T_{11}+T_{22})(X\otimes X+Y\otimes Y)$ $\displaystyle\hskip
56.9055pt\left.+\frac{1}{2}(T_{12}-T_{21})(X\otimes Y-Y\otimes X)\right]$ (97)

Now consider the more general $U(1)=\\{e^{i\mathbf{r}\cdot\sigma}:\
0\leq\theta<2\pi\\}$. This acts on a Bloch vector as

$\displaystyle
e^{i\theta\mathbf{r}\cdot\sigma}(\mathbf{u}\cdot\sigma)e^{-i\theta\mathbf{r}\cdot\sigma}$
$\displaystyle=\cos{2\theta}\ \mathbf{u}\cdot\sigma+\sin{2\theta}\
(\mathbf{u}\times\mathbf{r})\cdot\sigma+2\sin^{2}{\theta}\
(\mathbf{r}\cdot\mathbf{u})\ \mathbf{r}\cdot\sigma,$ (98)

therefore

$\displaystyle{\mathcal{P}}_{U(1)}(\mathbf{a}\cdot\sigma\otimes\mathds{1})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\
e^{i\theta\mathbf{r}\cdot\sigma}(\mathbf{a}\cdot\sigma)e^{-i\theta\mathbf{r}\cdot\sigma}\otimes\mathds{1}$
(99) $\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\ \cos{2\theta}\
\mathbf{a}\cdot\sigma\otimes\mathds{1}+\sin{2\theta}\
(\mathbf{a}\times\mathbf{r})\cdot\sigma\otimes\mathds{1}+2\sin^{2}{\theta}\
(\mathbf{r}\cdot\mathbf{a})\ \mathbf{r}\cdot\sigma\otimes\mathds{1}$
$\displaystyle=(\mathbf{r}\cdot\mathbf{a})\
\mathbf{r}\cdot\sigma\otimes\mathds{1}$ (100)

and similarly
$P_{U(1)}(\mathds{1}\otimes\mathbf{b}\cdot\sigma)=(\mathbf{r}\cdot\mathbf{b})\
\mathds{1}\otimes\mathbf{r}\cdot\sigma$. The group action on the correlation
terms is

$\displaystyle
e^{i\theta\mathbf{r}\cdot\sigma}(\mathbf{u}\cdot\sigma)e^{-i\theta\mathbf{r}\cdot\sigma}\otimes
e^{i\theta\mathbf{r}\cdot\sigma}(\mathbf{v}\cdot\sigma)e^{-i\theta\mathbf{r}\cdot\sigma}$
$\displaystyle=\cos^{2}{2\theta}\
\mathbf{u}\cdot\sigma\otimes\mathbf{v}\cdot\sigma+4\sin^{4}{\theta}(\mathbf{r}\cdot\mathbf{u})(\mathbf{r}\cdot\mathbf{v})\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+2\cos{2\theta}\sin{2\theta}\
[\mathbf{u}\cdot\sigma\otimes(\mathbf{v}\times\mathbf{r})\cdot\sigma$
$\displaystyle\qquad+(\mathbf{u}\times\mathbf{r})\cdot\sigma\otimes\mathbf{v}\cdot\sigma]+2\cos{2\theta}\sin^{2}{\theta}[(\mathbf{r}\cdot\mathbf{v})\mathbf{u}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+(\mathbf{r}\cdot\mathbf{u})\mathbf{r}\cdot\sigma\otimes\mathbf{v}\cdot\sigma]$
$\displaystyle\qquad+2\sin{2\theta}\sin^{2}{\theta}\
[(\mathbf{r}\cdot\mathbf{v})(\mathbf{u}\times\mathbf{r})\cdot\sigma\otimes\mathbf{r}\cdot\sigma+(\mathbf{r}\cdot\mathbf{u})\mathbf{r}\cdot\sigma\otimes(\mathbf{v}\times\mathbf{r})\cdot\sigma]$
$\displaystyle\qquad+\sin^{2}{2\theta}(\mathbf{u}\times\mathbf{r})\cdot\sigma\otimes(\mathbf{v}\times\mathbf{r})\cdot\sigma.$
(101)

This gives

$\displaystyle{\mathcal{P}}_{U(1)}(\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\
e^{i\theta\mathbf{r}\cdot\sigma}(\mathbf{c}_{i}\cdot\sigma)e^{-i\theta\mathbf{r}\cdot\sigma}\otimes
e^{i\theta\mathbf{r}\cdot\sigma}(\mathbf{c}_{j}\cdot\sigma)e^{-i\theta\mathbf{r}\cdot\sigma}$
(102) $\displaystyle=\frac{1}{2}[\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma-(\mathbf{r}\cdot\mathbf{c}_{j})\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{r}\cdot\sigma-(\mathbf{r}\cdot\mathbf{c}_{i})\mathbf{r}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma$
$\displaystyle\hskip
56.9055pt+(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma\otimes(\mathbf{c}_{j}\times\mathbf{r})\cdot\sigma+3(\mathbf{r}\cdot\mathbf{c}_{i})(\mathbf{r}\cdot\mathbf{c}_{j})\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma\
]$ (103) $\displaystyle=\frac{1}{2}[\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma-\delta_{1j}\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{1}\cdot\sigma-\delta_{1i}\mathbf{c}_{1}\cdot\sigma\otimes\mathbf{c}_{j}\cdot\sigma$
$\displaystyle\qquad+(\mathbf{c}_{i}\times\mathbf{c}_{1})\cdot\sigma\otimes(\mathbf{c}_{j}\times\mathbf{c}_{1})\cdot\sigma+3\
\delta_{1i}\delta_{1j}\
\mathbf{c}_{1}\cdot\sigma\otimes\mathbf{c}_{1}\cdot\sigma],$ (104)

therefore

$\displaystyle{\mathcal{P}}_{U(1)}(\rho)$
$\displaystyle=\frac{1}{4}\bigg{[}\mathds{1}\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{a})\
\mathbf{r}\cdot\sigma\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{b})\
\mathds{1}\otimes\mathbf{r}\cdot\sigma+T^{\prime}_{11}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma$ $\displaystyle\hskip
28.45274pt+\frac{1}{2}(T^{\prime}_{22}+T^{\prime}_{33})\
[\mathbf{c}_{2}\cdot\sigma\otimes\mathbf{c}_{2}\cdot\sigma+\mathbf{c}_{3}\cdot\sigma\otimes\mathbf{c}_{3}\cdot\sigma]$
$\displaystyle\hskip 28.45274pt+\frac{1}{2}(T^{\prime}_{23}-T^{\prime}_{32})\
[\mathbf{c}_{2}\cdot\sigma\otimes\mathbf{c}_{3}\cdot\sigma-\mathbf{c}_{3}\cdot\sigma\otimes\mathbf{c}_{2}\cdot\sigma]\bigg{]}$
(105)
$\displaystyle=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{a})\
\mathbf{r}\cdot\sigma\otimes\mathds{1}+(\mathbf{r}\cdot\mathbf{b})\
\mathds{1}\otimes\mathbf{r}\cdot\sigma+\tau_{1}\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\tau\sum_{i=2}^{3}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right]$ (106)

where $\\{\mathbf{r},\mathbf{c}_{i}\\}$ and $\\{\mathbf{r},\mathbf{d}_{i}\\}$
are orthonormal bases of $\mathbb{R}^{3}$.

###  C.5 $K_{2}\prec\rm{Iso}(\rho)$

Consider one such subgroup, $K_{2}=\\{\pm\mathds{1},\pm iX,\pm iY,\pm iZ\\}$.
For this subgroup,

${\mathcal{P}}_{K_{2}}(\rho)=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+{\mathcal{P}}_{K_{2}}(\mathbf{a}\cdot\sigma\otimes\mathds{1})+{\mathcal{P}}_{K_{2}}(\mathds{1}\otimes\mathbf{b}\cdot\sigma)+\sum_{i,j=1}^{3}T_{ij}\
{\mathcal{P}}_{K_{2}}(\sigma_{i}\otimes\sigma_{j})\ \right].$ (107)

Local Bloch vectors are eliminated, since
${\mathcal{P}}_{K_{2}}(X\otimes\mathds{1})={\mathcal{P}}_{K_{2}}(Y\otimes\mathds{1})={\mathcal{P}}_{{\mathcal{K}}_{2}}(Z\otimes\mathds{1})=0$,
and likewise for the other local Bloch vector. For the correlation terms,

$\displaystyle{\mathcal{P}}_{K_{2}}(X\otimes X)$
$\displaystyle=\frac{1}{4}[X\otimes X+XXX\otimes XXX+YXY\otimes YXY+ZXZ\otimes
ZXZ]=X\otimes X,$ (108)

and similarly ${\mathcal{P}}_{K_{2}}(Y\otimes Y)=Y\otimes Y$ and
${\mathcal{P}}_{K_{2}}(Z\otimes Z)=Z\otimes Z$. The remaining correlation
terms vanish since

$\displaystyle{\mathcal{P}}_{K_{2}}(X\otimes Y)$
$\displaystyle=\frac{1}{4}[X\otimes Y+XXX\otimes XYX+YXY\otimes YYY+ZXZ\otimes
ZYZ]$ (109) $\displaystyle=\frac{1}{4}[X\otimes Y-X\otimes Y-X\otimes
Y+X\otimes Y]=0$ (110)

and similarly for other $\sigma_{i}\otimes\sigma_{j}$ terms with $i\neq j$.
Therefore

${\mathcal{P}}_{K_{2}}(\rho)=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+\sum_{i=1}^{3}T_{ii}\
\sigma_{i}\otimes\sigma_{i}\right],$ (111)

which are the T-states. This proves that all T-states $\rho_{T}$ have
$K_{2}\prec\rm{Iso}(\rho_{T})$.

Consider the more general $K_{2}$ subgroup, $\\{\pm\mathds{1},\pm
i\mathbf{r}_{1}\cdot\sigma,\pm i\mathbf{r}_{2}\cdot\sigma,\pm
i\mathbf{r}_{3}\cdot\sigma\\}$ where
$\\{\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3}\\}$ forms an orthonormal
basis of $\mathbb{R}^{3}$. We use that same basis for the correlation terms,
such that 2-qubit states can be expressed

$\rho=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+\mathbf{a}\cdot\sigma\otimes\mathds{1}+\mathds{1}\otimes\mathbf{b}\cdot\sigma+\sum_{i,j=1}^{3}T^{\prime}_{ij}\
\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma\ \right].$ (112)

We first consider the action of ${\mathcal{P}}_{K_{2}}$ on the local Bloch
vectors, $\mathbf{a}\cdot\sigma=\sum_{i=1}^{3}(\mathbf{r}_{i}\cdot\mathbf{a})\
\mathbf{r}_{i}\cdot\sigma$,

$\displaystyle{\mathcal{P}}_{K_{2}}(\mathbf{a}\cdot\sigma\otimes\mathds{1})$
$\displaystyle=\frac{1}{4}\bigg{[}\mathbf{a}\cdot\sigma\otimes\mathds{1}+\sum_{i=1}^{3}(\mathbf{r}_{i}\cdot\sigma)(\mathbf{a}\cdot\sigma)(\mathbf{r}_{i}\cdot\sigma)\otimes\mathds{1}\bigg{]}$
(113)
$\displaystyle=\frac{1}{4}\bigg{[}\mathbf{a}\cdot\sigma\otimes\mathds{1}+\sum_{i=1}^{3}(2(\mathbf{r}_{i}\cdot\mathbf{a})\mathbf{r}_{i}\cdot\sigma-\mathbf{a}\cdot\sigma)\otimes\mathds{1}\bigg{]}$
(114) $\displaystyle=\frac{1}{2}\left[\
\sum_{i=1}^{3}(\mathbf{r}_{i}\cdot\mathbf{a})\
\mathbf{r}_{i}\cdot\sigma\otimes\mathds{1}-\mathbf{a}\cdot\sigma\otimes\mathds{1}\
\right]=0,$ (115)

and similarly
${\mathcal{P}}_{K_{2}}(\mathds{1}\otimes\mathbf{b}\cdot\sigma)=0$. The
correlation terms give

$\displaystyle{\mathcal{P}}_{K_{2}}(\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma)$
$\displaystyle=\frac{1}{4}\left[\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma+\sum_{k=1}^{3}(\mathbf{r}_{k}\cdot\sigma)(\mathbf{r}_{i}\cdot\sigma)(\mathbf{r}_{k}\cdot\sigma)\otimes(\mathbf{r}_{k}\cdot\sigma)(\mathbf{r}_{j}\cdot\sigma)(\mathbf{r}_{k}\cdot\sigma)\right]$
(116)
$\displaystyle=\frac{1}{4}\left[\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma+\sum_{k=1}^{3}\bigg{(}4(\mathbf{r}_{k}\cdot\mathbf{r}_{i})(\mathbf{r}_{k}\cdot\mathbf{r}_{j})\mathbf{r}_{k}\cdot\sigma\otimes\mathbf{r}_{k}\cdot\sigma+\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma\right.$
$\displaystyle\hskip
142.26378pt-2(\mathbf{r}_{k}\cdot\mathbf{r}_{i})\mathbf{r}_{k}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma-2(\mathbf{r}_{k}\cdot\mathbf{r}_{j})\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{k}\cdot\sigma\bigg{)}\bigg{]}$
$\displaystyle=\frac{1}{4}\left[\sum_{k=1}^{3}(4\delta_{ik}\delta_{jk}\mathbf{r}_{k}\cdot\sigma\otimes\mathbf{r}_{k}\cdot\sigma-2\delta_{ik}\mathbf{r}_{k}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma-2\delta_{jk}\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{k}\cdot\sigma)+4\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma\right]$
$\displaystyle=\delta_{ij}\
\mathbf{r}_{j}\cdot\sigma\otimes\mathbf{r}_{j}\cdot\sigma$ (117)

Therefore

${\mathcal{P}}_{K_{2}}(\rho)=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+\sum_{i=1}^{3}\tau_{i}\
\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{i}\cdot\sigma\right]$ (118)

where $\\{\mathbf{r}_{i}\\}$ is an orthonormal basis for $\mathbb{R}^{3}$.

###  C.6 $\rm{Iso}(\rho)=\mathbb{Z}_{4}$

We have now calculated ${\mathcal{P}}_{H}(\rho)$ for the two subgroups
directly above $\mathbb{Z}_{4}$ on the subgroup lattice that are possible
isotropy subgroups. This identifies the 2-qubit states in $C(\mathbb{Z}_{4})$.
They are:

  * •

States of the form

$\rho=\frac{1}{4}\bigg{[}\mathds{1}\otimes\mathds{1}+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i=2}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\bigg{]}$ (119)

where $\tau_{2}\neq\tau_{3}$ and $\mathbf{c}_{i}\neq\mathbf{d}_{i}$ for
$i=2,3$.

  * •

States of the form

$\rho=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+a\mathbf{r}\cdot\sigma\otimes\mathds{1}+b\mathds{1}\otimes\mathbf{r}\cdot\sigma+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sum_{i=2}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right]$ (120)

where $\tau_{2}\neq\tau_{3}$ and $\mathbf{c}_{i}\neq\mathbf{d}_{i}$ for
$i=2,3$, and $a,b\in\mathbb{R}$.

  * •

The states

$\rho=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+a\mathbf{c}_{1}\cdot\sigma\otimes\mathds{1}+b\mathds{1}\otimes\mathbf{c}_{1}\cdot\sigma+\sum_{i=1}^{3}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma\right]$ (121)

where $a,b\in\mathbb{R}$. This gives a $\mathbb{Z}_{4}$ isotropy subgroup when
$\tau_{1}=\tau_{2}\neq\tau_{3}$.

###  C.7 $K_{\infty}\prec\rm{Iso}(\rho)$

The next subgroup to consider is $K_{\infty}$, which contains the $K_{2}$
subgroup. Therefore we need only apply ${\mathcal{P}}_{K_{\infty}}$ to states
${\mathcal{P}}_{K_{2}}(\rho)$. We can also say the same about states
${\mathcal{P}}_{U(1)}(\rho)$, however the $K_{2}$ subgroup projects onto a
simpler subset of states.

As an illustrative example, consider the particular subgroup
$\\{(iX)^{\alpha}e^{i\theta Z}:\ 0\leq\theta<2\pi,\ \alpha=0,1,2,3\\}$. This
contains the $K_{2}$ subgroup $\\{\pm\mathds{1},\pm iX,\pm iY,\pm iZ\\}$, so
we need only apply ${\mathcal{P}}_{K_{\infty}}$ to the T-states. The CPTP map
${\mathcal{P}}_{K_{\infty}}$ can be split up as
${\mathcal{P}}_{K_{\infty}}={\mathcal{P}}_{U(1)}\circ{\mathcal{P}}_{\mathbb{Z}_{4}}$,
therefore

${\mathcal{P}}_{K_{\infty}}(\rho_{T})={\mathcal{P}}_{U(1)}(\rho_{T})=\frac{1}{4}\bigg{(}\mathds{1}\otimes\mathds{1}+\frac{1}{2}(\tau_{1}+\tau_{2})\
(X\otimes X+Y\otimes Y)+\tau_{3}\ Z\otimes Z\bigg{)}$ (122)

More generally, we consider

$K_{\infty}=\\{(i\ \mathbf{r}_{j}\cdot\sigma)^{\alpha}e^{i\theta\
\mathbf{r}_{k}\cdot\sigma}:\ 0\leq\theta<2\pi,\ \alpha=0,1,2,3\\}$ (123)

where $\\{\mathbf{r}_{i}\\}$ is an orthonormal basis of $\mathbb{R}^{3}$ and
$j\neq k$. The projector ${\mathcal{P}}_{K_{\infty}}$ on a general 2-qubit
state gives

$\displaystyle{\mathcal{P}}_{K_{\infty}}(\rho)$
$\displaystyle={\mathcal{P}}_{U(1)}\circ{\mathcal{P}}_{K_{2}}(\rho)={\mathcal{P}}_{U(1)}\left[\frac{1}{4}\left(\mathds{1}\otimes\mathds{1}+\sum_{i=1}^{3}\tau_{i}\
\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{i}\cdot\sigma\right)\right]$ (124)
$\displaystyle=\frac{1}{4}\left(\mathds{1}\otimes\mathds{1}+\frac{1}{2}\left[\sum_{i\neq
k}\tau_{i}\right]\left[\sum_{i\neq
k}\mathbf{r}_{i}\cdot\sigma\otimes\mathbf{r}_{i}\cdot\sigma\right]+\tau_{k}\
\mathbf{r}_{k}\cdot\sigma\otimes\mathbf{r}_{k}\cdot\sigma\right)$ (125)

###  C.8 $\rm{Iso}(\rho)=K_{2}$

These are the T-states and the states with maximally mixed marginals that can
be reached from the T-states through rigid $SU(2)$ rotations:

$\rho=\frac{1}{4}\bigg{[}\mathds{1}\otimes\mathds{1}+\sum_{i}\tau_{i}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma\bigg{]},$ (126)

with $\tau_{1}\neq\tau_{2}\neq\tau_{3}$. There cannot be any local Bloch
vectors.

###  C.9 $\rm{Iso}(\rho)=U(1)$

These are states of the form

$\rho=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\tau\sum_{i=2}^{3}\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right]$
(127)

where $\mathbf{c}_{i}\neq\mathbf{d}_{i}$ for $i=2,3$. Local Bloch vectors are
permitted so long as they are either aligned or anti-aligned with the shared
Bloch vector in the correlation terms:

$\rho=\frac{1}{4}\left[\mathds{1}\otimes\mathds{1}+a\mathbf{r}\cdot\sigma\otimes\mathds{1}+b\mathds{1}\otimes\mathbf{r}\cdot\sigma+\tau_{1}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\tau\sum_{i=2}^{3}\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{d}_{i}\cdot\sigma\right]$
(128)

where $\mathbf{c}_{i}\neq\mathbf{d}_{i}$ for $i=2,3$ and $a,b\in\mathbb{R}$.

###  C.10 $\rm{Iso}(\rho)=SU(2)$

Since $G=SU(2)$, when we apply ${\mathcal{P}}_{SU(2)}$ to a general 2-qubit
state we get the form of a general 2-qubit symmetric state. Since
$K_{\infty}\prec SU(2)$,

${\mathcal{P}}_{SU(2)}(\rho)={\mathcal{P}}_{SU(2)}\circ{\mathcal{P}}_{K_{\infty}}(\rho)=\frac{1}{4}\left(\mathds{1}\otimes\mathds{1}+\sum_{i=1}^{3}\tau_{i}\
{\mathcal{P}}_{SU(2)}\
[\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma]\right).$ (129)

A general $SU(2)$ group transformation is
$U(g)=e^{i\varphi\mathbf{r}\cdot\sigma}$. We parameterise
$\mathbf{r}=(\sin{\phi}\cos{\theta},\sin{\phi}\sin{\theta},\cos{\phi})$.
Considering each of the terms individually,

$\displaystyle{\mathcal{P}}_{SU(2)}(\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma)$
$\displaystyle=\frac{1}{2\pi^{2}}\int_{0}^{2\pi}d\theta\int_{0}^{\pi}d\phi\
\sin{\phi}\int_{0}^{\pi}d\varphi\ \sin^{2}{\varphi}\
\bigg{(}e^{i\varphi\mathbf{r}\cdot\sigma}\mathbf{c}_{i}\cdot\sigma
e^{-i\varphi\mathbf{r}\cdot\sigma}\otimes
e^{i\varphi\mathbf{r}\cdot\sigma}\mathbf{c}_{i}\cdot\sigma
e^{-i\varphi\mathbf{r}\cdot\sigma}\bigg{)}$ (130)
$\displaystyle=\frac{1}{2\pi^{2}}\int_{0}^{2\pi}d\theta\int_{0}^{\pi}d\phi\
\sin{\phi}\int_{0}^{\pi}d\varphi\ \sin^{2}{\varphi}\cos^{2}{2\varphi}\
\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma$
$\displaystyle\hskip
42.67912pt+2\sin^{2}{\varphi}\cos{2\varphi}\sin{2\varphi}\
[\mathbf{c}_{i}\cdot\sigma\otimes(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma+(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma]$
$\displaystyle\hskip 42.67912pt+2\cos{2\varphi}\sin^{4}{\varphi}\
(\mathbf{r}\cdot\mathbf{c}_{i})\
[\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\mathbf{r}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma]$
$\displaystyle\hskip 42.67912pt+2\sin{2\varphi}\sin^{4}{\varphi}\
(\mathbf{r}\cdot\mathbf{c}_{i})\
[(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\mathbf{r}\cdot\sigma\otimes(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma]$
$\displaystyle\hskip 42.67912pt+4\sin^{6}{\varphi}\
(\mathbf{r}\cdot\mathbf{c}_{i})^{2}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\sin^{2}{\varphi}\sin^{2}{2\varphi}\
(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma\otimes(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma$
(131) $\displaystyle=\frac{1}{8\pi}\int_{0}^{2\pi}d\theta\int_{0}^{\pi}d\phi\
\sin{\phi}\
\bigg{(}\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma-2(\mathbf{r}\cdot\mathbf{c}_{i})\
[\mathbf{c}_{i}\cdot\sigma\otimes\mathbf{r}\cdot\sigma+\mathbf{r}\cdot\sigma\otimes\mathbf{c}_{i}\cdot\sigma]$
$\displaystyle\hskip
142.26378pt+(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma\otimes(\mathbf{c}_{i}\times\mathbf{r})\cdot\sigma+5(\mathbf{r}\cdot\mathbf{c}_{i})^{2}\
\mathbf{r}\cdot\sigma\otimes\mathbf{r}\cdot\sigma\bigg{)}$ (132)

Performing the integrals over $\theta$ and $\phi$ gives

$\displaystyle{\mathcal{P}}_{SU(2)}(\rho)=\frac{1}{4}\bigg{[}\
\mathds{1}\otimes\mathds{1}+\tau\sum_{i=1}^{3}\sigma_{i}\otimes\sigma_{i}\
\bigg{]}.$ (133)