# Margolus-Levitin quantum speed limit for an arbitrary fidelity

Niklas Hörnedal  0000-0002-2005-8694 Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg  Ole
Sönnerborn  0000-0002-1726-4892<EMAIL_ADDRESS>Department of
Mathematics and Computer Science, Karlstad University, 651 88 Karlstad, Sweden
Department of Physics, Stockholm University, 106 91 Stockholm, Sweden

###### Abstract

The Mandelstam-Tamm and Margolus-Levitin quantum speed limits are two well-
known evolution time estimates for isolated quantum systems. These bounds are
usually formulated for fully distinguishable initial and final states, but
both have tight extensions to systems that evolve between states with an
arbitrary fidelity. However, the foundations of these extensions differ in
some essential respects. The extended Mandelstam-Tamm quantum speed limit has
been proven analytically and has a clear geometric interpretation.
Furthermore, which systems saturate the limit is known. The derivation of the
extended Margolus-Levitin quantum speed limit, on the other hand, is based on
numerical estimates. Moreover, the limit lacks a geometric interpretation, and
no complete characterization of the systems reaching it exists. In this paper,
we derive the extended Margolus-Levitin quantum speed limit analytically and
describe the systems that saturate the limit in detail. We also provide the
limit with a symplectic-geometric interpretation, which indicates that it is
of a different character than most existing quantum speed limits. At the end
of the paper, we analyze the maximum of the extended Mandelstam-Tamm and
Margolus-Levitin quantum speed limits and derive a dual version of the
extended Margolus-Levitin quantum speed limit. The maximum limit is tight
regardless of the fidelity of the initial and final states. However, the
conditions under which the maximum limit is saturated differ depending on
whether or not the initial state and the final state are fully
distinguishable. The dual limit is also tight and follows from a time reversal
argument. We describe the systems that saturate the dual quantum speed limit.

##  I Introduction

A quantum speed limit (QSL) is a lower bound on the time it takes to transform
a quantum system in some predetermined way. QSLs exist for all sorts of
transformations of both open and closed systems. Many involve statistical
quantities such as energy, entropy, fidelity, and purity [1, 2, 3, 4, 5].

A prominent QSL is attributed to Mandelstam and Tamm [6]. The Mandelstam-Tamm
QSL states that the time it takes for an isolated quantum system to evolve
between two fully distinguishable states is at least $\pi/2$ divided by the
energy uncertainty,111In this paper, “state” refers to a pure quantum state
unless otherwise is explicitly stated.,222Two states are fully distinguishable
if their fidelity is zero.,333All quantities are expressed in units such that
$\hbar=1$.

$\tau\geq\tau_{\textsc{mt}}(0),\quad\tau_{\textsc{mt}}(0)=\frac{\pi}{2\Delta
H}.$ (1)

Mandelstam and Tamm actually showed a more general result: If the fidelity
between the initial state and the final state is $\delta$, the evolution time
is bounded according to

$\tau\geq\tau_{\textsc{mt}}(\delta),\quad\tau_{\textsc{mt}}(\delta)=\frac{\arccos\sqrt{\delta}}{\Delta
H}.$ (2)

For $\delta=0$, estimate (2) agrees with estimate (1).

Anandan and Aharonov [7] provided the Mandelstam-Tamm QSL with an elegant
geometric interpretation when they showed that the Fubini-Study distance
between two states with fidelity $\delta$ is $\arccos\sqrt{\delta}$ and that
the Fubini-Study evolution speed equals the energy uncertainty. Inequality (2)
is thus saturated for isolated systems where the state evolves along a
shortest Fubini-Study geodesic. Anandan and Aharonov’s work inspired the
writing of Ref. [8], which extends the Mandelstam-Tamm QSL to systems in mixed
states in several different ways.

Margolus and Levitin [9] derived another QSL that is often mentioned together
with Mandelstam and Tamm’s. The Margolus-Levitin QSL states that the time it
takes for an isolated quantum system to evolve between two fully
distinguishable states is at least $\pi/2$ divided by the expected energy
shifted by the smallest energy,

$\tau\geq\tau_{\textsc{ml}}(0),\quad\tau_{\textsc{ml}}(0)=\frac{\pi}{2\langle
H-\epsilon_{0}\rangle}.$ (3)

Giovannetti _et al._ [10] extended Margolus and Levitin’s QSL to an arbitrary
fidelity. More precisely, they showed that the evolution time of an isolated
system is bounded from below according to

$\tau\geq\tau_{\textsc{ml}}(\delta),\quad\tau_{\textsc{ml}}(\delta)=\frac{\alpha(\delta)}{\langle
H-\epsilon_{0}\rangle},$ (4)

where $\alpha$ is a function that depends only on the fidelity $\delta$
between the initial and final states; see Sec. II. Giovannetti _et al._ also
showed that (4), like (2), is tight, which means that for each $\delta$ there
is a system for which the inequality in (4) is an equality. However, apart
from that, the situation is different from the case of the Mandelstam-Tamm
QSL: A closed formula for $\alpha$ does not exist, the derivation of (4) rests
on numerical estimates, there is no classification of the systems saturating
(4), and a geometric interpretation like that of (2) is lacking [1, 2, 11,
12].

In this paper, we derive the extended Margolus-Levitin QSL analytically and
characterize the systems that saturate this estimate (Sec. II). Furthermore,
we show that the extended Margolus-Levitin QSL has a symplectic-geometric
interpretation and is connected to the Aharonov-Anandan geometric phase [13]
(Sec. III). The Margolus-Levitin QSL thus differs fundamentally from most
existing QSLs. That the estimates in (3) and (4) are either invalid or correct
but not tight unless we shift the expected energy with the smallest energy
suggests that a ground state will play a central role in the derivation of the
extended Margolus-Levitin QSL and in its geometric interpretation.

The maximum of the Mandelstam-Tamm and the extended Margolus-Levitin QSL is
also a QSL. The characteristics of the maximum QSL differ depending on whether
or not the initial state and final state are fully distinguishable. We explain
why this is so, and we provide several QSLs that are related to but less sharp
than the extended Margolus-Levitin QSL. We also derive a dual version of the
extended Margolus-Levitin QSL involving the largest rather than the smallest
energy (Sec. IV). The paper concludes with a summary and a comment on the
difficulty of extending the Margolus-Levitin quantum speed limit to driven
systems (Sec. V). For a more detailed discussion of this difficulty, see Ref.
[14].

##  II The extended Margolus-Levitin quantum speed limit

The function $\alpha$ in Giovannetti _et al._ ’s estimate (4) is444The
$\alpha$ in Ref. [10] equals the $\alpha$ in (5) multiplied by $2/\pi$.

$\alpha(\delta)=\min_{z^{2}\leq\delta}\bigg{\\{}\frac{1+z}{2}\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}\bigg{\\}}.$
(5)

For each $\delta$, the minimum on the right-hand side is assumed for a unique
$z$. Figure 1 shows the graph of $\alpha$.

Figure 1: The graph of $\alpha$ as a function of the fidelity $\delta$ between
the initial state and the final state. If $\delta=0$, the initial and final
states are fully distinguishable, and $\alpha(\delta)=\pi/2$. In this case we
recover the Margolus-Levitin QSL (3). If $\delta=1$, the initial and final
states are the same, and $\alpha(\delta)=0$. This is reasonable since it takes
no time to remain in the initial state.

Note that $\alpha$ depends only on the fidelity $\delta$ between the initial
and final states. The fidelity, or overlap, between two pure states $\rho_{a}$
and $\rho_{b}$ is $\operatorname{tr}(\rho_{a}\rho_{b})$.

Although the Margolus-Levitin QSL (3) is quite surprising, its proof is
relatively simple [9]. Giovannetti _et al._ ’s estimate (4) reduces to the
Margolus-Levitin QSL for $\delta=0$, but the derivation in Ref. [10] for a
general $\delta$ is rather complicated. Moreover, it is partly based on
numerical calculations. In this section, we derive (4) analytically. We also
characterize the systems, that is, the states and Hamiltonians that saturate
(4).

For simplicity, we write $\langle H\rangle$ for the expected energy of a
system without reference to its state. Furthermore, we write $\langle
H-\epsilon_{0}\rangle$ for the difference between $\langle H\rangle$ and the
smallest eigenvalue $\epsilon_{0}$ of $H$. We call this quantity the
normalized expected energy. In this paper we only consider isolated systems,
that is, systems where a time-independent Hamiltonian governs the dynamics.
For such systems, the expected energy and normalized expected energy are
conserved quantities. The expected energy and normalized expected energy thus
depend on the initial state but do not change as the state evolves.

###  II.1 The extended Margolus-Levitin quantum speed limit for a two-
dimensional system

In this section we show that Giovannetti _et al._ ’s estimate (4) is valid and
tight for qubit systems. Derivations of the statements in this section can be
found in Appendix A.

Consider a qubit system with Hamiltonian

$H=\epsilon_{0}|0\rangle\langle 0|+\epsilon_{1}|1\rangle\langle
1|,\quad\epsilon_{0}<\epsilon_{1},$ (6)

the vectors $|0\rangle$ and $|1\rangle$ being orthonormal. We identify each
qubit state $\rho$ with a unit length vector $\mathbf{r}=(x,y,z)$ called the
Bloch vector of $\rho$ by defining

$\displaystyle x$ $\displaystyle=\langle 1|\rho|0\rangle+\langle
0|\rho|1\rangle,$ (7) $\displaystyle y$ $\displaystyle=i(\langle
1|\rho|0\rangle-\langle 0|\rho|1\rangle),$ (8) $\displaystyle z$
$\displaystyle=1-2\langle 0|\rho|0\rangle.$ (9)

The dynamics induced by $H$ causes the Bloch vectors to rotate about the $z$
axis with a constant inclination and an azimuthal angular speed
$\epsilon_{1}-\epsilon_{0}$. Furthermore, a state’s expected energy is
determined by, and determines, the $z$ coordinate of its Bloch vector:

$z=\frac{2\langle
H\rangle-\epsilon_{1}-\epsilon_{0}}{\epsilon_{1}-\epsilon_{0}}=2\frac{\langle
H-\epsilon_{0}\rangle}{\epsilon_{1}-\epsilon_{0}}-1.$ (10)

Two states thus have the same expected energy if and only if their Bloch
vectors have the same $z$ coordinate.

Let $\mathbf{r}_{a}$ and $\mathbf{r}_{b}$ be the Bloch vectors of two states
with a common $z$ coordinate $z$ and fidelity $\delta$, and suppose that $H$
causes $\mathbf{r}_{a}$ to rotate to $\mathbf{r}_{b}$ in time $\tau$. Using
(7)–(9) one can show that the inner product between $\mathbf{r}_{a}$ and
$\mathbf{r}_{b}$ relates to the fidelity $\delta$ as

$\mathbf{r}_{a}\cdot\mathbf{r}_{b}=2\delta-1.$ (11)

To determine the distance traveled by the rotating Bloch vector consider the
orthogonal projections $\bar{\mathbf{r}}_{a}$ and $\bar{\mathbf{r}}_{b}$ of
$\mathbf{r}_{a}$ and $\mathbf{r}_{b}$ on the $xy$ plane. During the evolution,
$\bar{\mathbf{r}}_{a}$ rotates to $\bar{\mathbf{r}}_{b}$ along the peripheral
arc of a circular sector in the $xy$ plane of radius
$|\bar{\mathbf{r}}_{a}|=(1-z^{2})^{1/2}$ and apex angle
$\arccos(\bar{\mathbf{r}}_{a}\cdot\bar{\mathbf{r}}_{b}/|\bar{\mathbf{r}}_{a}|^{2})=\arccos[(2\delta-1-z^{2})/(1-z^{2})]$.
The distance traveled by the rotating Bloch vector equals the length of the
peripheral arc and is, thus,

$\sqrt{1-z^{2}}\arccos\left(\frac{2\delta-1-z^{2}}{1-z^{2}}\right).$ (12)

The speed of the rotating Bloch vector is
$(\epsilon_{1}-\epsilon_{0})(1-z^{2})^{1/2}$, and hence, by (12), the
evolution time is

$\tau=\frac{1}{\epsilon_{1}-\epsilon_{0}}\arccos\left(\frac{2\delta-1-z^{2}}{1-z^{2}}\right).$
(13)

Combined with (10), this gives that

$\tau\langle
H-\epsilon_{0}\rangle=\frac{1+z}{2}\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}.$
(14)

The relation (11) implies that the $z$ coordinate squared of the Bloch vectors
of two states with the same expected energy is less than the fidelity between
the states:

$2\delta-1=\mathbf{r}_{a}\cdot\mathbf{r}_{b}\geq
2z^{2}-\mathbf{r}_{a}\cdot\mathbf{r}_{a}=2z^{2}-1.$ (15)

Conversely, each expected energy level corresponding to a $z$ such that
$z^{2}\leq\delta$ contains Bloch vectors of states with fidelity $\delta$. We
conclude that $\tau\geq\tau_{\textsc{ml}}(\delta)$ for a qubit. Equality holds
if and only if the $z$ coordinate of the Bloch vector of the initial state
minimizes the right side of (14) over the interval $-\sqrt{\delta}\leq
z\leq\sqrt{\delta}$ and thus is such that

$\frac{1+z}{2}\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}=\alpha(\delta).$
(16)

###  II.2 The extended Margolus-Levitin quantum speed limit for systems of
arbitrary dimension

Section II.1 shows that Giovannetti _et al._ ’s estimate (4) is valid and
tight for qubit systems. Section II.1 also shows that for an arbitrary
isolated system with Hamiltonian $H$ there is a state that evolves into one
with fidelity $\delta$ in such a way that (4) is an identity: Let $|0\rangle$
and $|1\rangle$ be eigenvectors of $H$ with eigenvalues $\epsilon_{0}$ and
$\epsilon_{1}$, with $\epsilon_{1}>\epsilon_{0}$. Choose a $\rho$ with support
in the span of $|0\rangle$ and $|1\rangle$ and such that $z$ defined by (9)
satisfies (16). Then $\rho$ will evolve to a state with fidelity $\delta$ in
time $\tau=\tau_{\textsc{ml}}(\delta)$. Conversely, for any system in a state
$\rho$, a Hamiltonian exists that transforms $\rho$ into a state with fidelity
$\delta$ in such a way that (4) is saturated: Take an $H$ whose sum of two
eigenspaces, one of which corresponds to its smallest eigenvalue
$\epsilon_{0}$, contains the support of $\rho$. Adjust $H$’s spectrum so that
$z$ defined by (9) satisfies (16). Then $\rho$ will evolve to a state with
fidelity $\delta$ in time $\tau=\tau_{\textsc{ml}}(\delta)$.

An _effective qubit_ for $H$ is a state with support in the sum of two
eigenspaces of $H$. It behaves like a genuine qubit in that its support
evolves in the linear span of two energy eigenvectors, one from each
eigenspace covering the initial support. The above discussion shows that (4)
holds for and can always be saturated by an effective qubit. We say that a
state is _partly grounded_ if the eigenspace corresponding to $\epsilon_{0}$
is not contained in the kernel of the state or, equivalently, if the
eigenspace of $\epsilon_{0}$ is not orthogonal to the support of the state. In
Appendix B we show the first main result of this paper: If $\tau\langle
H-\epsilon_{0}\rangle$ assumes its smallest possible value when evaluated for
all Hamiltonians $H$, states $\rho$, and $\tau\geq 0$ such that $H$ transforms
$\rho$ into a state with fidelity $\delta$ in time $\tau$, then $\rho$ is a
partly grounded effective qubit for $H$ and $\tau\langle
H-\epsilon_{0}\rangle=\alpha(\delta)$. The extended Margolus-Levitin QSL (4)
thus holds generally and is a tight estimate saturable in all dimensions. We
interpret (4) geometrically in Sec. III. There we see, among other things,
that one can interpret $\alpha(\delta)$ as an extremal dynamical phase. Notice
the contrast with the Mandelstam-Tamm QSL, where the numerator is a geodesic
distance.

######  Remark 1.

If we select a subset of the spectrum of $H$ and consider only initial states
with support in the sum of the eigenspaces of the eigenvalues in the subset,
then the support of the evolved state will remain in that sum. The proof in
Appendix B shows that the evolution time of each such state satisfies the
inequality

$\tau\geq\frac{\alpha(\delta)}{\langle H-\epsilon_{0}^{\prime}\rangle},$ (17)

with $\epsilon_{0}^{\prime}$ being the smallest eigenvalue in the subset.555To
avoid having to treat trivial cases separately, we assume that the subset
contains at least two different eigenvalues. Furthermore, by Sec. II.1,
inequality (17) can be saturated with an effective qubit that evolves in the
sum of the eigenspaces corresponding to two eigenvalues in the subset, one of
which is $\epsilon_{0}^{\prime}$. As a special case, we have that the
evolution time is bounded according to

$\tau\geq\frac{\alpha(\delta)}{\langle
H-\epsilon_{0}^{\prime\prime}\,\rangle},$ (18)

where $\epsilon_{0}^{\prime\prime}$ is the smallest occupied energy, that is,
the smallest eigenvalue of $H$ whose corresponding eigenspace is not
annihilated by $\rho$. Often, the Margolus-Levitin QSL is formulated with the
expected energy shifted by the smallest occupied energy rather than the
smallest energy. Mathematically, however, there is no difference because we
can always reduce the Hilbert space to an effective Hilbert space and consider
the smallest occupied energy as the smallest energy. (However, see Remark 2.)

######  Remark 2.

The state must evolve in the span of two eigenvectors of $H$ to saturate (4),
one of which has eigenvalue $\epsilon_{0}$. No requirements are placed on the
eigenvalue $\epsilon_{1}$ of the second eigenvector except that it must differ
from $\epsilon_{0}$. However, if we want the evolution time to be as short as
possible, $\epsilon_{1}$ must be the largest eigenvalue of $H$. This follows
from (10) since saturation of (4) implies that the quotient $\langle
H-\epsilon_{0}\rangle/(\epsilon_{1}-\epsilon_{0})$ is independent of
$\epsilon_{1}$. Thus, the maximum value of $\langle H-\epsilon_{0}\rangle$,
and consequently the minimum value of $\tau$, is obtained for the
$\epsilon_{1}$ maximizing the difference $\epsilon_{1}-\epsilon_{0}$. The
observation that the state that saturates (4) with the shortest possible
evolution time is an effective qubit with support in the sum of the
eigenspaces belonging to the largest and the smallest eigenvalue generalizes
the main result in Ref. [15] to arbitrary fidelity; see also Ref. [16]. A
corresponding statement holds if we restrict the Hilbert space as in Remark 1.

In contrast to the energy uncertainty, we cannot consider the normalized
expected energy $\langle H-\epsilon_{0}\rangle$ as a measure of a state’s rate
of change. Since for each state there exist Hamiltonians $H_{1}$ and $H_{2}$,
with smallest eigenvalues $\epsilon_{0}^{1}$ and $\epsilon_{0}^{2}$, that
identically evolve the state but for which $\langle
H_{1}-\epsilon_{0}^{1}\rangle$ and $\langle H_{2}-\epsilon_{0}^{2}\rangle$ are
different. Thus, unlike most QSLs, the extended Margolus-Levitin QSL is not a
quotient of a distance and a speed [1, 2, 11, 12].

##  III Geometry of the extended Margolus-Levitin quantum speed limit

Equations (7)–(9) describe a diffeomorphism between the projective Hilbert
space of qubit states and the Bloch sphere. (Thus, we can identify qubit
states with their corresponding Bloch vectors.) We push forward the Fubini-
Study Riemannian metric and symplectic form using this diffeomorphism.666The
Fubini-Study distance is equal to half of the standard distance function on
the unit sphere, and the Fubini-Study symplectic form is equal to half the
standard area form on the unit sphere. The expression in (14) is then the
negative of the symplectic area of a surface with a triangular boundary in the
Bloch sphere. The path traced out by the evolving state and the shortest
geodesics connecting the initial and final state to the lowest energy state
$|0\rangle\langle 0|$ form the boundary of the surface; see Fig. 2.

Figure 2: An oriented surface in the Bloch sphere bounded by the evolution
curve $\rho_{t}$ and the shortest geodesics connecting the initial and final
states $\rho_{0}$ and $\rho_{\tau}$ to the ground state $|0\rangle\langle 0|$.
The orientation of the surface is such that its symplectic area is negative.
The negative of the symplectic area is minimal if and only if the extended
Margolus-Levitin QSL is saturated.

Notice that we have oriented the boundary so that the surface has the reverse
orientation compared with the standard orientation of the Bloch sphere. In
Sec. III.2, we show that $\tau\langle H-\epsilon_{0}\rangle$ is equal to the
negative of the symplectic area of such a triangular surface also in the
general case.

The expected energy level to which the initial qubit state $\rho$ belongs is a
geodesic sphere centered at $|0\rangle\langle 0|$, that is, a sphere made up
of all states at a fixed distance from $|0\rangle\langle 0|$. The radius of
the geodesic sphere is

$r=\arccos\sqrt{\langle 0|\rho|0\rangle}=\arccos\sqrt{(1-z)/2}.$ (19)

Therefore, $z=-\cos(2r)$. If we substitute $z$ for $-\cos(2r)$ on the right-
hand side of (5) we get

$\alpha(\delta)=\min_{r}\bigg{\\{}\sin^{2}r\arccos\bigg{(}1-\frac{2(1-\delta)}{\sin^{2}(2r)}\bigg{)}\bigg{\\}},$
(20)

where $r$ ranges from $\frac{1}{2}\arccos\sqrt{\delta}$ to
$\frac{1}{2}\arccos(-\sqrt{\delta})$. Section III.3 shows that the expression
minimized on the right-hand side is an extremal dynamical phase in a gauge
specified by a stationary state with eigenvalue $\epsilon_{0}$. First, in Sec.
III.1, we interpret $\tau\langle H-\epsilon_{0}\rangle$ as a dynamical phase.

###  III.1 Evolution time times normalized expected energy as a dynamical
phase

Consider a quantum system modeled on a finite-dimensional Hilbert space
$\mathcal{H}$. Let $\SS$ be the unit sphere in $\mathcal{H}$ and $\mathcal{P}$
be the projective Hilbert space of orthogonal projection operators of rank $1$
on $\mathcal{H}$.777Such operators represent pure states. The Hopf bundle is
the $U(1)$-principal bundle $\eta$ that sends each $|\psi\rangle$ in $\SS$ to
the corresponding state $|\psi\rangle\langle\psi|$ in $\mathcal{P}$. The Berry
connection on the Hopf bundle is defined as

$\mathcal{A}|\dot{\psi}\rangle=i\langle\psi|\dot{\psi}\rangle$ (21)

on tangent vectors $|\dot{\psi}\rangle$ at $|\psi\rangle$.

Assume the system has a Hamiltonian $H$. Choose an eigenstate $\sigma$ of $H$
with eigenvalue $\epsilon$. Let $\Omega(\sigma)$ be the open neighborhood of
$\sigma$ consisting of all states that are not fully distinguishable from
$\sigma$. Hypersurfaces $Y_{|\phi\rangle}$ in $\SS$, one for each vector
$|\phi\rangle$ in the fiber over $\sigma$, foliate the preimage of
$\Omega(\sigma)$ under $\eta$; see Fig. 3 and Ref. [17].

Figure 3: The preimage under the Hopf projection $\eta$ of the set
$\Omega(\sigma)$ of states having nonzero fidelity with $\sigma$ is foliated
by hypersurfaces $Y_{|\phi\rangle}$; one for each $|\phi\rangle$ in $\SS$
being projected to $\sigma$. The hypersurface $Y_{|\phi\rangle}$ consists of
all the $|\psi\rangle$ in $\SS$ that are in phase with $|\phi\rangle$. The
gauge group permutes the hypersurfaces, and $\eta$ maps each hypersurface
diffeomorphically onto $\Omega(\sigma)$. We define a potential $A^{\sigma}$ on
$\Omega(\sigma)$ by pushing down the restriction of $\mathcal{A}$ to one of
the hypersurfaces. The potential depends on $\sigma$ but not on the choice of
hypersurface.

The hypersurface $Y_{|\phi\rangle}$ consists of all vectors $|\psi\rangle$ in
$\SS$ that are in phase with $|\phi\rangle$, that is, are such that
$\langle\phi|\psi\rangle>0$. The gauge group permutes the hypersurfaces, and
$\eta$ maps each hypersurface diffeomorphically onto $\Omega(\sigma)$. We can
thus define a gauge potential $A^{\sigma}$ on $\Omega(\sigma)$ by pushing down
the restriction of $\mathcal{A}$ to an arbitrary hypersurface
$Y_{|\phi\rangle}$. The potential depends on $\sigma$ but not on the choice of
$|\phi\rangle$ in the fiber over $\sigma$.

The second main result in the paper reads: If $\rho$ is a state in
$\Omega(\sigma)$, and $\rho_{t}$, where $0\leq t\leq\tau$, is the evolution
curve starting from $\rho$, then $\rho_{t}$ is contained in $\Omega(\sigma)$
and

$\tau\langle H-\epsilon\rangle=\int_{\rho_{t}}A^{\sigma}.$ (22)

In other words, $\tau\langle H-\epsilon\rangle$ is the dynamical phase of
$\rho_{t}$ in a gauge associated with $\sigma$.888Some authors call the
negative of the right-hand side of (22) the dynamical phase of $\rho_{t}$. To
prove (22) select a $|\phi\rangle$ in the fiber over $\sigma$, let
$|\psi\rangle$ be the vector over $\rho$ in phase with $|\phi\rangle$, and let
$|\psi_{t}\rangle$ be the curve that extends from $|\psi\rangle$ and has the
velocity field $|\dot{\psi}_{t}\rangle=-i(H-\epsilon)|\psi_{t}\rangle$. The
curve $|\psi_{t}\rangle$ is in phase with $|\phi\rangle$ and projects to
$\rho_{t}$:

$\displaystyle\langle\phi|\psi_{t}\rangle=\langle\phi|e^{-it(H-\epsilon)}|\psi\rangle=\langle\phi|\psi\rangle>0,$
(23)
$\displaystyle|\psi_{t}\rangle\langle\psi_{t}|=e^{-it(H-\epsilon)}|\psi\rangle\langle\psi|e^{it(H-\epsilon)}=\rho_{t}.$
(24)

Hence,

$\int_{\rho_{t}}A^{\sigma}=\int_{|\psi_{t}\rangle}\mathcal{A}=\tau\langle
H-\epsilon\rangle.$ (25)

Notice that $\epsilon=\epsilon_{0}$ if $\sigma$ is a ground state.

###  III.2 Evolution time times normalized expected energy as the negative of
a symplectic area

We equip the projective Hilbert space $\mathcal{P}$ with the Fubini-Study
Riemannian metric $g$ and symplectic form $\omega$. For tangent vectors
$\dot{\rho}_{a}$ and $\dot{\rho}_{b}$ at $\rho$,999We normalize $g$ and
$\omega$ asymmetrically as this gives rise to cleaner formulas.

$\displaystyle
g(\dot{\rho}_{a},\dot{\rho}_{b})=\frac{1}{2}\operatorname{tr}(\dot{\rho}_{a}\dot{\rho}_{b}),$
(26)
$\displaystyle\omega(\dot{\rho}_{a},\dot{\rho}_{b})=-i\operatorname{tr}([\dot{\rho}_{a},\dot{\rho}_{b}]\rho).$
(27)

The geodesic distance function associated with the Fubini-Study metric is

$\operatorname{dist}(\rho_{a},\rho_{b})=\arccos\sqrt{\operatorname{tr}(\rho_{a}\rho_{b})}.$
(28)

Suppose the system is in a state $\rho$ and a Hamiltonian $H$ governs its
dynamics. Let $\rho_{t}$, $0\leq t\leq\tau$, represent the evolving state.
Since $H$ is time-independent, the distances between $\rho_{t}$ and the
eigenstates of $H$ are preserved. Let $\sigma$ be an eigenstate with
eigenvalue $\epsilon$ located at a distance $r<\pi/2$ from $\rho$.
Furthermore, let $\gamma_{t}^{1}$ and $\gamma_{t}^{2}$ be the shortest unit
speed geodesics from $\sigma$ to $\rho_{0}$ and from $\rho_{\tau}$ to
$\sigma$, respectively. The $\sigma$ closure of $\rho_{t}$ is the
concatenation $\rho^{\sigma}_{t}=\gamma^{1}_{t}\ast\rho_{t}\ast\gamma^{2}_{t}$
defined as

$\rho^{\sigma}_{t}=\begin{cases}\gamma^{1}_{t}&\text{for}\quad 0\leq t\leq
r,\\\ \rho_{t-r}&\text{for}\quad r\leq t\leq\tau+r,\\\
\gamma^{2}_{t-\tau-r}&\text{for}\quad\tau+r\leq t\leq\tau+2r.\end{cases}$ (29)

The left part of Fig. 4 illustrates the $\sigma$ closure.

Figure 4: On the left, the $\sigma$ closure $\rho_{t}^{\sigma}$ made up of the
shortest geodesic $\gamma^{1}_{t}$ from $\sigma$ to the initial state
$\rho_{0}$, the evolution curve $\rho_{t}$, and the shortest geodesic
$\gamma^{2}_{t}$ from the final state $\rho_{\tau}$ to $\sigma$. On the right,
a Seifert surface $\Sigma$ for $\rho_{t}^{\sigma}$. The circular arrow
indicates the orientation of $\Sigma$ compatible with that of
$\rho_{t}^{\sigma}$.

Below we show the third main result of the paper:

$\tau\langle H-\epsilon\rangle=-\iint_{\Sigma}\omega,$ (30)

where $\Sigma$ is any Seifert surface for $\rho^{\sigma}_{t}$ that is
homologous to a Seifert surface for $\rho^{\sigma}_{t}$ in $\Omega(\sigma)$. A
Seifert surface for $\rho^{\sigma}_{t}$ is an oriented surface $\Sigma$ in
$\mathcal{P}$ whose boundary is parametrized by $\rho^{\sigma}_{t}$ as
illustrated in the right part of Fig. 4. For example, the ruled surface
obtained by connecting $\sigma$ and each $\rho_{t}$ with the shortest
arclength-parametrized geodesic is such a Seifert surface. In Appendix C we
provide formulas for the geodesics $\gamma_{t}^{1}$ and $\gamma_{t}^{2}$ and
explicitly construct a ruled Seifert surface for $\rho^{\sigma}_{t}$.

The pull-back of $\omega$ to $\SS$ by the Hopf projection is exact and equals
the negative of the Berry curvature, $\eta^{*}\omega=-\mathrm{d}\mathcal{A}$
[17]. We construct a lift $|\psi_{t}^{\sigma}\rangle$ of $\rho_{t}^{\sigma}$
to $\SS$ as follows. Let $|\phi\rangle$ be any vector in the fiber over
$\sigma$, let $|\psi_{t}\rangle$ be the lift of $\rho_{t}$ which is in phase
with $|\phi\rangle$, and for $0\leq t\leq r$ define $|\phi^{1}_{t}\rangle$ and
$|\phi^{2}_{t}\rangle$ as

$\displaystyle|\phi^{1}_{t}\rangle=\cos t|\phi\rangle+\frac{\sin t}{\sin
r}\big{(}|\psi_{0}\rangle-\cos r|\phi\rangle\big{)},$ (31)
$\displaystyle|\phi^{2}_{t}\rangle=\cos(r-t)|\phi\rangle+\frac{\sin(r-t)}{\sin
r}\big{(}|\psi_{\tau}\rangle-\cos r|\phi\rangle\big{)}.$ (32)

The curves $|\phi^{1}_{t}\rangle$ and $|\phi^{2}_{t}\rangle$ are lifts of
$\gamma_{t}^{1}$ and $\gamma_{t}^{2}$ connecting $|\phi\rangle$ to
$|\psi_{0}\rangle$ and $|\psi_{\tau}\rangle$ to $|\phi\rangle$, respectively;
see Appendix C. We define $|\psi_{t}^{\sigma}\rangle$ as the concatenation
$|\phi^{1}_{t}\rangle\ast|\psi_{t}\rangle\ast|\phi^{2}_{t}\rangle$. The curve
$|\psi_{t}^{\sigma}\rangle$ is a lift of $\rho_{t}^{\sigma}$ that is in phase
with $|\phi\rangle$.

Suppose that $\Sigma$ is homologous to a Seifert surface $\Sigma^{\prime}$ in
$\Omega(\sigma)$. The Hopf bundle restricts to a diffeomorphism from the
hypersurface $Y_{|\phi\rangle}$ of vectors in $\SS$ that are in phase with
$|\phi\rangle$ onto $\Omega(\sigma)$ [17]. Lift $\Sigma^{\prime}$ to a Seifert
surface for $|\psi_{t}^{\sigma}\rangle$ in $Y_{|\phi\rangle}$. By Stoke’s
theorem,

$\iint_{\Sigma}\omega=\iint_{\Sigma^{\prime}}\omega=-\iint_{\Sigma^{\prime}}\mathrm{d}\mathcal{A}=-\oint_{|\psi_{t}^{\sigma}\rangle}\mathcal{A}.$
(33)

The first identity is a consequence of $\Sigma-\Sigma^{\prime}$ being a
homological boundary and that $\omega$ is closed; cf. Remark 3 below. The
second identity results from $\eta$ being a diffeomorphism from
$Y_{|\phi\rangle}$ onto $\Omega(\sigma)$ and $-\mathrm{d}\mathcal{A}$ being
the pull-back of $\omega$. The third identity follows from
$|\psi_{t}^{\sigma}\rangle$ parametrizing the boundary of the lift of
$\Sigma^{\prime}$.

A direct calculation shows that the Berry connection annihilates the velocity
fields of $|\phi^{1}_{t}\rangle$ and $|\phi^{2}_{t}\rangle$:

$\mathcal{A}|\dot{\phi}^{j}_{t}\rangle=i\langle\phi^{j}_{t}|\dot{\phi}^{j}_{t}\rangle=0,\quad
j=1,2.$ (34)

Thus we have that

$\oint_{|\psi_{t}^{\sigma}\rangle}\mathcal{A}=\int_{|\psi_{t}\rangle}\mathcal{A}.$
(35)

Equations (25), (33), and (35) yield the third main result (30). Note that the
calculations rely on $\rho$ not being fully distinguishable from $\sigma$, cf.
Remark 1.

######  Remark 3.

The homology class of the projectivization of any two-dimensional subspace of
$\mathcal{H}$, that is, a Bloch sphere, generates the second singular homology
group of $\mathcal{P}$ with integer coefficients [18]. We choose such a Bloch
sphere that we orient so that its symplectic area is positive. The area is
$2\pi$, and $\omega/2\pi$ is thus of integral class.

The difference between two Seifert surfaces for $\rho_{t}^{\sigma}$ is a
$2$-cycle. The homology class of such a difference is an integer multiple of
the homology class for the Bloch sphere. It follows that the difference of the
symplectic areas of two Seifert surfaces for $\rho_{t}^{\sigma}$ is an integer
multiple of the symplectic area of the Bloch sphere. Hence, for an arbitrary
Seifert surface $\Sigma$ for $\rho_{t}^{\sigma}$,

$\tau\langle H-\epsilon\rangle=-\iint_{\Sigma}\omega\mod 2\pi.$ (36)

We can equivalently express this as $\tau\langle H-\epsilon\rangle$ being
equal to the Aharonov-Anandan geometric phase [13] of the $\sigma$ closure of
$\rho_{t}$ modulo $2\pi$. This connection to the Aharonov-Anandan phase is
utilized in Ref. [19] where QSLs for cyclic systems are derived.

###  III.3 $\alpha(\delta)$ as an extremal dynamical phase

Let $\sigma$ be any state and $S(\sigma,r)$ be the geodesic sphere of radius
$r$ centered at $\sigma$. The geodesic sphere consists of all states at
distance $r$ from $\sigma$. Fix a fidelity $\delta$ and choose $r$ such that
$\arccos\sqrt{\delta}\leq 2r\leq\arccos(-\sqrt{\delta})$. The geodesic sphere
$S(\sigma,r)$ is then contained in $\Omega(\sigma)$ and includes states
between which the fidelity is $\delta$.

Write $\Gamma(\sigma,r,\delta)$ for the space of smooth curves in
$S(\sigma,r)$ that extend between two states with fidelity $\delta$. Define
$\mathcal{J}^{\sigma}_{r,\delta}$ to be the functional that assigns the
dynamical phase to each $\rho_{t}$ in $\Gamma(\sigma,r,\delta)$ in the gauge
specified by $\sigma$,

$\mathcal{J}^{\sigma}_{r,\delta}[\rho_{t}]=\int_{\rho_{t}}A^{\sigma}.$ (37)

In Appendix D we show that $\rho_{t}$ is an extremal for
$\mathcal{J}^{\sigma}_{r,\delta}$ if and only if for every $|\phi\rangle$ in
the fiber over $\sigma$, the lift $|\psi_{t}\rangle$ of $\rho_{t}$ which is in
phase with $|\phi\rangle$ splits orthogonally as

$|\psi_{t}\rangle=\cos r\,|\phi\rangle+\sin r\,e^{i\Lambda_{t}}|w\rangle$ (38)

for some function $\Lambda_{t}$ that vanishes for $t=0$. The corresponding
extreme value is

$\mathcal{J}^{\sigma}_{r,\delta}[\rho_{t}]=\sin^{2}r\,\Lambda_{\tau}.$ (39)

The constraint on $\Lambda_{t}$ arising from the assumption that $\rho_{t}$
extends between two states with fidelity $\delta$ reads

$\cos\Lambda_{\tau}=1-\frac{2(1-\delta)}{\sin^{2}2r}.$ (40)

Thus,

$\mathcal{J}^{\sigma}_{r,\delta}[\rho_{t}]=\pm\sin^{2}r\,\arccos\bigg{(}1-\frac{2(1-\delta)}{\sin^{2}2r}\bigg{)}\mod
2\pi.$ (41)

According to (20), $\alpha(\delta)$ is equal to the smallest positive extreme
value of $\mathcal{J}^{\sigma}_{r,\delta}$ minimized over the interval
$\frac{1}{2}\arccos\sqrt{\delta}\leq r\leq\frac{1}{2}\arccos(-\sqrt{\delta})$.
This observation is the fourth main result of the paper.

######  Remark 4.

If $\rho_{t}$ is an extremal curve for $\mathcal{J}^{\sigma}_{r,\delta}$, then
so is $\rho_{\tau-t}$, and
$\mathcal{J}^{\sigma}_{r,\delta}[\rho_{\tau-t}]=-\mathcal{J}^{\sigma}_{r,\delta}[\rho_{t}]$.
Hence the $\pm$ in (41). This observation is related to the dual QSL derived
and discussed in Sec. IV.2.

##  IV Related quantum speed limits

The maximum of the Mandelstam-Tamm and the extended Margolus-Levitin QSLs is a
new QSL. Interestingly, the maximum QSL behaves differently for fully and not
fully distinguishable initial and final states. We describe this difference in
Sec. IV.1. In Sec. IV.2 we derive a QSL that extends the dual Margolus-Levitin
QSL studied in Ref. [20], and in Sec. IV.3 we provide three QSLs that are less
sharp but easier to calculate than the extended Margolus-Levitin QSL. These
three QSLs are not new but can be found in the cited papers.

###  IV.1 The maximum quantum speed limit

According to Anandan and Aharonov [7], the QSL of Mandelstam and Tamm (2) is
saturated if and only if the evolving state follows a shortest Fubini-Study
geodesic. Furthermore, by Brody [21], such a state is an effective qubit that
follows the equator of the Bloch sphere associated with the two eigenvectors
that have nonzero fidelity with the initial state; cf. Sec. II.1.101010None of
the eigenvectors need to be associated with the smallest eigenvalue of the
Hamiltonian. Levitin and Toffoli [16] showed that if we require the initial
and final states to be fully distinguishable, the same holds for the Margolus-
Levitin QSL (3). However, in that case, one of the eigenvectors must belong to
the smallest eigenvalue (or $\epsilon_{0}$ be replaced by the smallest
occupied energy, cf. Remark 1). Thus, if (3) is saturated, so is (1), and the
reverse holds if the support of the initial state is not perpendicular to the
eigenspace belonging to the smallest eigenvalue. These observations led
Levitin and Toffoli to conclude that the maximum of $\tau_{\textsc{mt}}(0)$
and $\tau_{\textsc{ml}}(0)$ is a tight lower bound for the evolution time only
reachable for states such that $\langle H-\epsilon_{0}\rangle=\Delta H$. We
can draw the same conclusion from the discussion in Sec. II.

Interestingly, the situation is almost the reverse for a nonzero fidelity
$\delta$ between initial and final states; the Mandelstam–Tamm and the
extended Margolus–Levitin QSLs can never saturate simultaneously: If the state
follows a geodesic, then (4) is not saturated by Sec. II.1, and if (4) is an
identity, the state does not follow a geodesic. However, the maximum QSL
$\max\\{\tau_{\textsc{mt}}(\delta),\tau_{\textsc{ml}}(\delta)\\}$ can be
reached: If the state follows a geodesic,
$\tau=\tau_{\textsc{mt}}(\delta)>\tau_{\textsc{ml}}(\delta)$, and if condition
(16) is satisfied,
$\tau=\tau_{\textsc{ml}}(\delta)>\tau_{\textsc{mt}}(\delta)$. In the latter
case, $\langle H-\epsilon_{0}\rangle$ must differ from $\Delta H$ since
$\alpha(\delta)$ is strictly less than $\arccos\sqrt{\delta}$ for
$0<\delta<1$; see Fig. 5.

Figure 5: The graphs of $\alpha(\delta)$ and $\arccos\sqrt{\delta}$. The two
functions agree only for $\delta=0$ and $\delta=1$. For all other fidelities,
$\arccos\sqrt{\delta}$ is strictly greater than $\alpha(\delta)$.

######  Remark 5.

The state does not follow a geodesic in a system that saturates the extended
Margolus-Levitin QSL for a nonzero fidelity. However, the state follows a sub-
Riemannian geodesic in a geodesic sphere centered at a ground state. See
Appendix C for details.

######  Remark 6.

In quite a few papers it is claimed that $\arccos\sqrt{\delta}/\langle
H-\epsilon_{0}\rangle$ is a QSL for isolated systems. Figure 5 and the fact
that the extended Margolus-Levitin QSL is tight show that this is not true in
general.

###  IV.2 The dual extended Margolus-Levitin quantum speed limit

Let $\rho_{a}$ and $\rho_{b}$ be states with fidelity $\delta$, and suppose
that $H$ evolves $\rho_{a}$ to $\rho_{b}$ in time $\tau$. Then $-H$ evolves
$\rho_{b}$ to $\rho_{a}$ in time $\tau$. The smallest eigenvalue of $-H$ is
$-\epsilon_{\mathrm{max}}$, where $\epsilon_{\mathrm{max}}$ is the largest
eigenvalue of $H$, and according to the extended Margolus-Levitin QSL,

$\tau\geq\tau_{\textsc{ml}}^{*}(\delta),\quad\tau_{\textsc{ml}}^{*}(\delta)=\frac{\alpha(\delta)}{\langle\epsilon_{\mathrm{max}}-H\rangle}.$
(42)

This estimate generalizes the main result in Ref. [20] to an arbitrary
fidelity between initial and final states. We adhere to the terminology in
Ref. [20] and call $\tau_{\textsc{ml}}^{*}(\delta)$ the dual extended
Margolus-Levitin QSL. Note that for some states,
$\tau_{\textsc{ml}}^{*}(\delta)$ is greater than $\tau_{\textsc{ml}}(\delta)$
and $\tau_{\textsc{mt}}(\delta)$.

Since the estimate in (42) is a consequence of applying the extended Margolus-
Levitin QSL to evolution generated by $-H$, it follows from Appendix B that
(42) can be saturated in all dimensions and that, when so, the state is an
effective qubit for $-H$ whose support is not orthogonal to the eigenspace
corresponding to $-\epsilon_{\mathrm{max}}$. But then the state is also an
effective qubit for $H$ whose support is not orthogonal to the eigenspace
corresponding to $\epsilon_{\mathrm{max}}$. We call such an effective qubit
_partly maximally excited_. To summarize, if
$\tau\langle\epsilon_{\mathrm{max}}-H\rangle$ assumes its smallest possible
value when evaluated for all Hamiltonians $H$, states $\rho$, and $\tau\geq 0$
such that $H$ transforms $\rho$ into a state with fidelity $\delta$ in time
$\tau$, then $\rho$ is a partly maximally excited effective qubit for $H$ and
$\tau\langle\epsilon_{\mathrm{max}}-H\rangle=\alpha(\delta)$.

######  Remark 7.

If we restrict the set of states as in Remark 1, $\epsilon_{\mathrm{max}}$ can
be replaced by the largest occupied energy.

The discussion in Sec. III, where we deliberately did not specify the
eigenvalue $\epsilon$, tells us that

$\tau\langle\epsilon_{\mathrm{max}}-H\rangle=-\int_{\rho_{t}}A^{\sigma}=\iint_{\Sigma}\omega,$
(43)

where $\sigma$ is an eigenstate of $H$ with eigenvalue
$\epsilon_{\mathrm{max}}$ that has nonzero fidelity with the initial state.
The surface $\Sigma$ is an arbitrary Seifert surface for the $\sigma$ closure
$\rho_{t}^{\sigma}$ in $\Omega(\sigma)$. In Fig. 6 we have illustrated such a
Seifert surface in the Bloch sphere for the same evolution as in Fig. 2.

Figure 6: A Seifert surface in the Bloch sphere for the $|1\rangle\langle 1|$
closure of an evolving qubit. The symplectic area of the surface is positive
and equal to $\tau\langle\epsilon_{\mathrm{max}}-H\rangle$. Note that the
surface is, in a sense, dual to the Seifert surface in Fig. 2. If the dual
extended Margolus-Levitin QSL is saturated, the surface assumes its smallest
possible symplectic area. For not fully distinguishable initial and final
states, the evolving state then has a strictly positive $z$ coordinate. This
is in contrast to the case when the extended Margolus-Levitin QSL is
saturated, in which case the $z$ coordinate is strictly negative. For fully
distinguishable initial and final states, both QSLs can be saturated
simultaneously. The magnitudes of the symplectic areas of the corresponding
Seifert surfaces are then equal.

In this case, a concatenation of the evolution curve with the shortest
geodesic connecting the initial and final states to the excited state
$|1\rangle\langle 1|$ parametrizes the boundary of the Seifert surface.
Furthermore, the orientation of the boundary is such that the Seifert surface
has a positive symplectic area, which is consistent with equation (43). The
$z$ coordinate of a qubit state that saturates the dual extended Margolus-
Levitin QSL satisfies the equation

$\frac{1-z}{2}\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}=\alpha(\delta).$
(44)

For $\delta\neq 0$, the $z$ coordinate of a saturating evolution is strictly
positive in contrast to an evolution that saturates the “original” extended
Margolus-Levitin QSL, in which case the $z$ coordinate is strictly negative,
as in Fig. 2. This observation lets us conclude that for not fully
distinguishable initial and final states, the extended Margolus-Levitin QSL
and its dual can never saturate simultaneously. Because if that were the case,
the state would be an effective qubit with support in the span of a ground
state and a highest energy state. In the projectivization of the span of these
eigenstates, the $z$ coordinate of the Bloch vector of the system’s state
would be strictly positive and strictly negative, which is contradictory. For
$\delta=0$, however, the QSLs can saturate simultaneously. The evolving state
then follows a geodesic and

$\langle\epsilon_{\mathrm{max}}-H\rangle=\Delta H=\langle
H-\epsilon_{0}\rangle.$ (45)

###  IV.3 Approximations of the extended Margolus-Levitin quantum speed limit

Consider a quantum system in a state $\rho$ with Hamiltonian $H$. Suppose the
system evolves into a state with fidelity $\delta$ relative to $\rho$ in time
$\tau$. Then

$\tau\geq\tau_{1}(\delta),\quad\tau_{1}(\delta)=\frac{1-\sqrt{\delta}}{\beta\langle
H-\epsilon_{0}\rangle},$ (46)

where the requirement that $y=1-\beta x$ is a tangent line to $\cos x$
specifies $\beta$ ($\beta\approx 0.724$). Also,

$\displaystyle\tau\geq\tau_{2}(\delta),\quad\tau_{2}(\delta)=\frac{4\arccos^{2}\\!\sqrt{\delta}}{\beta\pi^{2}\langle
H-\epsilon_{0}\rangle},$ (47)
$\displaystyle\tau\geq\tau_{3}(\delta),\quad\tau_{3}(\delta)=\frac{2\arccos^{2}\\!\sqrt{\delta}}{\pi\langle
H-\epsilon_{0}\rangle}.$ (48)

Derivations of (46) and (47) are found in Refs. [22, 23], and (48) is
mentioned in Ref. [10]. Where this latter QSL comes from is still unclear to
the authors.

Figure 7 displays the graphs of the extended Margolus-Levitin QSL
$\tau_{\textsc{ml}}(\delta)$ as well as the QSLs $\tau_{1}(\delta)$,
$\tau_{2}(\delta)$, and $\tau_{3}(\delta)$ multiplied by $\langle
H-\epsilon_{0}\rangle$.

Figure 7: The graphs of $\tau_{\textsc{ml}}(\delta)$, $\tau_{1}(\delta)$,
$\tau_{2}(\delta)$, and $\tau_{3}(\delta)$ multiplied by the normalized
expected energy $\langle H-\epsilon_{0}\rangle$. The graph belonging to
$\tau_{\textsc{ml}}(\delta)$ lies above the other graphs. Thus
$\tau_{\textsc{ml}}(\delta)$ is the strongest bound. The weakest bound is
$\tau_{2}(\delta)$. Since the graphs of $\tau_{1}(\delta)$ and
$\tau_{3}(\delta)$ intersect, their mutual strength depends on the fidelity.

Apparently, $\tau_{1}(\delta)$, $\tau_{2}(\delta)$, and $\tau_{3}(\delta)$ are
weaker than $\tau_{\textsc{ml}}(\delta)$, and $\tau_{2}(\delta)$ is weaker
than $\tau_{1}(\delta)$ and $\tau_{3}(\delta)$. However, which of
$\tau_{1}(\delta)$ and $\tau_{3}(\delta)$ is the stronger QSL depends on the
fidelity $\delta$, with $\tau_{1}(\delta)$ being greater than
$\tau_{3}(\delta)$ for large values of $\delta$ and $\tau_{3}(\delta)$ being
greater than $\tau_{1}(\delta)$ for small values of $\delta$.

##  V Summary and outlook

Giovannetti _et al._ [10] showed, partly numerically, that if an isolated
system evolves between two states with fidelity $\delta$, then the evolution
time is bounded from below by the extended Margolus-Levitin QSL (4).
Giovannetti _et al._ also showed that this QSL is tight in all dimensions.

In this paper, we have derived the extended Margolus-Levitin QSL analytically
and characterized the systems for which this QSL is saturated. Furthermore, we
have interpreted the extended Margolus-Levitin QSL geometrically as an
extremal dynamical phase in a gauge specified by the system’s ground state. We
have also shown that the maximum of the Mandelstam-Tamm and the extended
Margolus-Levitin QSLs is a tight QSL that behaves differently depending on
whether or not the initial state and the final state are fully
distinguishable. In addition, we have derived a tight dual version of the
extended Margolus-Levitin QSL using a straightforward time-reversal argument.
The dual QSL has similar properties as the extended Margolus-Levitin QSL and
saturates under similar circumstances but involves the largest rather than the
smallest occupied energy. We showed that the two QSLs can only be saturated
simultaneously if the start and end states are fully distinguishable. We
concluded the paper by reproducing three QSLs related to, but slightly weaker
than, the extended Margolus-Levitin QSL.

A recent paper on evolution time estimates for closed systems [14] suggests
that the Margolus-Levitin QSL does not straightforwardly extend to systems
whose dynamics are governed by time-dependent Hamiltonians, at least not
without limitations on the width of the energy spectrum.

The geometric analysis performed here shows that extended Margolus-Levitin QSL
is closely related to the Aharonov-Anandan geometric phase [13]. This
observation is further elaborated in Ref. [19] where QSLs for cyclically
evolving systems are derived.

Mixed state QSLs resembling the Margolus-Levitin QSL exist [24, 23, 22], and
Giovannetti _et al._ [10] showed that also the extended Margolus-Levitin QSL
can be generalized to a QSL for systems in mixed states, with $\delta$ being
the Uhlmann fidelity between the initial and final states [25]. Whether this
generalization has a symplectic-geometric interpretation similar to the one
presented here is an open question. So is the question whether the generalized
extended Margolus-Levitin QSL connects to a geometric phase for mixed states
[23, 28, 29, 27, 26]. The authors intend to investigate these questions in a
forthcoming paper.

## Acknowledgments

We thank Dan Allan for valuable discussions.

##  Appendix A The extended Margolus-Levitin quantum speed limit for qubit
systems

Let $H$ be the Hamiltonian in (6) and $\rho$ be a qubit state whose Bloch
vector $\mathbf{r}=(x,y,z)$ is defined by (7)–(9). Then

$\langle 0|\rho|0\rangle=\frac{1}{2}(1-z),\quad\langle
1|\rho|1\rangle=\frac{1}{2}(1+z),\quad\langle
0|\rho|1\rangle=\frac{1}{2}(x+iy),\quad\langle
1|\rho|0\rangle=\frac{1}{2}(x-iy).$ (49)

Equation (10) follows from

$2\langle H\rangle=2\langle 0|H\rho|0\rangle+2\langle
1|H\rho|1\rangle=2\epsilon_{0}\langle 0|\rho|0\rangle+2\epsilon_{1}\langle
1|\rho|1\rangle=\epsilon_{0}(1-z)+\epsilon_{1}(1+z)=\epsilon_{1}+\epsilon_{0}+(\epsilon_{1}-\epsilon_{0})z.$
(50)

To prove equation (11) consider two qubit states $\rho_{a}$ and $\rho_{b}$
with Bloch vectors $\mathbf{r}_{a}=(x_{a},y_{a},z_{a})$ and
$\mathbf{r}_{b}=(x_{b},y_{b},z_{b})$, respectively. Let $\delta$ be the
fidelity between $\rho_{a}$ and $\rho_{b}$. Then

$\begin{split}\delta&=\langle 0|\rho_{a}\rho_{b}|0\rangle+\langle
1|\rho_{a}\rho_{b}|1\rangle\\\ &=\langle 0|\rho_{a}|0\rangle\langle
0|\rho_{b}|0\rangle+\langle 0|\rho_{a}|1\rangle\langle
1|\rho_{b}|0\rangle+\langle 1|\rho_{a}|0\rangle\langle
0|\rho_{b}|1\rangle+\langle 1|\rho_{a}|1\rangle\langle 1|\rho_{b}|1\rangle\\\
&=\frac{1}{4}\big{(}(1-z_{a})(1-z_{b})+(x_{a}+iy_{a})(x_{b}-iy_{b})+(x_{a}-iy_{a})(x_{b}+iy_{b})+(1+z_{a})(1+z_{b})\big{)}\\\
&=\frac{1}{2}(1+x_{a}x_{b}+y_{a}y_{b}+z_{a}z_{b})\\\
&=\frac{1}{2}(1+\mathbf{r}_{a}\cdot\mathbf{r}_{b}).\end{split}$ (51)

This proves equation (11).

The dynamics caused by $H$ can be described as a rigid rotation of the Bloch
sphere about the $z$ axis. To prove (12) assume $z_{a}=z_{b}=z$ and that
$\mathbf{r}_{a}$ rotates to $\mathbf{r}_{b}$ during the evolution. Since the
$z$ coordinate remains fixed, $\mathbf{r}_{a}$ follows a path of the same
length as the orthogonal projection $\bar{\mathbf{r}}_{a}$ of $\mathbf{r}_{a}$
on the $xy$ plane. The projection $\bar{\mathbf{r}}_{a}$ rotates to the
projection $\bar{\mathbf{r}}_{b}$ of $\mathbf{r}_{b}$ along a circular arc in
the $xy$ plane. The radius of the corresponding circle sector is

$\sqrt{\bar{\mathbf{r}}_{a}\cdot\bar{\mathbf{r}}_{a}}=\sqrt{\mathbf{r}_{a}\cdot\mathbf{r}_{a}-z^{2}}=\sqrt{1-z^{2}}.$
(52)

Furthermore, according to (11), the circle sector has apex angle

$\arccos\bigg{(}\frac{\bar{\mathbf{r}}_{a}\cdot\bar{\mathbf{r}}_{b}}{\bar{\mathbf{r}}_{a}\cdot\bar{\mathbf{r}}_{a}}\bigg{)}=\arccos\bigg{(}\frac{\mathbf{r}_{a}\cdot\mathbf{r}_{b}-z^{2}}{\mathbf{r}_{a}\cdot\mathbf{r}_{a}-z^{2}}\bigg{)}=\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}.$
(53)

Thus, the length of the circular arc is

$\sqrt{1-z^{2}}\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}.$ (54)

Finally, to prove (13) we first determine the speed of the rotating Bloch
vector. Write $x_{t}$, $y_{t}$, and $z_{t}=z$ for the coordinates of the
rotating Bloch vector. We have

$\dot{x}_{t}+i\dot{y}_{t}=2\langle 0|\dot{\rho}_{t}|1\rangle=-2i\langle
0|[H,\rho_{t}]|1\rangle=-i(\epsilon_{0}-\epsilon_{1})(x_{t}+iy_{t}).$ (55)

Hence $\dot{x}_{t}=(\epsilon_{0}-\epsilon_{1})y_{t}$ and
$\dot{y}_{t}=(\epsilon_{1}-\epsilon_{0})x_{t}$. The rotating Bloch vector thus
has the speed

$\sqrt{\dot{x}_{t}^{2}+\dot{y}_{t}^{2}}=(\epsilon_{1}-\epsilon_{0})\sqrt{x_{t}^{2}+y_{t}^{2}}=(\epsilon_{1}-\epsilon_{0})\sqrt{1-z^{2}}.$
(56)

This equation also says that $H$ causes the Bloch sphere to rotate with the
angular speed $\epsilon_{1}-\epsilon_{0}$ about the $z$ axis. The evolution
time equals the distance traveled by the Bloch vector divided by the speed of
the Bloch vector,

$\tau=\frac{\sqrt{1-z^{2}}\arccos\big{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\big{)}}{(\epsilon_{1}-\epsilon_{0})\sqrt{1-z^{2}}}=\frac{1}{\epsilon_{1}-\epsilon_{0}}\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}.$
(57)

This is the statement in equation (13).

##  Appendix B Characterization of time-optimal systems

Consider a quantum system of dimension $n\geq 2$. Let $0\leq\delta<1$ and set

$\alpha(n,\delta)=\inf_{(H,\rho,\tau)}\tau\langle H-\epsilon_{0}\rangle,$ (58)

where the infimum is over all triples consisting of a Hamiltonian $H$, a state
$\rho$, and a time $\tau\geq 0$ such that

$\operatorname{tr}(\rho e^{-i\tau H}\rho e^{i\tau H})=\delta.$ (59)

As before, $\epsilon_{0}$ is the smallest eigenvalue of $H$. In this appendix
we show that some triple $(H,\rho,\tau)$ realizes the infimum
$\alpha(n,\delta)$ and that for any such triple, $\rho$ is a partly grounded
effective qubit for $H$. Furthermore, we show that $\alpha(n,\delta)$ is the
same for all $n$ and thus equals $\alpha(\delta)$ defined in equation (5).
Note that condition (59) says that if $H$ controls the dynamics of the system,
then $\rho$ will evolve to a state with fidelity $\delta$ in time $\tau$.

###  B.1 Reduction to admissible pairs

We begin by establishing that

$\alpha(n,\delta)=\inf_{(H,\rho)}\langle H\rangle,$ (60)

where the infimum is over all pairs consisting of a Hamiltonian $H$ with
spectrum in $[0,2\pi]$ and a state $\rho$ such that

$\operatorname{tr}(\rho e^{-iH}\rho e^{iH})=\delta.$ (61)

We call such a pair _admissible_.

Take an arbitrary triple $(H,\rho,\tau)$ that satisfies condition (59). Let
$\rho^{\prime}=e^{-i\tau H}\rho e^{i\tau H}$ and set $H_{1}=H-\epsilon_{0}$.
The Hamiltonian $H_{1}$ is positive, has the minimum eigenvalue $0$, and gives
rise to dynamics identical to that caused by $H$. Hence, $(H_{1},\rho,\tau)$
satisfies condition (59). Since $H_{1}$ and $H$ have the same eigenspaces and,
in particular, they have the same eigenspace for their smallest eigenvalues,
$\rho$ is a partly grounded effective qubit for $H_{1}$ if and only if $\rho$
is a partly grounded effective qubit for $H$.

Next, we normalize the evolution time by setting $H_{2}=\tau H_{1}$. The
Hamiltonian $H_{2}$ is positive, has the minimum eigenvalue $0$, and evolves
$\rho$ to $\rho^{\prime}$ along the same path as $H_{1}$ but in time $1$. Thus
$(H_{2},\rho,1)$ satisfies condition (59). Since $H_{2}$ and $H_{1}$ have the
same eigenspaces, especially for the eigenvalue $0$, $\rho$ is a partly
grounded effective qubit for $H_{2}$ if and only if $\rho$ is a partly
grounded effective qubit for $H_{1}$.

Lastly, we define $H_{3}$ as the Hamiltonian obtained from $H_{2}$ by
replacing each eigenvalue $\epsilon$ of $H_{2}$ with the $\epsilon^{\prime}$
in $[0,2\pi]$ satisfying $\epsilon^{\prime}=\epsilon\;\mathrm{mod}\,2\pi$. The
Hamiltonian $H_{3}$ is positive, has the smallest eigenvalue $0$, and evolves
$\rho$ to $\rho^{\prime}$ in time $1$. The expectation value of $H_{3}$ is not
greater than that of $H_{2}$. We thus have that

$\langle H_{3}\rangle\leq\langle H_{2}\rangle=\tau\langle
H_{1}\rangle=\tau\langle H-\epsilon_{0}\rangle.$ (62)

This shows that $\inf_{(H,\rho)}\langle H\rangle\leq\alpha(n,\delta)$, where
the infimum is over all admissible pairs. The reverse inequality is
automatically satisfied. Since the space of admissible pairs is compact,
$\alpha(n,\delta)$ is attained for an admissible pair.

Now, suppose that $\rho$ is a partly grounded effective qubit for $H$ and thus
for $H_{2}$. Then $\rho$ is also a partly grounded effective qubit for
$H_{3}$. For eigenspaces of $H_{2}$ can merge when forming $H_{3}$, but not in
such a way that the support of $\rho$ is covered by the eigenspace of $H_{3}$
that belongs to the eigenvalue $0$. The reverse, however, is not necessarily
true. If $\rho$ is a partly grounded effective qubit for $H_{3}$, $\rho$ need
not be a partly grounded effective qubit for $H_{2}$. But in that case, the
inequality in (62) is strict, and the original triple $(H,\rho,\tau)$ does not
saturate (58).

######  Remark 8.

One can choose the state arbitrarily and further restrict the infimum to
Hamiltonians that form an admissible pair with the selected state. For if
$(H,\rho,\tau)$ satisfies (59), $\rho^{\prime}$ is an arbitrary state, $U$ is
a unitary such that $U\rho U^{\dagger}=\rho^{\prime}$, and
$H^{\prime}=UHU^{\dagger}$, then also $(H^{\prime},\rho^{\prime},\tau)$
satisfies (59). Furthermore, $H^{\prime}$ and $H$ have the same spectrum, the
expectation value of $H^{\prime}$ when the state is $\rho^{\prime}$ is the
same as the expectation value of $H$ when the state is $\rho$, and
$\rho^{\prime}$ is a partly grounded effective qubit for $H^{\prime}$ if and
only if $\rho$ is a partly grounded effective qubit for $H$.

###  B.2 Not fully distinguishable initial and final states

Assume that $\delta>0$. (For the sake of completeness, we treat the case
$\delta=0$ in Appendix B.3, although this case has been dealt with earlier
[15, 16].) Let $(\mathbf{p},\bm{\epsilon})$ denote a point
$(p_{1},\dots,p_{n},\epsilon_{1},\dots,\epsilon_{n})$ in the compact
rectangular block $[0,1]^{n}\times[0,2\pi]^{n}$. Define the functions $f$,
$g$, and $h$ on $[0,1]^{n}\times[0,2\pi]^{n}$ as

$\displaystyle f(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{n}p_{j}\epsilon_{j},$
(63) $\displaystyle
g(\mathbf{p},\bm{\epsilon})=\Big{(}\sum_{j=1}^{n}p_{j}\cos\epsilon_{j}\Big{)}^{2}+\Big{(}\sum_{j=1}^{n}p_{j}\sin\epsilon_{j}\Big{)}^{2},$
(64) $\displaystyle h(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{n}p_{j}.$ (65)

Furthermore, let $M$ be the subset of $[0,1]^{n}\times[0,2\pi]^{n}$ defined by
the constraints $g(\mathbf{p},\bm{\epsilon})=\delta$ and
$h(\mathbf{p},\bm{\epsilon})=1$. Since $M$ is compact and $f$ is continuous,
the restriction of $f$ to $M$, denoted $f|_{M}$, assumes a minimum value. This
minimum value is $\alpha(n,\delta)$. To see this suppose that $(H,\rho)$ is an
admissible pair such that $\langle H\rangle=\alpha(n,\delta)$. Let
$\epsilon_{1},\epsilon_{2},\dots,\epsilon_{n}$ be the eigenvalues of $H$, with
corresponding eigenvectors $|1\rangle,\dots,|n\rangle$, and set $p_{j}=\langle
j|\rho|j\rangle$. The point
$(\mathbf{p},\bm{\epsilon})=(p_{1},\dots,p_{n},\epsilon_{1},\dots,\epsilon_{n})$
belongs to $M$, since $g(\mathbf{p},\bm{\epsilon})=\operatorname{tr}(\rho
e^{-iH}\rho e^{iH})=\delta$ and
$h(\mathbf{p},\bm{\epsilon})=\operatorname{tr}(\rho)=1$, and
$f(\mathbf{p},\bm{\epsilon})=\langle H\rangle=\alpha(n,\delta)$. Thus,
$f|_{M}$ assumes the value $\alpha(n,\delta)$. To see that $\alpha(n,\delta)$
is the smallest value of $f|_{M}$ let $(\mathbf{p},\bm{\epsilon})$ is an
arbitrary point in $M$. Then, let $|1\rangle,\dots,|n\rangle$ be an arbitrary
orthonormal basis for the Hilbert space of the system and define
$H=\sum_{j}\epsilon_{j}|j\rangle\langle j|$ and
$\rho=\sum_{i,j}\sqrt{p_{i}p_{j}}|i\rangle\langle j|$. The pair $(H,\rho)$ is
admissible and $f(\mathbf{p},\bm{\epsilon})=\langle H\rangle$. Hence,
$f(\mathbf{p},\bm{\epsilon})\geq\alpha(n,\delta)$.

Let $(\mathbf{p},\bm{\epsilon})$ be a point in $M$ at which $f|_{M}$ assumes
its minimum value and let $J$ be the set of indices $j$ for which $p_{j}\neq
0$. The minimum point $(\mathbf{p},\bm{\epsilon})$ has the following
properties:

  * (i)

There are indices $j_{1}$ and $j_{2}$ in $J$ such that $\epsilon_{j_{1}}\neq
0$ and $\epsilon_{j_{2}}=0$.

  * (ii)

For all indices $j_{1}$ and $j_{2}$ in $J$ such that $\epsilon_{j_{1}}\neq 0$
and $\epsilon_{j_{2}}\neq 0$ we have that $\epsilon_{j_{1}}=\epsilon_{j_{2}}$.

Before showing that $(\mathbf{p},\bm{\epsilon})$ has these properties let us
discuss their implications for the extended Margolus-Levitin QSL.

Assume that $(H,\rho)$ is an admissible pair such that $\langle
H\rangle=\alpha(n,\delta)$. Let $\epsilon_{1},\epsilon_{2},\dots,\epsilon_{n}$
be the spectrum of $H$ and let $p_{j}$ be the probability of obtaining
$\epsilon_{j}$ when measuring $H$ in the state $\rho$ divided by the
degeneracy of $\epsilon_{j}$. The point
$(\mathbf{p},\bm{\epsilon})=(p_{1},\dots,p_{n},\epsilon_{1},\dots,\epsilon_{n})$
belongs to $M$ since $g(\mathbf{p},\bm{\epsilon})=\delta$ according to the
normed evolution time condition (61) and $h(\mathbf{p},\bm{\epsilon})=1$
according to the law of total probability. Furthermore, $f|_{M}$ assumes its
minimum value at $(\mathbf{p},\bm{\epsilon})$ since
$f(\mathbf{p},\bm{\epsilon})=\langle H\rangle$. Property (i) says that $\rho$
is partly grounded; that is, its support is not orthogonal to the eigenspace
belonging to the smallest eigenvalue of $H$, which in this case is $0$.
Property (ii) says that $\rho$ is an effective qubit because the support of
$\rho$ is only non-orthogonal to one eigenspace other than that belonging to
$0$.

Let us now prove properties (i) and (ii). For clarity let
$(\mathbf{p},\bm{\epsilon})=(p_{1},\dots,p_{n},\epsilon_{1},\dots,\epsilon_{n})$
denote an arbitrary, unspecified point in $[0,1]^{n}\times[0,2\pi]^{n}$ and
let
$(\mathbf{p}^{*},\bm{\epsilon}^{*})=(p_{1}^{*},\dots,p_{n}^{*},\epsilon_{1}^{*},\dots,\epsilon_{n}^{*})$
be a point in $M$ at which $f|_{M}$ assumes its minimum value. Let $k$ be the
number of nonzero $p_{j}^{*}$s. Since the values of $f$, $g$, and $h$ are
invariant under rearrangements of the form
$(p_{1},\dots,p_{n},\epsilon_{1},\dots,\epsilon_{n})\to(p_{\kappa(1)},\dots,p_{\kappa(n)},\epsilon_{\kappa(1)},\dots,\epsilon_{\kappa(n)})$,
where $\kappa$ is a permutation of the set $\\{1,2,\dots,n\\}$, we can assume
that the $k$ nonzero $p_{j}^{*}$s are placed first in the sequence
$\mathbf{p}^{*}$, so that
$\mathbf{p}^{*}=(p_{1}^{*},\dots,p_{k}^{*},\bar{\mathbf{0}}^{*})$, where
$\bar{\mathbf{0}}^{*}$ is the vector consisting of $n-k$ zeros, and that the
corresponding $\epsilon_{j}^{*}$s are arranged in descending order of
magnitude. We assume that the number of nonzero $\epsilon_{j}^{*}$s among the
first $k$ is $l$. These constitute the first $l$ elements of
$\bm{\epsilon}^{*}$. We denote the remaining $n-l$ $\epsilon_{j}^{*}$s by
$\bar{\bm{\epsilon}}^{*}$, so that
$\bm{\epsilon}^{*}=(\epsilon_{1}^{*},\dots,\epsilon_{l}^{*},\bar{\bm{\epsilon}}^{*})$.
The minimum point has the following properties:

  * (a)

$\epsilon_{1}^{*},\dots,\epsilon_{l}^{*}$ are strictly less than $2\pi$. For
suppose that some $\epsilon_{j}^{*}=2\pi$. If we change the value of this
$\epsilon_{j}^{*}$ to $0$, $g$ and $h$ will not change their values, and hence
we are still in $M$. However, since $p_{j}\neq 0$, such a change lowers the
value of $f$. This contradicts that $f|_{M}$ assumes its minimum value at
$(\mathbf{p}^{*},\bm{\epsilon}^{*})$.

  * (b)

There exists a $j$ such that $p_{j}^{*}\neq 0$ and $\epsilon_{j}^{*}=0$, and
thus $k>l$. For suppose that $k=l$. The functions $g$ and $h$ assume the same
values at $(\mathbf{p}^{*},\bm{\epsilon}^{*})$ and
$(\mathbf{p}^{*},\epsilon_{1}^{*}-\epsilon_{k}^{*},\dots,\epsilon_{k}^{*}-\epsilon_{k}^{*},0,\dots,0)$,
so both points belong to $M$. However, since $p^{*}_{k}\neq 0$, $f$ has a
lower value at the latter point. This contradicts that $f|_{M}$ assumes its
minimum value in $(\mathbf{p}^{*},\bm{\epsilon}^{*})$. Since not all the
$\epsilon_{j}^{*}$s can be zero, statement (i) follows.

It remains to prove statement (ii), that is, that all the nonzero
$\epsilon_{1}^{*},\dots,\epsilon_{l}^{*}$ have the same value. Since we have
assumed that $\delta>0$, there exists a unique function $\theta$ on $M$ with
values in the interval $(-\pi,\pi]$ such that

$\displaystyle\sqrt{\delta}\cos\theta(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{n}p_{j}\cos\epsilon_{j},$
(66)
$\displaystyle\sqrt{\delta}\sin\theta(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{n}p_{j}\sin\epsilon_{j}.$
(67)

This follows immediately from the observation that
$g(\mathbf{p},\bm{\epsilon})=\delta$ if and only if
$\sum_{j}p_{j}e^{i\epsilon_{j}}=\sqrt{\delta}e^{i\theta(\mathbf{p},\bm{\epsilon})}$
for a unique $\theta(\mathbf{p},\bm{\epsilon})$ in $(-\pi,\pi]$. Now, consider
the restrictions of $f$, $g$, and $h$ to
$F=[0,1]^{k}\times\\{\bar{\mathbf{0}}^{*}\\}\times[0,2\pi]^{l}\times\\{\bar{\bm{\epsilon}}^{*}\\}$:

$\displaystyle
f|_{F}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{l}p_{j}\epsilon_{j},$ (68)
$\displaystyle
g|_{F}(\mathbf{p},\bm{\epsilon})=\Big{(}\sum_{j=1}^{l}p_{j}\cos\epsilon_{j}+\sum_{j=l+1}^{k}p_{j}\Big{)}^{2}+\Big{(}\sum_{j=1}^{l}p_{j}\sin\epsilon_{j}\Big{)}^{2},$
(69) $\displaystyle h|_{F}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{k}p_{j}.$
(70)

The point $(\mathbf{p}^{*},\bm{\epsilon}^{*})$ lies in the intersection of $M$
and the interior of $F$. Let
$\theta^{*}=\theta(\mathbf{p}^{*},\bm{\epsilon}^{*})$. The gradient vectors of
$f|_{F}$, $g|_{F}$, and $h|_{F}$ at $(\mathbf{p}^{*},\bm{\epsilon}^{*})$ are

$\displaystyle\nabla f|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$
$\displaystyle=\sum_{j=1}^{l}\epsilon_{j}^{*}\frac{\partial}{\partial
p_{j}}+\sum_{j=1}^{l}p_{j}^{*}\frac{\partial}{\partial\epsilon_{j}},$ (71)
$\displaystyle\nabla g|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$
$\displaystyle=2\sqrt{\delta}\Big{(}\sum_{j=1}^{l}\cos(\theta^{*}-\epsilon_{j}^{*})\frac{\partial}{\partial
p_{j}}+\sum_{j=l+1}^{k}\cos\theta^{*}\frac{\partial}{\partial
p_{j}}+\sum_{j=1}^{l}p_{j}^{*}\sin(\theta^{*}-\epsilon_{j}^{*})\frac{\partial}{\partial\epsilon_{j}}\Big{)},$
(72) $\displaystyle\nabla h|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$
$\displaystyle=\sum_{j=1}^{k}\frac{\partial}{\partial p_{j}}.$ (73)

None of these are zero. According to the method of Lagrange multipliers, there
are real and nonzero constants $\lambda$ and $\mu$ such that $\nabla
f|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})=\lambda\nabla
g|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})+\mu\nabla
h|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$. We thus have that

$\displaystyle\epsilon_{j}^{*}$
$\displaystyle=2\sqrt{\delta}\lambda\cos(\theta^{*}-\epsilon_{j}^{*})+\mu,$
(74) $\displaystyle 1$
$\displaystyle=2\sqrt{\delta}\lambda\sin(\theta^{*}-\epsilon_{j}^{*}),$ (75)
$\displaystyle 0$ $\displaystyle=2\sqrt{\delta}\lambda\cos\theta^{*}+\mu.$
(76)

Equations (74) and (75) hold for $j=1,2,\dots,l$. These equations imply that
all the $\epsilon_{j}^{*}$s are solutions to the quadratic equation
$(x-\mu)^{2}=4\delta\lambda^{2}-1$. For if we move $\mu$ to the left side in
(74), square both (74) and (75), and then add them, we obtain the equation
$(\epsilon_{j}^{*}-\mu)^{2}=4\delta\lambda^{2}-1$. Thus, the
$\epsilon_{j}^{*}$s can assume at most two values,

$\displaystyle\bar{\epsilon}_{1}=\mu+\sqrt{4\delta\lambda^{2}-1},$ (77)
$\displaystyle\bar{\epsilon}_{2}=\mu-\sqrt{4\delta\lambda^{2}-1}.$ (78)

We assume that both of these are present among the $\epsilon_{j}^{*}$s.
(Otherwise, we are done.) According to (77) and (78), $\mu$ is the arithmetic
mean of $\bar{\epsilon}_{1}$ and $\bar{\epsilon}_{2}$. Furthermore, (74) and
(75) imply that

$\displaystyle
2\cos\Big{(}\theta^{*}-\frac{\bar{\epsilon}_{1}+\bar{\epsilon}_{1}}{2}\Big{)}$
$\displaystyle\cos\Big{(}\frac{\bar{\epsilon}_{2}-\bar{\epsilon}_{1}}{2}\Big{)}=\cos(\theta^{*}-\bar{\epsilon}_{1})+\cos(\theta^{*}-\bar{\epsilon}_{2})=0,$
(79) $\displaystyle
2\cos\Big{(}\theta^{*}-\frac{\bar{\epsilon}_{1}+\bar{\epsilon}_{1}}{2}\Big{)}$
$\displaystyle\sin\Big{(}\frac{\bar{\epsilon}_{2}-\bar{\epsilon}_{1}}{2}\Big{)}=\sin(\theta^{*}-\bar{\epsilon}_{1})-\sin(\theta^{*}-\bar{\epsilon}_{2})=0.$
(80)

Thus,

$\theta^{*}=\frac{\bar{\epsilon}_{1}+\bar{\epsilon}_{2}}{2}+n\cdot\frac{\pi}{2}$
(81)

for an odd integer $n$. Consequently,

$\displaystyle\sin(\theta^{*}-\bar{\epsilon}_{1})=(-1)^{\frac{n-1}{2}}\cos\left(\frac{\bar{\epsilon}_{1}-\bar{\epsilon}_{2}}{2}\right),$
(82)
$\displaystyle\cos(\theta^{*}-\bar{\epsilon}_{1})=(-1)^{\frac{n-1}{2}}\sin\left(\frac{\bar{\epsilon}_{1}-\bar{\epsilon}_{2}}{2}\right).$
(83)

Together with (74) och (75), these equations imply

$\frac{\bar{\epsilon}_{1}-\bar{\epsilon}_{2}}{2}=\frac{2\sqrt{\delta}\lambda\cos(\theta^{*}-\bar{\epsilon}_{1})}{2\sqrt{\delta}\lambda\sin(\theta^{*}-\bar{\epsilon}_{1})}=\tan\left(\frac{\bar{\epsilon}_{1}-\bar{\epsilon}_{2}}{2}\right).$
(84)

Since $-\pi\leq(\bar{\epsilon}_{1}-\bar{\epsilon}_{2})/2\leq\pi$ and $x=\tan
x$ has the unique solution $x=0$ in this interval, we can conclude that
$\bar{\epsilon}_{1}=\bar{\epsilon}_{2}$.

###  B.3 Fully distinguishable initial and final states

If $\delta=0$ we must modify the previous section’s arguments slightly. We
start by replacing $g$ with the two functions

$\displaystyle
g_{1}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{n}p_{j}\cos\epsilon_{j},$ (85)
$\displaystyle
g_{2}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{n}p_{j}\sin\epsilon_{j},$ (86)

and define $M$ as the subset of $[0,1]^{n}\times[0,2\pi]^{n}$ given by the
constraints $g_{1}(\mathbf{p},\bm{\epsilon})=0$,
$g_{2}(\mathbf{p},\bm{\epsilon})=0$, and $h(\mathbf{p},\bm{\epsilon})=1$. We
again let
$(\mathbf{p}^{*},\bm{\epsilon}^{*})=(p_{1}^{*},\dots,p_{n}^{*},\epsilon_{1}^{*},\dots,\epsilon_{n}^{*})$
be a point in $M$ at which $f|_{M}$ assumes its minimum value, which in this
case is $\alpha(n,0)=\pi/2$. We arrange the coordinates in
$(\mathbf{p}^{*},\bm{\epsilon}^{*})$ as in Appendix B.2. Properties (a) and
(b) of $(\mathbf{p}^{*},\bm{\epsilon}^{*})$ also apply in this case. We
consider the restrictions of $f$, $g_{1}$, $g_{2}$, and $h$ to
$F=[0,1]^{k}\times\\{\bar{\mathbf{0}}^{*}\\}\times[0,2\pi]^{l}\times\\{\bar{\bm{\epsilon}}^{*}\\}$:

$\displaystyle
f|_{F}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{l}p_{j}\epsilon_{j},$ (87)
$\displaystyle
g_{1}|_{F}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{l}p_{j}\cos\epsilon_{j}+\sum_{j=l+1}^{k}p_{j},$
(88) $\displaystyle
g_{2}|_{F}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{l}p_{j}\sin\epsilon_{j},$
(89) $\displaystyle h|_{F}(\mathbf{p},\bm{\epsilon})=\sum_{j=1}^{k}p_{j}.$
(90)

The point $(\mathbf{p}^{*},\bm{\epsilon}^{*})$ lies in the intersection of $M$
and the interior of $F$, and the gradient vectors of the restrictions of $f$,
$g$, and $h$ to $F$ at $(\mathbf{p}^{*},\bm{\epsilon}^{*})$ are

$\displaystyle\nabla f|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$
$\displaystyle=\sum_{j=1}^{l}\epsilon_{j}^{*}\frac{\partial}{\partial
p_{j}}+\sum_{j=1}^{l}p_{j}^{*}\frac{\partial}{\partial\epsilon_{j}},$ (91)
$\displaystyle\nabla g_{1}|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$
$\displaystyle=\sum_{j=1}^{l}\cos\epsilon_{j}^{*}\frac{\partial}{\partial
p_{j}}+\sum_{j=l+1}^{k}\frac{\partial}{\partial
p_{j}}-\sum_{j=1}^{l}p_{j}^{*}\sin\epsilon_{j}^{*}\frac{\partial}{\partial\epsilon_{j}},$
(92) $\displaystyle\nabla g_{2}|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$
$\displaystyle=\sum_{j=1}^{l}\sin\epsilon_{j}^{*}\frac{\partial}{\partial
p_{j}}+\sum_{j=1}^{l}p_{j}^{*}\cos\epsilon_{j}^{*}\frac{\partial}{\partial\epsilon_{j}},$
(93) $\displaystyle\nabla h|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$
$\displaystyle=\sum_{j=1}^{k}\frac{\partial}{\partial p_{j}}.$ (94)

None of these are zero. According to the method of Lagrange multipliers there
exist real nonzero constants $\lambda_{1}$, $\lambda_{2}$, and $\mu$ such that
$\nabla f|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})=\lambda_{1}\nabla
g_{1}|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})+\lambda_{2}\nabla
g_{2}|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})+\mu\nabla
h|_{F}(\mathbf{p}^{*},\bm{\epsilon}^{*})$. We thus have that

$\displaystyle\epsilon_{j}^{*}$
$\displaystyle=\lambda_{1}\cos\epsilon_{j}^{*}+\lambda_{2}\sin\epsilon_{j}^{*}+\mu,$
(95) $\displaystyle 1$
$\displaystyle=\lambda_{2}\cos\epsilon_{j}^{*}-\lambda_{1}\sin\epsilon_{j}^{*},$
(96) $\displaystyle 0$ $\displaystyle=\lambda_{1}+\mu.$ (97)

Equations (95) and (96) hold for $j=1,2,\dots,l$. If we move $\mu$ to the left
side in (95), square both (95) and (96), and then add them, we get the
equation $(\epsilon_{j}^{*}-\mu)^{2}=\lambda_{1}^{2}+\lambda_{2}^{2}$. Thus
there are only two possibilities for the $\epsilon_{j}^{*}$s,

$\displaystyle\bar{\epsilon}_{1}=\mu+\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}},$
(98)
$\displaystyle\bar{\epsilon}_{2}=\mu-\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}.$
(99)

We assume both are present among the $\epsilon_{j}^{*}$s and, therefore, are
strictly positive. The multiplier $\mu$, and hence $-\lambda_{1}$, is the
arithmetic mean of $\bar{\epsilon}_{1}$ and $\bar{\epsilon}_{2}$. It follows
that

$\lambda_{2}^{2}=(\bar{\epsilon}_{1}-\mu)^{2}-\lambda_{1}^{2}=\bigg{(}\frac{\bar{\epsilon}_{1}-\bar{\epsilon}_{2}}{2}\bigg{)}^{2}-\bigg{(}\frac{\bar{\epsilon}_{1}+\bar{\epsilon}_{2}}{2}\bigg{)}^{2}=-\bar{\epsilon}_{1}\bar{\epsilon}_{2}.$
(100)

This equation cannot be satisfied for positive $\bar{\epsilon}_{1}$ and
$\bar{\epsilon}_{2}$. Thus, we have reached a contradiction. The conclusion is
that only one of $\bar{\epsilon}_{1}$ and $\bar{\epsilon}_{2}$ is present
among the $\epsilon_{j}^{*}$s and, therefore, all the $\epsilon_{j}^{*}$s have
the same value.

###  B.4 Conclusion

Appendices B.2 and B.3 show that if $\tau\langle H-\epsilon_{0}\rangle$
assumes its smallest possible value under the requirement that the state
evolves between two states with fidelity $\delta$, then the state is a partly
grounded effective qubit. Consequently, the case reduces to that treated in
Sec. II.1, regardless of the dimension of the system. We conclude that
$\alpha(n,\delta)=\alpha(\delta)$ and thus that the extended Margolus-Levitin
QSL (4) is valid in all dimensions.

##  Appendix C Geodesics and ruled Seifert surfaces

A curve $\rho_{t}$ in $\mathcal{P}$ is a geodesic if its acceleration
vanishes, $\nabla_{\dot{\rho}_{t}}\dot{\rho}_{t}=0$. For the Fubini-Study
metric,
$\nabla_{\dot{\rho}_{t}}\dot{\rho}_{t}=[[\ddot{\rho}_{t},\rho_{t}],\rho_{t}]$;
see, e.g., Ref. [8]. Let $\sigma$ and $\rho$ be two states at a distance
$r<\pi/2$ apart. There is a unique shortest unit speed geodesic from $\sigma$
to $\rho$: Choose a unit vector $|\phi\rangle$ over $\sigma$ and let
$|\psi\rangle$ be the unit vector over $\rho$ that is in phase with
$|\phi\rangle$. Then $\langle\phi|\psi\rangle=\cos r$. Define $|w\rangle$ by
the condition $|\psi\rangle=\cos r|\phi\rangle+\sin r|w\rangle$, and let
$|\phi_{t}\rangle=\cos t|\phi\rangle+\sin t|w\rangle$. The curve

$\gamma_{t}=|\phi_{t}\rangle\langle\phi_{t}|=\cos^{2}t|\phi\rangle\langle\phi|+\cos
t\sin t\big{(}|\phi\rangle\langle
w|+|w\rangle\langle\phi|\big{)}+\sin^{2}t|w\rangle\langle w|,\quad 0\leq t\leq
r,$ (101)

is the shortest unit speed geodesic that extends from $\sigma$ to $\rho$;
straightforward calculations show that
$[[\ddot{\gamma}_{t},\gamma_{t}],\gamma_{t}]=0$, that
$g(\dot{\gamma}_{t},\dot{\gamma}_{t})=1$, and that $\gamma_{t}$ has the length
$r=\operatorname{dist}(\sigma,\rho)$.

###  C.1 Optimal evolution curves are sub-Riemannian geodesics

If the extended Margolus-Levitin QSL is saturated, the state is a partly
grounded effective qubit:

$\rho=\cos^{2}r|0\rangle\langle 0|+\cos r\sin r\big{(}|0\rangle\langle
1|+|1\rangle\langle 0|\big{)}+\sin^{2}r|1\rangle\langle 1|.$ (102)

Here $|0\rangle$ and $|1\rangle$ are eigenvectors of the Hamiltonian with
eigenvalues $\epsilon_{0}$ and $\epsilon_{1}$, respectively. This state
evolves on the geodesic sphere $S(\sigma,r)$ around $\sigma=|0\rangle\langle
0|$ with radius $r=\operatorname{dist}(\sigma,\rho)$. The evolution curve is

$\rho_{t}=\cos^{2}r|0\rangle\langle 0|+\cos r\sin
r\big{(}e^{it(\epsilon_{1}-\epsilon_{0})}|0\rangle\langle
1|+e^{-it(\epsilon_{1}-\epsilon_{0})}|1\rangle\langle
0|\big{)}+\sin^{2}r|1\rangle\langle 1|.$ (103)

This curve has the acceleration

$\nabla_{\dot{\rho}_{t}}\dot{\rho}_{t}=\frac{(\epsilon_{1}-\epsilon_{0})^{2}}{4}\sin(4r)\Big{(}\sin(2r)\big{(}|0\rangle\langle
0|-|1\rangle\langle
1|\big{)}-\cos(2r)\big{(}e^{it(\epsilon_{1}-\epsilon_{0})}|0\rangle\langle
1|+e^{-it(\epsilon_{1}-\epsilon_{0})}|1\rangle\langle 0|\big{)}\Big{)}.$ (104)

The acceleration vanishes identically if and only if $r=\pi/4$. In this case,
the state moves along the equator in the projectivization of the span of
$|0\rangle$ and $|1\rangle$. Otherwise, the evolution curve is not a geodesic
in $\mathcal{P}$. However, since the acceleration is everywhere perpendicular
to $S(\sigma,r)$, the evolution curve is a sub-Riemannian geodesic in
$S(\sigma,r)$, that is, a geodesic when considered a curve in $S(\sigma,r)$
with the sub-Riemannian geometry. To see this, fix a $t$ and let $\gamma_{s}$,
$0\leq s\leq r$, be the shortest, arclength-parametrized geodesic from
$\sigma$ to $\rho_{t}$. According to Gauss’s lemma [30], its velocity vector
at $s=r$ is perpendicular to $S(\sigma,r)$. Explicitly, we have that

$\dot{\gamma}_{r}=\sin(2r)\big{(}|1\rangle\langle 1|-|0\rangle\langle
0|\big{)}+\cos(2r)\big{(}e^{it(\epsilon_{1}-\epsilon_{0})}|0\rangle\langle
1|+e^{-it(\epsilon_{1}-\epsilon_{0})}|1\rangle\langle 0|\big{)}.$ (105)

By equations (104) and (105),
$\nabla_{\dot{\rho}_{t}}\dot{\rho}_{t}=-\frac{1}{4}(\epsilon_{1}-\epsilon_{0})^{2}\sin(4r)\dot{\gamma}_{r}$,
which implies that $\nabla_{\dot{\rho}_{t}}\dot{\rho}_{t}$ is perpendicular to
$S(\sigma,r)$ at $\rho_{t}$. Consequently, $\rho_{t}$ is a sub-Riemannian
geodesic.

###  C.2 Ruled Seifert surfaces

Let $\rho_{t}$, $0\leq t\leq\tau$, be a curve in $\mathcal{P}$ which for each
$t$ is at the distance $r<\pi/2$ from $\sigma$. We can construct a Seifert
surface for its $\sigma$ closure in the following way. Let $|\phi\rangle$ be a
lift of $\sigma$, let $|\psi_{t}\rangle$ be the lift of $\rho_{t}$ that is in
phase with $|\phi\rangle$, define $|w_{t}\rangle$ by the condition
$|\psi_{t}\rangle=\cos r|\phi\rangle+\sin r|w_{t}\rangle$, and set
$|\psi_{s,t}\rangle=\cos s|\phi\rangle+\sin s|w_{t}\rangle$ for $0\leq s\leq
r$ and $0\leq t\leq\tau$. Then
$\Sigma(s,t)=|\psi_{s,t}\rangle\langle\psi_{s,t}|$ is a Seifert surface for
$\rho_{t}^{\sigma}$ consisting of geodesics starting from $\sigma$ and ending
at points on $\rho_{t}$.

##  Appendix D Extreme values of $\mathcal{J}^{\sigma}_{r,\delta}$

Let $\sigma$ be an arbitrary state and $\Gamma(\sigma,r,\delta)$ be the set of
smooth curves in $S(\sigma,r)$ that stretches between two states with fidelity
$\delta$. Consider the functional $\mathcal{J}^{\sigma}_{r,\delta}$ on
$\Gamma(\sigma,r,\delta)$ defined as

$\mathcal{J}^{\sigma}_{r,\delta}[\rho_{t}]=\int_{\rho_{t}}A^{\sigma}.$ (106)

We calculate the extreme values of $\mathcal{J}^{\sigma}_{r,\delta}$.

Fix an arbitrary lift $|\phi\rangle$ of $\sigma$ and recall that
$Y_{|\phi\rangle}$ is the hypersurface in $\SS$ consisting of all vectors in
phase with $|\phi\rangle$. Let $|\phi\rangle$ be the first vector in an
orthonormal basis $|\phi\rangle,|1\rangle,\dots,|n-1\rangle$ for
$\mathcal{H}$. Every lift $|\psi_{t}\rangle$ of a curve in $S(\sigma,r)$ to
$Y_{|\phi\rangle}$ has a decomposition of the form $|\psi_{t}\rangle=\cos
r|\phi\rangle+\sin r|w_{t}\rangle$, where

$|w_{t}\rangle=\sum_{j=1}^{n-1}(u_{j;t}+iv_{j;t})|j\rangle$ (107)

for some real-valued functions $u_{j;t}$ and $v_{j;t}$ satisfying

$\sum_{j=1}^{n-1}(u_{j;t}^{2}+v_{j;t}^{2})=1.$ (108)

Conversely, any such curve in $Y_{|\phi\rangle}$ projects to a curve in
$S(\sigma,r)$. Let $\rho_{t}=|\psi_{t}\rangle\langle\psi_{t}|$ and assume that
$0\leq t\leq\tau$. Then

$\mathcal{J}^{\sigma}_{r,\delta}[\rho_{t}]=\sum_{j=1}^{n-1}\int_{0}^{\tau}\mathrm{d}t\,(u_{j;t}\dot{v}_{j;t}-\dot{u}_{j;t}v_{j;t}).$
(109)

Suppose that the fidelity between $\rho_{0}$ and $\rho_{\tau}$ is $\delta$ and
that $\rho_{t}$ is extremal for $\mathcal{J}^{\sigma}_{r,\delta}$. The
$u_{j;t}$s and $v_{j;t}$s then satisfy the Euler-Lagrange equations for the
augmented Lagrangian

$L(u_{j},\dot{u}_{j},v_{j},\dot{v}_{j},\lambda)=\sum_{j=1}^{n-1}(u_{j}\dot{v}_{j}-\dot{u}_{j}v_{j})-\lambda\bigg{(}\sum_{j=1}^{n-1}(u^{2}_{j}+v^{2}_{j})-1\bigg{)}.$
(110)

The Euler-Lagrange equations read

$\displaystyle\dot{u}_{j}=-\lambda v_{j},$ (111)
$\displaystyle\dot{v}_{j}=\lambda u_{j},$ (112)
$\displaystyle\sum_{j=1}^{n-1}(u^{2}_{j}+v^{2}_{j})=1.$ (113)

These equations imply that
$u_{j;t}+iv_{j;t}=e^{i\Lambda_{t}}(u_{j;0}+iv_{j;0})$, with $\Lambda_{t}$
being the integral of the Lagrange multiplier,

$\Lambda_{t}=\int_{0}^{t}\mathrm{d}s\,\lambda_{s}.$ (114)

Thus, $|\psi_{t}\rangle=\cos r\,|\phi\rangle+\sin
r\,e^{i\Lambda_{t}}|w_{0}\rangle$. That the fidelity between $\rho_{0}$ and
$\rho_{\tau}$ is $\delta$ translates to

$\delta=1+\frac{1}{2}\sin^{2}2r(\cos\Lambda_{\tau}-1).$ (115)

We conclude that

$\Lambda_{\tau}=\pm\arccos\left(1-\frac{2(1-\delta)}{\sin^{2}2r}\right)\mod
2\pi$ (116)

and, hence, that

$\mathcal{J}^{\sigma}_{r,\delta}[\rho_{t}]=\pm\sin^{2}r\,\arccos\bigg{(}1-\frac{2(1-\delta)}{\sin^{2}2r}\bigg{)}\mod
2\pi.$ (117)

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