# Multi-copy adaptive local discrimination: Strongest possible two-qubit
nonlocal bases

Manik Banik  School of Physics, IISER Thiruvananthapuram, Vithura, Kerala
695551, India.  Tamal Guha  Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 B.T. Road, Kolkata 700108, India.  Mir Alimuddin
Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T.
Road, Kolkata 700108, India.  Guruprasad Kar  Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India.
Saronath Halder  Harish-Chandra Research Institute, HBNI, Chhatnag Road,
Jhunsi, Allahabad 211019, India.  Some Sankar Bhattacharya  Department of
Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong.

###### Abstract

Ensembles of composite quantum states can exhibit nonlocal behaviour in the
sense that their optimal discrimination may require global operations. Such an
ensemble containing $N$ pairwise orthogonal pure states, however, can always
be perfectly distinguished under adaptive local scheme if $(N-1)$ copies of
the state are available. In this letter, we provide examples of orthonormal
bases in two-qubit Hilbert space whose adaptive discrimination require $3$
copies of the state. For this composite system we analyze multi-copy adaptive
local distinguishability of orthogonal ensembles in full generality which in
turn assigns varying nonlocal strength to different such ensembles. We also
come up with ensembles whose discrimination under adaptive separable scheme
require less number of copies than adaptive local schemes. Our construction
finds important application in multipartite secret sharing tasks and indicates
towards an intriguing super-additivity phenomenon for locally accessible
information.

Introduction.– Second quantum revolution aims to harness individual quantum
system for storing, transferring, and manipulating information Chuang . In
many of the information protocols the elementary step is reliable decoding of
the classical information encoded in some physical system. When involved
systems are quantum, several interesting observations appear that otherwise
are not present in classical world. For instance, classical message encoded in
the states of a composite quantum system may not be completely retrieved under
local quantum operations and classical communications (LOCC) among the
spatially separated subsystems. Such a set of states is called nonlocal as
their optimal discrimination require global operation(s) on the composite
system. In a seminal paper Bennett99 , Bennett et al. provided examples of
orthonormal bases in $(\mathbb{C}^{3})^{\otimes 2}$ and
$(\mathbb{C}^{2})^{\otimes 3}$ that are not perfectly distinguishable under
LOCC. These examples are quite striking as they contain only product states
and hence introduced the phenomenon of ‘quantum nonlocality without
entanglement’. For the simplest multipartite system $(\mathbb{C}^{2})^{\otimes
2}$, example of LOCC indistinguishable ensemble was first identified in
Ghosh01 . Unlike Bennett et al.’s examples, an orthogonal ensemble in
$(\mathbb{C}^{2})^{\otimes 2}$ must contain entangled state(s) for local
indistinguishability Walgate02 . Historically, a difference between global and
local distinguishability for a two-qubit ensemble containing only product
states was conjectured by Peres & Wootters Peres91 . Importantly, Peres-
Wootters’ ensemble contains non-orthogonal states and recently their
conjecture has been proven to be true Chitambar13 . The results in Bennett99 ;
Ghosh01 ; Walgate02 ; Peres91  initiated a plethora of studies on the local
state discrimination problem that turn out to be deeply interlinked with
quantum entanglement theory (see Bennett99(1) ; Walgate00 ; Horodecki03 ;
DiVincenzo03 ; Ghosh04 ; Watrous05 ; Niset06 ; Duan07 ; Calsamiglia10 ;
Bandyopadhyay11 ; Chitambar14 ; Halder18 ; Demianowicz18 ; Halder19 ;
Agrawal19 ; Rout19 ; Rout20 ; Horodecki09  and references there in).
Subsequently, local indistinguishability has been identified as crucial
primitive for cryptography protocols, such as quantum data hiding Terhal01 ;
Eggeling02 ; Matthews09  and quantum secret sharing Hillery99 ; Markham08 ;
Rahaman15 ; it also underlies the non-zero gap between single-shot and multi-
shot classical capacities of noisy quantum channels Fuchs , and has been shown
to have foundational appeal Pusey12 ; Bandyopadhyay17 ; Bhattacharya20 .
Mathematical difficulty of characterizing the class of LOCC operations
Chitambar14(1)  motivates researchers to address the state discrimination
problem with a larger class of operations, namely separable super-operator
(SEP) and/or the set of operations that map positive partial transposition
(PPT) states to PPT states Duan09 ; Yu12 ; Yu14 ; Bandyopadhyay15 ; Cohen15 .

The problem of local discrimination discussed so far considers identifying the
unknown state from a single copy of the system. Given many copies of the state
$\ket{\psi_{i}}^{\otimes k}$ the probability of knowing the state
$\ket{\psi_{i}}$ and consequently guessing the classical message ‘$i$’
increases; $\ket{\psi}_{i}$ is sampled from some known ensemble
$\mathcal{E}_{N}\equiv\\{p_{i},\ket{\psi_{i}}\leavevmode\nobreak\
|\leavevmode\nobreak\ p_{i}>0\leavevmode\nobreak\ \&\leavevmode\nobreak\ \sum
p_{i}=1\\}_{i=1}^{N}$. If there is no limitation on the number $k$, then the
state can be identified exactly even without any prior knowledge from which
ensemble it is sampled. This fact, known as local tomography, has been
axiomatized by many authors in physical reconstruction of Hilbert space
quantum theory Chiribella11 ; Barnum14 ; Niestegge20 . However, given the
prior knowledge of the orthogonal pure states ensemble $\mathcal{E}_{N}$, only
$(N-1)$ copies are sufficient for perfect local discrimination Walgate00 ;
Bandyopadhyay11 . In this multi-copy scenario, the parties are allowed to
invoke adaptive as well as non-adaptive strategies for discrimination.

######  Definition 1.

(Local adaptive strategy) Given multiple copies of the state in an adaptive
strategy each of the copies are addressed individually and the maximal
possible information regarding the unknown state is extracted through LOCC.
Depending on the knowledge obtained from the former copies strategies on the
later copies are modified adaptively.

Such adaptive strategies have already been studied extensively in quantum
channel discrimination Hayashi09 ; Harrow10 ; Pirandola19  and in noise
estimation Pirandola17 ; Cope17 . Perfect local discrimination of any
orthogonal ensemble $\mathcal{E}_{N}$ is possible with $(N-1)$ copies under
such adaptive schemes. The known examples of the nonlocal ensemble, however,
do not even require $(N-1)$ copies for perfect discrimination. In fact, the
authors in Ref. Bennett99  found no example where more than two copies of the
unknown state are needed for perfect discrimination and thus they laid down
the question: “Are there any sets of states, entangled or not, for which some
finite number (greater than $2$) of copies of the state is necessary for
distinguishing the states reliably?" Even after more than two decades, to the
best of our knowledge, there is no conclusive answer to this question. In the
present letter we address this question and show that there indeed exist
ensembles of orthogonal pure states for composite system that require more
than two copies of the state for perfect local discrimination under adaptive
strategies. Our explicit constructions form orthonormal bases for two-qubit
Hilbert space. We, in-fact, completely characterize the two-qubit ensembles
that require three copies of the state for perfect local discrimination under
adaptive strategies. Interestingly, we also find ensembles that require three
copies of the state for perfect discrimination under adaptive LOCC whereas two
copies suffice if adaptive SEP protocols are followed. Muli-copy consideration
of local state discrimination problem establishes the ‘more nonlocality with
less entanglement’ phenomenon for $(\mathbb{C}^{2})^{\otimes 2}$ ensembles
which was earlier reported for ensembles in $(\mathbb{C}^{3})^{\otimes 2}$
Horodecki03 . Our study indicates an intriguing super-additivity phenomenon
for the locally accessible information of multi-site quantum ensembles, and we
discuss a practical application of the present constructions in multipartite
secret sharing task.

Results.– Throughout the paper we follow the standard notations used in
quantum information community. For instance, $\\{\ket{0},\ket{1}\\}$ is the
computational bases (eigenkets of Pauli $\sigma_{z}$ operator) of
$\mathbb{C}^{2}$ and $\ket{ab}$ stands for the short hand notation of the
bipartite state
$\ket{a}_{A}\otimes\ket{b}_{B}\in\mathbb{C}^{2}_{A}\otimes\mathbb{C}^{2}_{B}$.
Given single copy of the state, the authors in Walgate02  have analyzed LOCC
distinguishability of orthogonal ensembles in $(\mathbb{C}^{2})^{\otimes 2}$.
Here we will analyse their multi-copy local distinguishability. We will
explore the case where Alice and Bob follow adaptive protocols for
discrimination. An ensemble of three orthogonal pure states (of arbitrary
composite system) is always two-copy distinguishable under adaptive LOCC
Walgate00 ; Bandyopadhyay11 . We will therefore focus on the ensembles
$\mathcal{E}_{4}=\\{\ket{\psi_{i}}\\}_{i=1}^{4}\subset(\mathbb{C}^{2})^{\otimes
2}$, where $\langle\psi_{i}|\psi_{j}\rangle=\delta_{ij}$. Throughout the paper
we will consider that the states are uniformly sampled from the ensemble.
Given multiple copies, while discriminating such a set under adaptive
protocol, following three situations can arise: (a) the states can be
discriminated after acting on the first copy at an early stage of the
protocol, (b) protocol on the first copy results in a conclusion that the
given state belongs to the group $\mathcal{G}_{l}\equiv\\{\ket{\psi_{l}}\\}$
or in its complement group
$\mathcal{G}_{l}^{\mathsf{C}}:=\mathcal{E}_{4}\setminus\mathcal{G}_{l}$, for
some $l\in\\{1,\cdots,4\\}$, and (c) protocol on the first copy results in a
conclusion that the state belongs to the group
$\mathcal{G}_{ij}\equiv\\{\ket{\psi_{i}},\ket{\psi_{j}}\\}$ for some
$i,j\in\\{1,\cdots,4\\}$ with $i\neq j$. In the second and third cases the
discrimination has to be completed from the subsequent copies. This leads us
to our first result.

######  Theorem 1.

An orthogonal ensemble $\mathcal{E}_{4}\subset(\mathbb{C}^{2})^{\otimes 2}$ is
two-copy distinguishable under adaptive LOCC protocol if and only if one of
the following is satisfied:

  * (i)

the set is one-copy distinguishable and second copy is not at all required
(trivial case);

  * (ii)

protocol on the first copy leads to a conclusion that the given state belongs
to a group $\mathcal{G}_{l}$ or $\mathcal{G}_{l}^{\mathsf{C}}$ for some
$l\in\\{1,\cdots,4\\}$ such that $\mathcal{G}_{l}^{\mathsf{C}}$ contains no
more than one entangled state.

  * (iii)

protocol on the first copy leads to a conclusion that the given state belongs
to the a group $\mathcal{G}_{ij}$ for some $i,j\in\\{1,\cdots,4\\}$ with
$i\neq j$, such that the projectors $\mathbb{P}_{ij}$ and
$\mathbb{P}_{ij}^{\mathsf{C}}$ are separable, where
$\mathbb{P}_{ij}:=\ket{\psi_{i}}\bra{\psi_{i}}+\ket{\psi_{j}}\bra{\psi_{j}}\leavevmode\nobreak\
\&\leavevmode\nobreak\
\mathbb{P}_{ij}^{\mathsf{C}}:=\mathbf{I}_{4}-\mathbb{P}_{ij}$.

###### Proof.

For perfect discrimination, after the protocol on first copy either the state
is discriminated straightaway [case (i)] or some states (at least one) must be
eliminated [the other two cases]. In case (ii), if the protocol on the first
copy leads to the conclusion that that the given set is in some group
$\mathcal{G}_{l}$ then the discrimination is done. If it leads to a conclusion
in the complement group $\mathcal{G}_{l}^{\mathsf{C}}$ (which will be the case
with some non-zero probability) then the unknown state needs to be
distinguished from the second copy. Recall that three orthogonal states in
$(\mathbb{C}^{2})^{\otimes 2}$ can be exactly locally distinguished if and
only if at least two of those states are product states Walgate02 , and hence
leading us to the assertion (ii). In case (iii), if the protocol on first copy
leads to a conclusion that the unknown state belongs to either in some group
$\mathcal{G}_{ij}$ or in its complement group $\mathcal{G}_{ij}^{\mathsf{C}}$
then protocol on the second copy perfectly succeed as two orthogonal states
(in any dimension) can always be exactly locally distinguished Walgate00 .
Suppose that there is some LOCC protocol on the first copy that leads us to
this desired conclusion. Such a protocol will perfectly distinguish the
density operators $\rho:=\frac{1}{2}\mathbb{P}_{ij}$ and
$\rho^{\perp}:=\frac{1}{2}\mathbb{P}_{ij}^{\mathsf{C}}$. Recall that two rank-
two orthogonal density operators
$\rho,\rho^{\perp}\in\mathcal{D}(\mathbb{C}^{2}\otimes\mathbb{C}^{2})$ are
LOCC distinguishable if and only if the projectors onto the supports of each
of the mixed states are separable Chitambar14 , and hence leads us to the
assertion (iii).

To complete the argument, let us consider that after the protocol on first
copy, none of the states is eliminated, rather the knowledge regarding the
ensemble gets updated, i.e. the uniform ensemble
$\mathcal{E}_{4}\equiv\\{1/4,\ket{\psi_{i}}\\}$ ends up into some non-uniform
ensemble
$\mathcal{E}^{\prime}_{4}\equiv\\{p_{i},\ket{\psi_{i}}\leavevmode\nobreak\
|\leavevmode\nobreak\ p_{i}\neq 0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\\}$. In such a case, the ensemble
$\mathcal{E}^{\prime}$ must be discriminated from the second copy of the
state. According to the Theorem 4 of Ref. Walgate02 , this can be done if and
only if all the four states are product. This is because Theorem 4 of Ref.
Walgate02  (and also the other theorems therein) is independent of the prior
probability distributions. This completes the proof. ∎

Given the above theorem, naturally the question arises which orthogonal
ensembles of $(\mathbb{C}^{2})^{\otimes 2}$ are two-copy distinguishable under
adaptive LOCC. As an immediate corollary of Theorem 1 we first have the
following result (Proof provided in the Supplemental).

######  Corollary 1.

Any orthogonal ensemble of $(\mathbb{C}^{2})^{\otimes 2}$ containing no more
than two entangled states is two-copy distinguishable under adaptive LOCC.

Ensembles having more than two entangled states can not be in category (i) or
category (ii) of Theorem 1. However, they can be in category (iii). For
instance, consider the orthonormal basis $\mathcal{B}^{[\theta]}$ with basis
states:

$\displaystyle\ket{\phi^{+}_{\theta}}$
$\displaystyle:=S_{\theta}\ket{00}+C_{\theta}\ket{11},\leavevmode\nobreak\
\ket{\phi^{-}_{\theta}}:=C_{\theta}\ket{00}-S_{\theta}\ket{11},$
$\displaystyle\ket{\psi^{+}_{\theta}}$
$\displaystyle:=S_{\theta}\ket{01}+C_{\theta}\ket{10},\leavevmode\nobreak\
\ket{\psi^{-}_{\theta}}:=C_{\theta}\ket{01}-S_{\theta}\ket{10},$

where $S_{\theta}\equiv\sin\theta\leavevmode\nobreak\ \&\leavevmode\nobreak\
C_{\theta}\equiv\cos\theta$ with $0\leq\theta\leq\pi/2$. All the states are
entangled whenever $\theta\neq 0,\pi/2$ and in particular $\theta=\pi/4$
corresponds to the maximally entangled basis/ Bell basis. These entangled
bases are two-copy distinguishable under adaptive LOCC. The protocol goes as
follows: on the first copy both Alice and Bob perform $\sigma_{z}$ measurement
on their part of the composite system and compare their measurement results.
Correlated outcomes imply that the given state belongs to the group
$\mathcal{G}_{\ket{\phi}}\equiv\\{\ket{\phi^{+}_{\theta}},\ket{\phi^{-}_{\theta}}\\}$
whereas anti-correlated outcomes imply that it is from the group
$\mathcal{G}_{\ket{\psi}}\equiv\\{\ket{\psi^{+}_{\theta}},\ket{\psi^{-}_{\theta}}\\}$.
The result of Walgate et al. Walgate00  assures prefect local
distinguishability of the states from the second copy. Let us now construct
another orthonormal basis

$\displaystyle\rotatebox[origin={c}]{0.0}{$\mathcal{A}_{\gamma}^{[\alpha,\beta]}\equiv$}\left\\{\\!\begin{aligned}
\ket{a_{1}}&:=\ket{\phi^{-}_{\alpha}},\leavevmode\nobreak\
\leavevmode\nobreak\
\ket{a_{3}}:=S_{\gamma}\ket{\phi^{+}_{\alpha}}+C_{\gamma}\ket{\psi^{+}_{\beta}}\\\
\ket{a_{2}}&:=\ket{\psi^{-}_{\beta}},\leavevmode\nobreak\ \leavevmode\nobreak\
\ket{a_{4}}:=C_{\gamma}\ket{\phi^{+}_{\alpha}}-S_{\gamma}\ket{\psi^{+}_{\beta}}\end{aligned}\right\\},$

with $\alpha,\beta,\gamma\in[0,\pi/2]$. Whenever $\alpha,\beta\neq 0,\pi/2$,
the states $\ket{a_{1}}\leavevmode\nobreak\ \&\leavevmode\nobreak\
\ket{a_{2}}$ are entangled. However entanglement of the other two states
demand further restrictions on the parameters. For instance, the states
$\ket{a_{3}},\ket{a_{4}}\in\mathcal{A}_{\pi/4}^{[\alpha,\beta]}$ are entangled
iff $\alpha\neq\beta,\leavevmode\nobreak\ \pi/2-\beta$. In the rest of the
paper we will mainly analyze the case $\gamma=\pi/4$. For generic
consideration of $\gamma$ see the supplementary material Supple . We also
consider the cases where all the states in
$\mathcal{A}_{\pi/4}^{[\alpha,\beta]}$ are entangled. Next we report an
interesting feature of the set $\mathcal{A}_{\pi/4}^{[\alpha,\beta]}$.

######  Proposition 1.

$\mathbb{V}[ij]:=Span\\{\ket{a_{i}},\ket{a_{j}}\\}$ with $i\neq j$ and
$\ket{a_{i}},\ket{a_{j}}\in\mathcal{A}_{\pi/4}^{[\alpha,\beta]}$;
$\alpha\neq\beta,\leavevmode\nobreak\ \pi/2-\beta$. For every choice of
$i,j\in\\{1,2,3,4\\}$ the projector onto the two dimensional subspace
$\mathbb{V}[ij]$ is entangled.

See supplementary material Supple  for the proof. Coming back to local
distinguishability of the set $\mathcal{A}_{\pi/4}^{[\alpha,\beta]}$, it is
neither in category (i) nor in category (ii) of Theorem 1. Furthermore,
Proposition 1 obstructs it to be in category (iii) too. These altogether guide
us to the following result.

######  Theorem 2.

The set $\mathcal{A}_{\pi/4}^{[\alpha,\beta]}$, with
$\alpha\neq\beta,\leavevmode\nobreak\ \pi/2-\beta$, is perfectly
distinguishable under adaptive LOCC protocol if and only if three copies of
the state are available.

Under the adaptive local discrimination scheme we therefore have a conclusive
answer to the Bennett et al.’ question Bennett99  discussed in introduction –
there indeed exists ensemble of states that requires more than two copies of
the state for perfect discrimination under adaptive local scheme. Under such
schemes the set $\mathcal{A}_{\pi/4}^{[\alpha,\beta]}$ can be considered as
the strongest nonlocal ensemble (of orthogonal pure states) in
$(\mathbb{C}^{2})^{\otimes 2}$. Naturally the question arises whether such a
nonlocal ensemble requires all of its states to be entangled. Our next theorem
(proof provided in supplementary material Supple ) answers this question.

######  Theorem 3.

Any orthonormal basis of $(\mathbb{C}^{2})^{\otimes 2}$ containing exactly
three entangled states requires three copies of the state for perfect
discrimination under adaptive LOCC.

Theorem 3 is quite interesting in a sense. It tells that any basis containing
three entangled states is two-copy indistinguishable under adaptive LOCC,
whereas some bases with all entangled states (e.g. Bell basis) are two-copy
distinguishable under local adaptive scheme. This mimics the phenomenon of
more nonlocality with less entanglement Horodecki03  as a set with less number
of entangled states turns out to be harder to discriminate. Importantly, the
ensemble in Ref. Horodecki03  exhibiting this feature lives in
$(\mathbb{C}^{3})^{\otimes 2}$ and with single-copy consideration such a
phenomenon is not possible in $(\mathbb{C}^{2})^{\otimes 2}$. Multi-copy
adaptive discrimination makes this phenomenon possible for two-qubit ensembles
and in-fact introduces a hierarchical strength in the more nonlocality with
less entanglement phenomenon.

Next we consider multi-copy discrimination of nonlocal ensembles under the
adaptive SEP. Recall that this set of operations strictly includes the set of
LOCC operations Chitambar14(1) , even for two-qubit Hilbert space Duan09 . Our
next result, however, establishes that given two copies of the state even this
larger class of operations fails to erase the nonlocal feature of some
ensembles.

######  Theorem 4.

There exist suitable choices of the parameters $\alpha,\beta$, and $\gamma$
such that under adaptive SEP protocol perfect discrimination of the set
$\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ requires three copies of the state.

For instance, $\alpha,\beta\neq 0,\pi/2$ and $\gamma=\pi/6$ constitute such
choices. Generic analysis is referred to the supplementary material Supple .
Similar to Theorem 1, any protocol on the first copy that just updates the
prior probability distribution without eliminating state(s) does not work good
in this case too. Theorem 4 is also important from another perspective.
Mathematical characterization of local operations is extremely hard in
general. While implementation of such an operation may be done by some finite
round of LOCC protocol, its implementation may also demand infinite round of
protocol. One can also define topological closure of different such sets.
However all these different classes of local operations are strictly included
within the set of SEP protocol Chitambar14(1) . Therefore the ensembles in
Theorem 4 remain two copy indistinguishable under adaptive protocol even under
infinite round of LOCC. Arguably, these are the strongest two-qubit nonlocal
ensembles under any adaptive discrimination scheme - LOCC and/or SEP.
Considering general $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ we, indeed, find
ensembles that are perfectly two-copy distinguishable under adaptive SEP,
whereas adaptive LOCC requires three copies for perfect discrimination
(ensembles in Theorem 2 are such examples).

Figure 1: (Color online) Alice and Bob share multiple copies of an unknown
state $|\psi_{i}\rangle$ belonging to a known ensemble $\mathcal{E}_{N}$.
Availability of additional entanglement resource across the vertical (party-
cut) and/or horizontal (copy-cut) dotted lines give rise to different sets of
allowed operations performed by Alice and Bob- a) no additional entanglement
across the vertical and horizontal lines gives adaptive LOCC, b) adaptive SEP
(more generally entangling operation) requires additional entanglement across
vertical line but no entanglement across horizontal line, c) additional
entanglement across horizontal line but no entanglement across vertical line
gives non-adaptive LOCC, d) non-adaptive SEP requires entanglement across both
vertical and horizontal lines.

We are therefore left with the only possibility whether the aforesaid
ensembles are two-copy distinguishable under a non-adaptive local protocol. In
such a protocol Alice (Bob) addresses both the systems at her (his) end
simultaneously to perform some global measurement and communicates the
outcomes with the other party. Such measurement includes SEP as well as
entangled basis measurements. Important to note that, existence of an ensemble
that is distinguishable under multi-copy local non-adaptive scheme but
indistinguishable under adaptive local scheme will establish an intriguing
super-additivity behaviour for the locally accessible information of multi-
site quantum ensembles Badziag03 . Our constructions are therefore quite
interesting – either they will establish two-copy indistinguishability (under
generic protocol) or it will demonstrate super-additivity of locally
accessible information. In this regard, super-additivity phenomena of
classical capacity of a noisy quantum channel is worth mentioning Fuchs97 ;
King01 ; Schumacher01 ; King02 . There also entangled decoding on multiple
copies of noisy channels turns out to be more efficient than its multiple
single-copy use.

An ensemble indistinguishable under local non-adaptive scheme will also be
indistinguishable under adaptive LOCC. However, such an ensemble may be
perfectly distinguished with adaptive SEP. Note that, physical realization of
both the local non-adaptive scheme as well as adaptive SEP scheme require
entanglement. While in the first case entanglement across the cut between
different copies, in the later case entanglement is consumed across the cut
between parties (see Fig.1). However, at present we have no intuition for
construction of such ensembles and leave this question open for further
research. In the following we rather focus on possible application of the
present constructions.

Multiparty secret sharing.– Secret sharing is an important cryptographic
primitive Shamir79 . In $(k,m)$ secret sharing scheme an administrator wishes
to distribute $k$-bit classical message among $m$ spatially separated parties
in such a way that perfect revelation of the message requires (classical)
collaboration among all the $m$ parties. In quantum scenario, administrator
encodes $k$-bit massage in some $m$-partite state, i.e.
$i\mapsto\rho_{i}\in\mathcal{D}(\mathcal{H}^{\otimes m})$ such that
$\\{\rho_{i}\\}_{i=1}^{2^{k}}$ is a mutually orthogonal set of states
Hillery99 . Our construction turns out to be efficient for a $(2,6)$ pure
quantum protocol (encoded states are pure). The administrator encodes her
$2$-bit message into an ensemble
$\mathcal{E}_{4}\equiv\\{\ket{\psi_{i}}\\}_{i=1}^{4}$ which is
indistinguishable under two-copy adaptive LOCC, prepares three copies of the
state and distributes those among the six separated parties. In short,
$i\mapsto\ket{\psi_{i}}\mapsto\ket{\psi_{i}}_{A_{1}B_{1}}\otimes\ket{\psi_{i}}_{A_{2}B_{2}}\otimes\ket{\psi_{i}}_{A_{3}B_{3}}$.
No four parties can exactly decode the message through classical collaboration
among them Self .

In a stronger variant of the aforesaid task even the classical collaborations
among the parties are insufficient for perfect revelation of the message – the
parties need to come together to decode the information. It is also possible
to come up with such a $(1,2)$ protocol from our construction. Recall that the
result of Walgate et al. Walgate02  prohibits any such $(1,m)$ pure quantum
protocol. So, the administrator choose some pair of index $i,j$ as in
Proposition 1 and then encodes her massage into the density operators
$\sigma(\lambda):=\lambda\ket{a_{i}}\bra{a_{i}}+(1-\lambda)\ket{a_{j}}\bra{a_{j}}$
supported on the subspace $\mathbb{V}[ij]$ and its orthogonal pair
$\sigma^{\perp}(\mu)$ supported on the complementary subspace. Whenever
$\lambda,\mu\in(0,1)$, the perfect revelation of the massage demands both the
parties to come together to apply a global discrimination measurement. At this
point it may be interesting to find the values of $\lambda\leavevmode\nobreak\
\&\leavevmode\nobreak\ \mu$ that will ensure least amount of information
accessible by LOCC.

Discussion.– We have discoursed the problem of local state discrimination when
multiple copies of the unknown state are available. Our study is a major
advancement in this direction as it establishes that there exist ensembles of
pure composite orthogonal states that are not two copy distinguishable under
adaptive LOCC protocol. Under multi-copy adaptive scheme we have completely
characterize the nonlocal behavior of the orthogonal ensembles of two-qubit
Hilbert space. Such two-copy indistinguishable sets find useful applications
in multi-party secret sharing tasks.

Our study also raises a number of interesting questions. A conclusive answer
to the possible super-additivity behavior for locally accessible information
is worth deserving. A non-adaptive protocol is still missing for the example
of ensembles where adaptive SEP protocol extract more classical information
than local. Study of multi-copy state discrimination for higher dimensional
and multipartite Hilbert spaces might reveal several other interesting
features. One can consider more exotic adaptive protocols that allow to
interact back and forth with the different copies by means of non-projective
measurements. We believe that even under this more exotic protocol our two-
copy local indistinguishable ensembles will remain indistinguishable. Formal
proof of this assertion demands further research.

###### Acknowledgements.

MB acknowledges research grant of INSPIRE-faculty fellowship from the
Department of Science and Technology, Government of India. MA acknowledges
support from the CSIR project 09/093(0170)/2016- EMR-I. SSB is supported by
the National Natural Science Foundation of China through grant 11675136, the
Foundational Questions Institute through grant FQXiRFP3-1325, the Hong Kong
Research Grant Council through grant 17300918, and the ID#61466 grant from the
John Templeton Foundation, as part of the “The Quantum Information Structure
of Spacetime (QISS)" Project (qiss.fr). The opinions expressed in this
publication are those of the author(s) and do not necessarily reflect the
views of the John Templeton Foundation.

## References

  * (1) I. Chuang and M. Nielsen; Quantum Computation and Quantum Information, Cambridge University Press.
  * (2) C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters; Quantum nonlocality without entanglement, Phys. Rev. A 59, 1070 (1999).
  * (3) S. Ghosh, G. Kar, A. Roy, A. Sen(De), and U. Sen; Distinguishability of Bell States, Phys. Rev. Lett. 87, 277902 (2001).
  * (4) J. Walgate and L. Hardy; Nonlocality, Asymmetry, and Distinguishing Bipartite States, Phys. Rev. Lett. 89, 147901 (2002).
  * (5) A. Peres and W. K. Wootters; Optimal detection of quantum information, Phys. Rev. Lett. 66, 1119 (1991).
  * (6) E. Chitambar and Min-Hsiu Hsieh; Revisiting the optimal detection of quantum information, Phys. Rev. A 88, 020302(R) (2013).
  * (7) C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal; Unextendible Product Bases and Bound Entanglement, Phys. Rev. Lett. 82, 5385 (1999).
  * (8) J. Walgate, A. J. Short, L. Hardy, and V. Vedral; Local Distinguishability of Multipartite Orthogonal Quantum States, Phys. Rev. Lett. 85, 4972 (2000).
  * (9) M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki; Local Indistinguishability: More Nonlocality with Less Entanglement, Phys. Rev. Lett. 90, 047902 (2003).
  * (10) D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, B. M. Terhal; Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement, Comm. Math. Phys. 238, 379 (2003).
  * (11) S. Ghosh, G. Kar, A. Roy, and D. Sarkar; Distinguishability of maximally entangled states, Phys. Rev. A 70, 022304 (2004).
  * (12) J. Watrous; Bipartite Subspaces Having No Bases Distinguishable by Local Operations and Classical Communication, Phys. Rev. Lett. 95, 080505 (2005).
  * (13) J. Niset and N. J. Cerf; Multipartite nonlocality without entanglement in many dimensions, Phys. Rev. A 74, 052103 (2006).
  * (14) R. Duan, Y. Feng, Z. Ji, and M. Ying; Distinguishing Arbitrary Multipartite Basis Unambiguously Using Local Operations and Classical Communication, Phys. Rev. Lett. 98, 230502 (2007).
  * (15) J. Calsamiglia, J. I. de Vicente, R. Muñoz-Tapia, and E. Bagan; Local Discrimination of Mixed States, Phys. Rev. Lett. 105, 080504 (2010).
  * (16) S. Bandyopadhyay; More Nonlocality with Less Purity, Phys. Rev. Lett. 106, 210402 (2011).
  * (17) E. Chitambar, R. Duan, and Min-Hsiu Hsieh; When do Local Operations and Classical Communication Suffice for Two-Qubit State Discrimination? IEEE Trans. Inform. Theory 60, 1549 (2014).
  * (18) S. Halder; Several nonlocal sets of multipartite pure orthogonal product states, Phys. Rev. A 98, 022303 (2018).
  * (19) M. Demianowicz and R. Augusiak; From unextendible product bases to genuinely entangled subspaces, Phys. Rev. A 98, 012313 (2018).
  * (20) S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay; Strong Quantum Nonlocality without Entanglement, Phys. Rev. Lett. 122, 040403 (2019).
  * (21) S. Agrawal, S. Halder, and M. Banik; Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition, Phys. Rev. A 99, 032335 (2019).
  * (22) S. Rout, A. G. Maity, A. Mukherjee, S. Halder, and M. Banik; Genuinely nonlocal product bases: Classification and entanglement-assisted discrimination, Phys. Rev. A 100, 032321 (2019).
  * (23) S. Rout, A. G. Maity, A. Mukherjee, S. Halder, and M. Banik; Local State Discrimination and Ordering of Multipartite Entangled States, arXiv:1910.14308 [quant-ph].
  * (24) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki; Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  * (25) B. M. Terhal, D. P. DiVincenzo, and D. W. Leung; Hiding Bits in Bell States, Phys. Rev. Lett. 86, 5807 (2001).
  * (26) T. Eggeling and R. F. Werner; Hiding Classical Data in Multipartite Quantum States, Phys. Rev. Lett. 89, 097905 (2002).
  * (27) W. Matthews, S. Wehner, and A. Winter; Distinguishability of Quantum States Under Restricted Families of Measurements with an Application to Quantum Data Hiding, Commun. Math. Phys. 291, 813–843 (2009).
  * (28) M. Hillery, V. Bužek, and A. Berthiaume; Quantum secret sharing, Phys. Rev. A 59, 1829 (1999).
  * (29) D. Markham and B. C. Sanders; Graph states for quantum secret sharing, Phys. Rev. A 78, 042309 (2008).
  * (30) R. Rahaman and Matthew G. Parker; Quantum scheme for secret sharing based on local distinguishability, Phys. Rev. A 91, 022330 (2015).
  * (31) C. A. Fuchs; Just Two Nonorthogonal Quantum States, In: P. Kumar, G. M. D’Ariano, O. Hirota O. (eds) Quantum Communication, Computing, and Measurement 2. Springer, Boston, MA.
  * (32) M. F. Pusey, J. Barrett, and T. Rudolph; On the reality of the quantum state, Nature Physics 8, 475 (2012).
  * (33) S. Bandyopadhyay, M. Banik, S. S. Bhattacharya, S. Ghosh, G. Kar, A. Mukherjee, and A. Roy; Reciprocal ontological models show indeterminism of the order of quantum theory, Found Phys 47, 265 (2017).
  * (34) S. S. Bhattacharya, S. Saha, T. Guha, and M. Banik; Nonlocality without entanglement: Quantum theory and beyond, Phys. Rev. Research 2, 012068(R) (2020).
  * (35) E. Chitambar, D. Leung, L. Mancinska, M. Ozols, and A. Winter; Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask), Commun. Math. Phys. 328, 303 (2014).
  * (36) R. Duan, Y. Feng, Y. Xin, and M. Ying; Distinguishability of quantum states by separable operations, IEEE Trans. Inf. Theory 55, 1320 (2009).
  * (37) N. Yu, R. Duan, and M. Ying; Four Locally Indistinguishable Ququad-Ququad Orthogonal Maximally Entangled States, Phys. Rev. Lett. 109, 020506 (2012).
  * (38) N. Yu, R. Duan, and M. Ying; Distinguishability of Quantum States by Positive Operator-Valued Measures With Positive Partial Transpose, IEEE Trans. Inform. Theory 60, 2069 (2014).
  * (39) S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Watrous, and N. Yu; Limitations on separable measurements by convex optimization, IEEE Trans. Inf. Theory 61, 3593 (2015).
  * (40) S. M. Cohen; Class of unambiguous state discrimination problems achievable by separable measurements but impossible by local operations and classical communication, Phys. Rev. A 91, 012321 (2015).
  * (41) G. Chiribella, G. Mauro D’Ariano, and P. Perinotti; Informational derivation of quantum theory, Phys. Rev. A 84, 012311 (2011).
  * (42) H. Barnum and A. Wilce; Local Tomography and the Jordan Structure of Quantum Theory, Found Phys 44, 192 (2014).
  * (43) G. Niestegge; Local tomography and the role of the complex numbers in quantum mechanics, Proc. R. Soc. A. 476, 20200063 (2020).
  * (44) M. Hayashi; Discrimination of two channels by adaptive methods and its application to quantum system, IEEE Trans. Inf. Theory 55, 3807 (2009).
  * (45) A. W. Harrow, A. Hassidim, D. W. Leung, and J. Watrous; Adaptive versus nonadaptive strategies for quantum channel discrimination, Phys. Rev. A 81, 032339 (2010).
  * (46) S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira; Fundamental limits to quantum channel discrimination, npj Quantum Inf 5, 50 (2019).
  * (47) S. Pirandola and C. Lupo; Ultimate Precision of Adaptive Noise Estimation, Phys. Rev. Lett. 118, 100502 (2017).
  * (48) T. P. W. Cope and S. Pirandola; Adaptive estimation and discrimination of Holevo Werner channels, Quantum Meas. Quantum Metrol. 4, 44 (2017).
  * (49) See Supplemental Material.
  * (50) P. Badziag, M. Horodecki, A. Sen(De), and U. Sen; Locally Accessible Information: How Much Can the Parties Gain by Cooperating? Phys. Rev. Lett. 91, 117901 (2003).
  * (51) C. A. Fuchs; Nonorthogonal Quantum States Maximize Classical Information Capacity, Phys. Rev. Lett. 79, 1162 (1997).
  * (52) C. King and M. B. Ruskai; Capacity of quantum channels using product measurements, J. Math. Phys. 42, 87 (2001).
  * (53) B. Schumacher and M. D. Westmoreland; Optimal signal ensembles, Phys. Rev. A 63, 022308 (2001).
  * (54) C. King, M. Nathanson, and M. B. Ruskai; Qubit Channels Can Require More Than Two Inputs to Achieve Capacity, Phys. Rev. Lett. 88, 057901 (2002).
  * (55) A. Shamir; How to share a secret, Commun. ACM 22, 612 (1979).
  * (56) Collaboration among certain groups of the four parties can reveal at-most $1$-bit data which is only half of the desired message. It is also not hard to argue that collaboration among any five parties cannot reveal $2$-bit message exactly. It should be noted that security of the protocol is assured under the adaptive protocol considered in Definition 1. A more generic protocol might further demand limitation on the LOCC operations among different groups of the parties.
  * (57) S. Bandyopadhyay; More Nonlocality with Less Purity, Phys. Rev. Lett. 106, 210402 (2011).
  * (58) W. K. Wootters; Entanglement of Formation of an Arbitrary State of Two Qubits, Phys. Rev. Lett. 80, 2245 (1998).
  * (59) P. Horodecki, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal; Rank two bipartite bound entangled states do not exist, Theor. Comput. Sci. 292, 589 (2003).

Supplemental Material

##  I Proof of Corollary 1

###### Proof.

Ensembles containing no entangled state are in category (i) of Theorem 1.
There can not be any orthogonal basis in $(\mathbb{C}^{2})^{\otimes 2}$ that
contain only one entangled state Bennett99(1) . For the ensembles containing
two entangled states the elimination round at the first copy level has to be
done in a way so that one entangled states gets eliminated. Note that this can
always be done (see Theorem 1 of Ref. Bandyopadhyay11 ) which consequently
leaves these ensembles in category (ii) of Theorem 1. ∎

##  II Analysis of the general case: $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$

Recall the set of four orthogonal states
$\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ as introduced in the main text. Among
them the states $\ket{a_{1}}\equiv\ket{\phi^{-}(\alpha)}$ and
$\ket{a_{2}}\equiv\ket{\psi^{-}(\beta)}$ are entangled
$\forall\leavevmode\nobreak\ \alpha,\beta\in(0,\frac{\pi}{2})$. Consider the
state $\ket{a_{3}}$ given by,

$\displaystyle\ket{a_{3}}$
$\displaystyle=S_{\gamma}\ket{\phi^{+}_{\alpha}}+C_{\gamma}\ket{\psi^{+}_{\beta}}$
(1)
$\displaystyle=\sin\gamma\sin\alpha\ket{00}+\sin\gamma\cos\alpha\ket{11}+\cos\gamma\sin\beta\ket{01}+\cos\gamma\cos\beta\ket{10}$
$\displaystyle=\ket{0}\otimes(\sin\gamma\sin\alpha\ket{0}+\cos\gamma\sin\beta\ket{1})+\ket{1}\otimes(\sin\gamma\cos\alpha\ket{1}+\cos\gamma\cos\beta\ket{0})$
$\displaystyle=\mathcal{N}(\eta_{0})\ket{0\eta_{0}}+\mathcal{N}(\eta_{1})\ket{1\eta_{1}},$

where,
$\mathcal{N}(\eta_{0}):=\sqrt{\sin^{2}\gamma\sin^{2}\alpha+\cos^{2}\gamma\sin^{2}\beta},\leavevmode\nobreak\
\mathcal{N}(\eta_{1}):=\sqrt{\sin^{2}\gamma\cos^{2}\alpha+\cos^{2}\gamma\cos^{2}\beta}$
and $\ket{\eta_{0}},\leavevmode\nobreak\ \ket{\eta_{1}}$ are the normalized
states. The state in Eq.(1) is separable, if and only if
$\ket{\eta_{0}}\equiv\ket{\eta_{1}}$ up to a global phase, which implies

$\displaystyle\frac{\sin\gamma\sin\alpha}{\sqrt{\sin^{2}\gamma\sin^{2}\alpha+\cos^{2}\gamma\sin^{2}\beta}}$
$\displaystyle=$
$\displaystyle\frac{\cos\gamma\cos\beta}{\sqrt{\sin^{2}\gamma\cos^{2}\alpha+\cos^{2}\gamma\cos^{2}\beta}},$
$\displaystyle\implies\tan^{2}\gamma$ $\displaystyle=$
$\displaystyle\frac{\sin 2\beta}{\sin 2\alpha}.$ (2)

Similarly, separability of
$\ket{a_{4}}=C_{\gamma}\ket{\phi^{+}_{\alpha}}-S_{\gamma}\ket{\psi^{+}_{\beta}}$
yields,

$\tan^{2}\gamma=\frac{\sin 2\alpha}{\sin 2\beta}.$ (3)

Notably, for $\gamma=\frac{\pi}{4}$ the conditions $\alpha=\beta$ or,
$\alpha+\beta=\frac{\pi}{2}$ makes both of $\ket{a_{3}}$ and $\ket{a_{4}}$
separable. For generic $\gamma\in[0,\frac{\pi}{2}]$ and
$\alpha,\beta\in(0,\frac{\pi}{2})$, separability of the states
$\ket{a_{3}}\leavevmode\nobreak\ \&\leavevmode\nobreak\ \ket{a_{4}}$ is
depicted in the Fig.2.

Figure 2: (Color online) Parametric plot for $\\{\alpha,\beta,\gamma\\}$. (a)
Every tuple of $(\alpha,\beta,\gamma)$ lying on the curved surface makes
$\ket{a_{3}}$ separable. (b) Every tuple of $(\alpha,\beta,\gamma)$ lying on
the curved surface makes $\ket{a_{4}}$ separable.

Further in this section, we will calculate concurrence Wootters1998  of the
states $\ket{a_{i}}\in\mathcal{A}_{\gamma}^{[\alpha,\beta]}$. Any two-qubit
pure state $\ket{\psi}$ can be written as
$\ket{\psi}=(\mathbf{I}\otimes\Psi)\ket{\phi^{+}}$ where
$\ket{\phi^{+}}:=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$ and $\Psi$ is a
$2\times 2$ matrix. The concurrence for $\ket{\psi}$ is then given by
$C(\psi)=|\text{det}(\Psi)|$. Representing the states as
$\ket{a_{i}}=(\mathbf{I}\otimes\mathbf{a_{i}})\ket{\phi^{+}}$ a simple
numerical exercise yields,

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{a_{1}}:=\begin{pmatrix}\sqrt{2}\cos\alpha&0\\\
0&-\sqrt{2}\sin\alpha\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{a_{2}}:=\begin{pmatrix}0&-\sqrt{2}\sin\beta\\\
\sqrt{2}\cos\beta&0\end{pmatrix},$ (4)
$\displaystyle\mathbf{a_{3}}:=\begin{pmatrix}\sqrt{2}\sin\gamma\sin\alpha&\sqrt{2}\cos\gamma\cos\beta\\\
\sqrt{2}\cos\gamma\sin\beta&\sqrt{2}\sin\gamma\cos\alpha\end{pmatrix},\leavevmode\nobreak\
\leavevmode\nobreak\
\mathbf{a_{4}}:=\begin{pmatrix}\sqrt{2}\cos\gamma\sin\alpha&-\sqrt{2}\sin\gamma\sin\beta\\\
-\sqrt{2}\sin\gamma\cos\beta&\sqrt{2}\cos\gamma\cos\alpha\end{pmatrix}.$

Thus we have,

$\displaystyle C(a_{1})=|\det(\mathbf{a_{1}})|=\sin 2\alpha,$ (5)
$\displaystyle C(a_{2})=|\det(\mathbf{a_{2}})|=\sin 2\beta,$ $\displaystyle
C(a_{3})=|\det(\mathbf{a_{3}})|=|\cos^{2}\gamma\sin 2\beta-\sin^{2}\gamma\sin
2\alpha|,$ $\displaystyle C(a_{4})=|\det(\mathbf{a_{4}})|=|\cos^{2}\gamma\sin
2\alpha-\sin^{2}\gamma\sin 2\beta|.$

Evidently, for $\alpha,\beta\in(0,\frac{\pi}{2})$ the concurrences $C(a_{1})$
and $C(a_{2})$ are both positive. However, depending upon the triplet
$(\alpha,\beta,\gamma)$ we can consider following four different regions for
Eq.(5) (see Fig3 for better clarification):

  * Region I ($\mathcal{R}_{I}$): $\frac{\sin 2\beta}{\sin 2\alpha}\geq\tan^{2}\gamma\geq\frac{\sin 2\alpha}{\sin 2\beta}$. Here, $C(a_{3})=\cos^{2}\gamma\sin 2\beta-\sin^{2}\gamma\sin 2\alpha$ and $C(a_{4})=\sin^{2}\gamma\sin 2\beta-\cos^{2}\gamma\sin 2\alpha$.

  * Region II ($\mathcal{R}_{II}$): $\frac{\sin 2\alpha}{\sin 2\beta}\geq\tan^{2}\gamma\geq\frac{\sin 2\beta}{\sin 2\alpha}$. Here, $C(a_{3})=\sin^{2}\gamma\sin 2\alpha-\cos^{2}\gamma\sin 2\beta$ and $C(a_{4})=\cos^{2}\gamma\sin 2\alpha-\sin^{2}\gamma\sin 2\beta$.

  * Region III ($\mathcal{R}_{III}$): $\tan^{2}\gamma\geq\max\left\\{\frac{\sin 2\alpha}{\sin 2\beta},\frac{\sin 2\beta}{\sin 2\alpha}\right\\}$. Here, $C(a_{3})=\sin^{2}\gamma\sin 2\alpha-\cos^{2}\gamma\sin 2\beta$ and $C(a_{4})=\sin^{2}\gamma\sin 2\beta-\cos^{2}\gamma\sin 2\alpha$.

  * Region IV ($\mathcal{R}_{IV}$): $\tan^{2}\gamma\leq\min\left\\{\frac{\sin 2\alpha}{\sin 2\beta},\frac{\sin 2\beta}{\sin 2\alpha}\right\\}$. Here, $C(a_{3})=\cos^{2}\gamma\sin 2\beta-\sin^{2}\gamma\sin 2\alpha$ and $C(a_{4})=\cos^{2}\gamma\sin 2\alpha-\sin^{2}\gamma\sin 2\beta$.

Notably, the saturation of either inequality implies the corresponding state
is product, as mentioned earlier.

Figure 3: (Color online) All the four regions depending upon
$\\{\alpha,\beta,\gamma\\}$. Green stands for the region $\mathcal{R}_{I}$,
blue for $\mathcal{R}_{II}$, yellow for $\mathcal{R}_{III}$, and red for
$\mathcal{R}_{IV}$.

##  III Proof of the Theorem(s) & Proposition(s)

###  III.1 Proof of Proposition 1

To prove the Proposition 1 we need to check NPT-ness of the projectors
$\mathbb{P}_{ij}=\ket{a_{i}}\bra{a_{i}}+\ket{a_{j}}\bra{a_{j}},\forall\leavevmode\nobreak\
i\neq j$. In the following we do this checking for general consideration, i.e.
for the set $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$. For instance, for
arbitrary $\gamma$, eigenvalues for $\mathbb{P}_{12}^{\mathsf{T}_{p}}$ and
$\mathbb{P}_{34}^{\mathsf{T}_{p}}$ turn out to be,

$\displaystyle e_{1}=\frac{1}{4}\left(2-\sqrt{2}\left[2+\cos 4\alpha-\cos
4\beta\right]^{\frac{1}{2}}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
e_{2}=\frac{1}{4}\left(2+\sqrt{2}\left[2+\cos 4\alpha-\cos
4\beta\right]^{\frac{1}{2}}\right),$ (6a) $\displaystyle
e_{3}=\frac{1}{4}\left(2-\sqrt{2}\left[2-\cos 4\alpha+\cos
4\beta\right]^{\frac{1}{2}}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
e_{4}=\frac{1}{4}\left(2+\sqrt{2}\left[2-\cos 4\alpha+\cos
4\beta\right]^{\frac{1}{2}}\right).$ (6b)

Here $\mathsf{T}_{p}$ denotes partial transposition. Clearly,
$e_{2}\leavevmode\nobreak\ \&\leavevmode\nobreak\ e_{4}$ are always positive.
However, almost everywhere in parameters space $(\alpha,\beta,\gamma)$ one of
the eigenvalues either $e_{1}$ or $e_{3}$ is negative since $e_{1}e_{3}<0$
unless $\cos 4\alpha=\cos 4\beta$ (see Fig.4) and hence ensures NPT-ness of
the projectors $\mathbb{P}_{12}$ and $\mathbb{P}_{34}$.

Figure 4: (color online) Green corresponds to the zero ($z=0$) surface. (a)
Eigenvalues $e_{1}$ (blue surface) and $e_{3}$ (yellow surface) of Eq.(6).
Almost everywhere one of $e_{1},e_{3}$ is negative. (b) Eigenvalue $e_{3}$ of
Eq.(7) (yellow surface).

For the other choices of the sub-spaces the generic expression for the
eigenvalues of the partial transposition of the projectors onto them is much
complicated. For instance, $\gamma=\pi/4$, eigenvalues of
$\mathbb{P}_{13}^{\mathsf{T}_{p}},\leavevmode\nobreak\
\mathbb{P}_{14}^{\mathsf{T}_{p}},,\leavevmode\nobreak\
\mathbb{P}_{23}^{\mathsf{T}_{p}}$, and $\mathbb{P}_{24}^{\mathsf{T}_{p}}$ are
given by

$\displaystyle e_{1}$
$\displaystyle=\frac{1}{8}\left(4-\left(16-2\sqrt{2}\left[(\sin(2\alpha)+\sin(2\beta))^{2}(18-12\sin(2\alpha)\sin(2\beta)-\cos(4\alpha)-\cos(4\beta))\right]^{\frac{1}{2}}\right)^{\frac{1}{2}}\right),$
(7a) $\displaystyle e_{2}$
$\displaystyle=\frac{1}{8}\left(4+\left(16-2\sqrt{2}\left[(\sin(2\alpha)+\sin(2\beta))^{2}(18-12\sin(2\alpha)\sin(2\beta)-\cos(4\alpha)-\cos(4\beta))\right]^{\frac{1}{2}}\right)^{\frac{1}{2}}\right),$
(7b) $\displaystyle e_{3}$
$\displaystyle=\frac{1}{8}\left(4-\left(16+2\sqrt{2}\left[(\sin(2\alpha)+\sin(2\beta))^{2}(18-12\sin(2\alpha)\sin(2\beta)-\cos(4\alpha)-\cos(4\beta))\right]^{\frac{1}{2}}\right)^{\frac{1}{2}}\right),$
(7c) $\displaystyle e_{4}$
$\displaystyle=\frac{1}{8}\left(4+\left(16+2\sqrt{2}\left[(\sin(2\alpha)+\sin(2\beta))^{2}(18-12\sin(2\alpha)\sin(2\beta)-\cos(4\alpha)-\cos(4\beta))\right]^{\frac{1}{2}}\right)^{\frac{1}{2}}\right).$
(7d)

Eigenvalue $e_{3}$ is negative for almost all values of $\alpha,\beta$ as
depicted in Fig.4 ($e_{3}<0$ unless $\sin 2\alpha=-\sin 2\beta$). Furthermore,
numerical verification (in MATHEMATICA) ensures NPT-ness of these projectors
almost everywhere in the parameter space $(\alpha,\beta,\gamma)$.

###  III.2 Proof of Theorem 3

###### Proof.

An orthogonal basis $\mathcal{B}^{[3E1P]}$ containing three entangled states
and one product state cannot be in category (i) of the Theorem 1. Furthermore
$\mathcal{B}^{[3E1P]}$ cannot be in category (ii) of the Theorem 1, as any
$1\leavevmode\nobreak\ vs\leavevmode\nobreak\ 3$ grouping ends up with atleast
two entangled states in the three-state group. Finally, $\mathcal{B}^{[3E1P]}$
cannot be in category (iii) of the Theorem 1 too. Any $2\leavevmode\nobreak\
vs\leavevmode\nobreak\ 2$ grouping ends up with a group containing one product
and one entangled state making the projector on this spanning subspace
entangled (see Theorem 2 of Ref. Horodecki03(1) ). This completes the proof. ∎

For the set $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$, if we choose
$\gamma=\tan^{-1}\left(\sqrt{\frac{\sin 2\alpha}{\sin 2\beta}}\right)$ or
$\gamma=\tan^{-1}\left(\sqrt{\frac{\sin 2\beta}{\sin 2\alpha}}\right)$ and
$\alpha,\beta\in(0,\pi/2)$, then the set contains exactly one product state
and three entangled states. These constitute example of ensembles stated in
Theorem 3.

##  IV (In)distinguishability under adaptive SEP

###  IV.1 Proof of Theorem 4

###### Proof.

An orthonormal basis of $(\mathbb{C}^{2})^{\otimes 2}$ is single-copy
distinguishable under SEP if and only if all the basis states are product
states Duan09  (see also Proposition 3 of Ref.Chitambar14 ). For a basis
containing entangled state(s), an adaptive discrimination protocol of SEP on
the first copy will lead to one of the following two conclusions: (i) the
state is from some group $\mathcal{G}_{ij}$ or from its complement group
$\mathcal{G}^{\mathsf{C}}_{ij}$, (ii) it is from some group $\mathcal{G}_{l}$
or from its complement group $\mathcal{G}^{\mathsf{C}}_{l}$;
$i,j,l\in\\{1,2,3,4\\}\leavevmode\nobreak\ \&\leavevmode\nobreak\ i\neq j$.
For the first case the protocol will discriminate the corresponding density
operator $\rho_{ij}$ and $\rho_{ij}^{\perp}$. The condition for perfect
discrimination of two such rank two density operators under SEP is same as
LOCC, i.e. they can be distinguished under SEP if and only if the projectors
onto the supports of each of the density operators are separable Chitambar14 .
Proposition 1, however, restrains this possibility. In the second case the
protocol will sometime lead to the conclusion that the state is in the group
$\mathcal{G}^{\mathsf{C}}_{l}$. In that case, perfect discrimination needs to
be done from the second copy. An ensemble $\\{\ket{\psi_{i}}\\}_{i=1}^{3}$ of
three orthogonal states of $(\mathbb{C}^{2})^{\otimes 2}$ is perfectly
distinguishable by separable operations if and only if: (a)
$\psi_{k}\phi^{-1}$ has two anti-parallel eigenvalues for each entangled state
$\psi_{i}$, and (b) $\sum_{i=1}^{3}C(\psi_{i})=C(\phi)$ Duan09 . Here
$\ket{\phi}$ is the state orthogonal to the subspace spanned by
$\\{\ket{\psi_{i}}\\}_{i=1}^{3}$, and $C(\psi)=|\det(\psi)|$ denotes the
concurrence of the state
$\ket{\psi}:=(\mathbb{I}\otimes\psi)\ket{\phi^{+}_{\frac{\pi}{4}}}$, with
$\psi$ being a $2\times 2$ matrix. In the following we check these conditions.

To complete the argument, let us consider that after the protocol on first
copy, none of the states is eliminated, rather the knowledge regarding the
ensemble gets updated, i.e. the uniform ensemble
$\mathcal{E}_{4}\equiv\\{1/4,\ket{\psi_{i}}\\}$ ends up into some non-uniform
ensemble
$\mathcal{E}^{\prime}_{4}\equiv\\{p_{i},\ket{\psi_{i}}\leavevmode\nobreak\
|\leavevmode\nobreak\ p_{i}\neq 0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\\}$. In such a case, the ensemble
$\mathcal{E}^{\prime}$ must be discriminated from the second copy of the
state. However this cannot be done as the argument presented above is
independent of the prior probability distributions. This completes the proof.
∎

We consider the generic ensemble $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ and
analyze its distinguishability under adaptive SEP.

Region I: Straightforward calculation yields
$\mathbf{a_{k}}\mathbf{a_{2}}^{-1}$ has anti-parallel eigenvalues for all
$k\in\\{1,3,4\\}$. Furthermore,

$C(a_{1})+C(a_{3})+C(a_{4})=\sin 2\beta=C(a_{2}).$

Thus in the adaptive scheme, from the first copy it is determined whether the
given state is $\ket{a_{2}}$ or not. In fact this can be done by LOCC
Bandyopadhyay11 . From the second copy the rest three state can be perfectly
discriminated under SEP Duan09 .

Region II: This case is same as Region I with $\ket{a_{2}}$ interchanged with
$\ket{a_{1}}$.

Region III: Note that,

$\displaystyle C(a_{1})=\sum_{i\neq 1}C(a_{i})$
$\displaystyle\implies\tan^{2}\gamma=\frac{\sin 2\alpha}{\sin 2\beta},$ (8a)
$\displaystyle C(a_{2})=\sum_{i\neq 2}C(a_{i})$
$\displaystyle\implies\tan^{2}\gamma=\frac{\sin 2\beta}{\sin 2\alpha},$ (8b)
$\displaystyle C(a_{3})=\sum_{i\neq 3}C(a_{i})$
$\displaystyle\implies\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \beta=0,$ (8c) $\displaystyle C(a_{4})=\sum_{i\neq
4}C(a_{i})$ $\displaystyle\implies\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \alpha=0.$ (8d)

None of the above conditions can be satisfied in $\mathcal{R}_{III}$ with all
of states in $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ entangled. Hence, none of
the possible $1:3$ partition for the set
$\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ can be discriminated perfectly under
adaptive SEP and therefore under adaptive LOCC.

Region IV:

$\displaystyle C(a_{1})=\sum_{i\neq 1}C(a_{i})$
$\displaystyle\implies\tan^{2}\gamma=\frac{\sin 2\beta}{\sin 2\alpha},$ (9a)
$\displaystyle C(a_{2})=\sum_{i\neq 2}C(a_{i})$
$\displaystyle\implies\tan^{2}\gamma=\frac{\sin 2\alpha}{\sin 2\beta},$ (9b)
$\displaystyle C(a_{3})=\sum_{i\neq 3}C(a_{i})$
$\displaystyle\implies\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \alpha=0,$ (9c) $\displaystyle C(a_{4})=\sum_{i\neq
4}C(a_{i})$ $\displaystyle\implies\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \beta=0.$ (9d)

These imply impossibility of discriminating the set of the entangled states
ensemble $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ in $\mathcal{R}_{IV}$ under
adaptive SEP.

Figure 5: (color online) (a) The blue corresponds to
$\mathcal{R}_{III}(\alpha,\beta)\subset\mathcal{R}_{III}$ where the ensemble
$\mathcal{A}_{\gamma^{\star}}^{[\alpha,\beta]}$ can be perfectly distinguished
under adaptive SEP. (b) Similar region
$\mathcal{R}_{IV}(\alpha,\beta)\subset\mathcal{R}_{IV}$.

The parametric class $\mathcal{A}_{\gamma}^{[\alpha,\beta]}$ also possess some
interesting consequences. Note that, in $\mathcal{R}_{I}$ and
$\mathcal{R}_{II}$ no adaptive LOCC scheme can accomplish the discrimination
task perfectly while adaptive SEP can do so. For
$\gamma=\gamma^{\star}=\tan^{-1}\left(\sqrt{\frac{\sin 2\alpha}{\sin
2\beta}}\right)$, the state $\ket{a_{4}}$ is a product state. Now for some
values of $(\alpha,\beta)$ there will be some region in $\mathcal{R}_{III}$,
let denote this region by $\mathcal{R}_{III}(\alpha,\beta)$, where the set
$\mathcal{A}_{\gamma^{\star}}^{[\alpha,\beta]}$ can be discriminated under
adaptive SEP. If can be checked that $C(a_{1})=\sum_{i\neq 1}C(a_{i})$ and
also $\forall k\in\\{2,3\\},\leavevmode\nobreak\
\mathbf{a_{k}}\mathbf{a_{1}}^{-1}$ has two anti-parallel eigenvalues in
$\mathcal{R}_{III}$. For same $\gamma^{\star}$ in the region
$\mathcal{R}^{\mathsf{C}}_{III}(\alpha,\beta)$ complimentary to
$\mathcal{R}_{III}(\alpha,\beta)$ the set
$\mathcal{A}_{\gamma^{\star}}^{[\alpha,\beta]}$ can not be discriminated under
adaptive SEP (see Fig. 5). A similar analysis can be done in
$\mathcal{R}_{IV}$ and also for $\gamma=\tan^{-1}\left(\sqrt{\frac{\sin
2\beta}{\sin 2\alpha}}\right)$.