Necessary Condition for Local Distinguishability of Maximally Entangled States: Beyond Orthogonality Preservation

# Necessary Condition for Local Distinguishability of Maximally Entangled States: Beyond Orthogonality Preservation

Tanmay Singal Department of Applied Mathematics, Hanyang University (ERICA), 55 Hanyangdaehak-ro, Ansan, Gyeonggi-do 426-791, Korea    Ramij Rahaman Department of Mathematics, University of Allahabad, Allahabad 211002, U.P., India    Sibasish Ghosh Optics & Quantum Information Group, The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai, 600 113, India    Guruprasad Kar Physics & Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata - 700108, India
###### Abstract

The (im)possibility of local distinguishability of orthogonal multipartite quantum states still remains an intriguing question. Beyond , the problem remains unsolved even for maximally entangled states (MES). So far, the only known condition for the local distinguishability of states is the well-known orthogonality preservation (OP). Using an upper bound on the locally accessible information for bipartite states, we derive a very simple necessary condition for any set of pairwise orthogonal MES in to be perfectly locally distinguishable. This condition is seen to be stronger than the OP condition. This is particularly so for any set of number of pairwise orthogonal MES in . When testing this condition for the local distinguishability of all sets of four generalized Bell states in , we find that it is not only necessary but also sufficient to determine their local distinguishability. This demonstrates that the aforementioned upper-bound may play a significant role in the general scenario of local distinguishability of bipartite states.

local distinguishability, LOCC, maximally entangled states, locally accessible information
###### pacs:
03.67.Hk, 03.67.Mn

## I Introduction

Various quantum information processing tasks are often restricted by local operations and classical communication (LOCC) protocols. Such LOCC protocols are natural requirements when two or more physically separated parties have to accomplish a quantum information processing task. Local discrimination of pairwise orthogonal quantum states is one such task. This topic in quantum information theory has received considerable attention in recent years to understand more deeply the role of entanglement and non-locality in quantum information processing.

When restricted to LOCC, two distant parties Alice and Bob generally cannot distinguish among bipartite states as efficiently as they can with global operations. This is also true when the states are pairwise orthogonal. As is well-known, pairwise orthogonal states can be perfectly distinguished by global operations. Such a projective measurement is generally a global operation, and cannot be achieved through LOCC protocols. Hence generally, pairwise orthogonal bipartite states aren’t perfectly distinguishable by LOCC.

Walgate et al. W00 () showed that by using LOCC only, any two pure orthogonal multipartite states can be perfectly distinguished. The local indistinguishability of pairwise orthogonal multipartite states is a signature of the non-locality expressed by these states. Since entanglement is deeply connected with non-locality one would be well placed to assume that pairwise orthogonal product states would be perfectly distinguishable by LOCC. This assumption, however, is wrong, for instance, Bennett et al. BD99 () showed that there exist a set of nine unentangled pure orthogonal states in which cannot be perfectly distinguished by LOCC. Therefore, the local distinguishability problem is not so simple, particularly when the number of states is more than two.

Fan F04 () showed that when is a prime number and is a positive integer such that , then any number of mutually orthogonal generalized Bell states in (described in the teleportation paper of Bennet et al B93 () and defined in definition (1) of the present paper) are perfectly distinguishable by LOCC itself. The question of perfect local discrimination of pairwise orthogonal generalized Bell states in was later raised by Ghosh et al. G04 () for general . They showed that if , no set of number of generalized Bell states in can be perfectly distinguished by LOCC. In the context of general maximally entangled states (MES) in , it is known that no set of pairwise orthogonal MES in are perfectly distinguishable by LOCC N05 (). Moreoever, as a general result, it has been shown in N05 () that any three pairwise orthogonal MES in are perfectly distinguishable by LOCC. Yu et al. Y12 () provided the first example of a set of four pairwise orthogonal MES in (of the form , where ) which are not perfectly distinguisable by LOCC. In fact, they showed that these four states are not distinguisable when the POVMs are restricted to PPT-POVM type, i.e., each effect of the POVM is PPT. PPT-POVMs are more general than LOCC. Imperfect distinguishability of these states was later discussed by Cosentino C13 (). In N13 (), Nathanson gives an example of three pairwise orthogonal MES in which are not perfectly distinguishable by one-way LOCC, but are so by two-way LOCC.

In most of the aforementioned results, the sets of states exhibited a certain symmetry which was used to conclude whether the states were locally distinguishable or not. For general sets of orthogonal bipartite states, the only known condition for their local distinguishability is the existence of orthogonality preserving (OP) measurements on Alice’s and Bob’s sides. A measurement is OP if, for any measurement outcome, the corresponding post-measurement selected states remain pairwise orthogonal. The existence of OP measurements is a necessary (but not sufficient) condition for local distinguishability because for the post-measurement selected states to remain distinguishable it is required that they be pairwise orthogonal.

Locally accessible information of a set of states is the amount of classical information one can retrieve from the states using LOCC. For any quantum states to be locally distinguishable, it is necessary that the locally accessible information of these states should be at least bits. When performing a measurement on the states, it is possible that the locally accessible information of the post-measurement selected states decreases below bits. In such a case, the states become locally indistinguishable. Thus, the condition to maintain the locally accessible information to at least bits, is a constraint on the measurements, just like OP is a constraint on the measurements. In fact this condition is at least as strong as the OP condition because it subsumes the OP condition.

In this paper we use the Holevo-like upper bound on locally accessible information B03 () to give a necessary condition for the local distinguishability of MES. In the LOCC protocol for the local distinguishability of MES, when this upper bound (for the post-measurement selected states) becomes smaller than bits, then said states are locally indistinguishable. To show that our necessary condition is indeed stronger than OP, we show that there exist some sets of four generalized Bell states in , for which the OP condition isn’t strong enough to conclusively determined the states’ local indistinguishability whereas our condition is strong enough to do so. We also analyze the local (in)distinguishability of all sets of four generalized Bell states in . For this, we first partition such distinct sets into equivalence classes, where all sets within an equivalence class are related to each other by a local unitary transformation (see Appendix A for the details). We represent each equivalence class by a constituent set of (generalized Bell) states. We then use our necessary condition to isolate those sets which fail the necessary condition test which means that these sets of states are locally indistinguishable. Of the total , such sets are in number. Surprisingly, we find that states in each of the remaining sets are distinguishable by one-way LOCC. This shows that our condition is not only necessary but also sufficient to determine the local distinguishability of four generalized Bell states in . This signifies the role of the Holevo-like upper upper for locally accessible information in the context of local distinguishability of quantum states.

The paper is divided into the following sections: in section II we obtain the necessary condition for local distinguishability, in section III we provide an example of how to use the necessary condition. This also demonstrates that the OP condition is weaker than the necessary condition we derived in section II. Also, after the example, we apply the necessary condition for the local distinguishability of all sets of four generalized Bell states in . Section IV concludes the paper with future directions. In Appendix A, describe how to find out the equivalence classes of sets of four generalized Bell states in . In Appendix B we list the LOCC protocols used to distinguish sets of four generalized Bell states (in ), which satisfy the necessary condition.

## Ii Necessary Condition for Local Distinguishability of Ensemble of Maximally Entangled States

Consider a set of pairwise orthogonal MES , , , . Let Alice control one subsystem and Bob the other. Let and be the -th reduced state on Alice’s subsystem and Bob’s subsystem respectively. Let Alice start the LOCC protocol with some measurement, whose Kraus operators are , where and , where is the identity operator acting on Alice’s subsystem. Let the measurement yield the -th outcome. Thus the post-measurement state is given by

 |ψi⟩⟶|ψi,α⟩=Kα⊗1B√⟨ψi|K†αKα⊗1B|ψi⟩|ψi⟩, (1)

for all , and where is the identity operator acting on Bob’s subsystem. Let , and be the post measurement reduced states (PMRS) on Alice’s and Bob’s sides respectively. Then the average PMRS on Alice’s and Bob’s sides are and , respectively, where the factor denotes the probability which with each state appears in the ensemble.

###### Lemma 1.

If Alice starts the measurement protocol to distinguish MES by LOCC, the post measurement reduced states (PMRS) on her side are completely indistinguishable.

###### Proof.

Since are MES, the corresponding reduced states on Alice’s subsystem are maximally mixed, i.e., . As a result of the measurement, the states on Alice’s subsystem transform as , . This implies that (even) after the first measurement, the PMRS on Alice’s side are completely indistinguishable. ∎

For the post-measurement joint states to still be distinguishable, the indistinguishability of PMRS on Alice’s side imposes constraints on the average PMRS on Bob’s side. This is made clear in theorem 1.

###### Theorem 1.

If the PMRS on Alice’s side are completely indistinguishable, the von Neuman entropy of the average PMRS on Bob’s side has to be at least for the states to be perfectly distinguishable by LOCC.

###### Proof.

The Holevo-like upper bound for the locally accessible information of the set of states is given by B03 ()

 ILOCCacc≤ S(ρ(A)α)+S(ρ(B)α) −Max{1mm∑i=1S(ρ(X)i,α):X=A,B}. (2)

Since, the post-measurement states are all pure, we have the following:

 S(ρ(A)i,α)=S(ρ(B)i,α),∀1≤i≤m.

Also, that Alice’s PMRS are completely indistinguishable (Lemma 1) implies that , . This implies that . Since we need to distinguish between different states, we require that be at least bits. Then the aforementioned inequality tells us that has to be at least bits. ∎

With respect to the standard ONB of Alice’s system, every MES from the shared ensemble can be expressed as

 |ψi⟩=1√dd∑j=1|j⟩A|b(i)j⟩B, (3)

where is an ONB for Bob’s system for each .

The -th PMRS on Bob’s side is then given by

 ρ(B)i,α =1Tr(K†αKα)d∑j,k=1⟨j|K†αKα|k⟩|b(i)k⟩⟨b(i)j| =U(B)iKTαK∗αTr(KTαK∗α)U(B)i†, (4)

where are local unitaries on Bob’s side, such that , for , where , and where are operators on Bob’s system, whose matrix elements with respect to the ONB are the same as the complex conjugate of matrix elements of Alice’s POVM effect with respect to the ONB . The average PMRS corresponding to the set on Bob’s side is thus given by

 ρ(B)α=m∑i=11mU(B)iKTαK∗αTr(KTαK∗α)U(B)i†. (5)

We require that satisfies theorem (1). This requirement puts a constraint on Alice’s starting measurement.

We already know one constraint on Alice’s starting measurement, i.e., it should be OP. Hence whenever ,

 ⟨ψi|K†αKα⊗1|ψj⟩=0. (6)

It is easy to see that condition (6) should be subsumed in the requirement that should satisfy theorem (1).

Consider the special case when .

###### Corollary 2.

If in Theorem (1), then the average PMRS on Bob’s side has to be maximally mixed.

###### Proof.

When , we require of information to distinguish between states. The maximal value that can take is and it can take this value only when is maximally mixed. ∎

Thus, requiring that be at least implies that has to be a maximally mixed state, i.e., we require that

 d∑i=11dU(B)iKTαK∗αTr(KTαK∗α)U(B)i†=1d1d. (7)

The matrix equation (7) is linear in the matrix elements of , which are also the unknowns of the equation. Hence, we solve equation (7) to obtain the solution space for the matrix elements of .

Note that before any measurement, for the states is bits. The solution of condition (7) provides us those , for which the of the post-measurement states is also bits. Hence, if the only solution of equation (7) is that

 K†αKα∝1A,

then that implies that any non-trivial measurement by Alice will decrease of the post-measurement states lower than bits, which implies that the post-measurements states aren’t locally distinguishable anymore. Thus the necessary condition for local indistinguishability is given as follows.

R (Necessary Condition): If the solution of the condition (7) is that , the states are indistinguishable by LOCC. If not, then the states may still be distinguishable by LOCC.

## Iii Perfect Local Distinguishability of Four Generalized Bell States in C4⊗C4

The necessary condition, R (7) has to be tested for protocols initiated by both Alice and Bob, separately. Consider the case of generalized Bell states.

###### Definition 1.

Generalized Bell states are MES in of the form

 |ψ(d)nm⟩≡1√dd−1∑j=0e2πijnd|j⟩A|j⊕dm⟩B, (8)

where and where is an ONB for Alice’s subsystem and is an ONB for Bob’s subsystem. Here .

Note that , .

When , sets of four generalized Bell states can be partitioned into distinct equivalence classes, where all sets in the same equivalence class differ from each other by action of a local unitary (see Appendix A for details). Thus, if states in any one set in an equivalence class are locally (in)distinguishable, then so are the states of any other set in the same equivalence class. We tested the condition R (7) on these equivalence classes, and found that of them fail the necessary condition, and are hence locally indistinguishable. An explicit proof of one such a set is given in the following example 1.

###### Example 1.

The states , , , are locally indistinguishable.

###### Proof.

In reference S11 () (example 1, p 6), it has already been shown that the given set of states are indistinguishable by one-way LOCC when only projective measurements are used in the LOCC protocol. Here we will generalize the result for all possible LOCC protocols. Also, we show that the condition (7) is stronger than the OP condition (6).

Let Alice commense the protocol by applying a measurement, whose Kraus operators are on her subsystem, and obtain the -th outcome. We impose the conditions (6) on .

The orthogonality preserving condition (6) is given by:

 ⟨ψ(4)00|(K†αKα⊗1)|ψ(4)11⟩=0, (9a) ⟨ψ(4)00|(K†αKα⊗1)|ψ(4)31⟩=0, (9b) ⟨ψ(4)00|(K†αKα⊗1)|ψ(4)32⟩=0, (9c) ⟨ψ(4)11|(K†αKα⊗1)|ψ(4)31⟩=0, (9d) ⟨ψ(4)11|(K†αKα⊗1)|ψ(4)32⟩=0, (9e) ⟨ψ(4)31|(K†αKα⊗1)|ψ(4)32⟩=0. (9f)

Let the spectral decomposition of be given by

 K†αKα=|u⟩⟨u|+|v⟩⟨v|+|w⟩⟨w|+|x⟩⟨x|, (10)

where , but , , and aren’t normalized.

Additionally, let’s , , and have the following expansions in the standard ONB.

 |u⟩=3∑i=0ui|i⟩, (11a) |v⟩=3∑i=0vi|i⟩, (11b) |w⟩=3∑i=0wi|i⟩, (11c) |x⟩=3∑i=0xi|i⟩. (11d)

Then equations (9a)-(9f) respectively become (using the definition (1))

 (u1u∗0+v1v∗0+w1w∗0+x1x∗0) + i(u2u∗1+v2v∗1+w2w∗1+x2x∗1) − (u3u∗2+v3v∗2+w3w∗2+x3x∗2) − i(u0u∗3+v0v∗3+w0w∗3+x0x∗3)=0, (12a) (u1u∗0+v1v∗0+w1w∗0+x1x∗0) − i(u2u∗1+v2v∗1+w2w∗1+x2x∗1) − (u3u∗2+v3v∗2+w3w∗2+x3x∗2) + i(u0u∗3+v0v∗3+w0w∗3+x0x∗3)=0, (12b) (u2u∗0+v2v∗0+w2w∗0+x2x∗0) − i(u3u∗1+v3v∗1+w3w∗1+x3x∗1) − (u0u∗2+v0v∗2+w0w∗2+x0x∗2) + i(u1u∗3+v1v∗3+w1w∗3+x1x∗3)=0, (12c) (u0u∗0+v0v∗0+w0w∗0+x0x∗0) − (u1u∗1+v1v∗1+w1w∗1+x1x∗1) + (u2u∗2+v2v∗2+w2w∗2+x2x∗2) − (u3u∗3+v3v∗3+w3w∗3+x3x∗3)=0, (12d) (u1u∗0+v1v∗0+w1w∗0+x1x∗0) − (u2u∗1+v2v∗1+w2w∗1+x2x∗1) + (u3u∗2+v3v∗2+w3w∗2+x3x∗2) − (u0u∗3+v0v∗3+w0w∗3+x0x∗3)=0, (12e) (u1u∗0+v1v∗0+w1w∗0+x1x∗0) + (u2u∗1+v2v∗1+w2w∗1+x2x∗1) + (u3u∗2+v3v∗2+w3w∗2+x3x∗2) + (u0u∗3+v0v∗3+w0w∗3+x0x∗3)=0. (12f)

Expand in the basis; then equations (11a)-(11d) allow us to define

 ξij≡(K†αKα)ij=uiu∗j+viv∗j+wiw∗j+xix∗j,∀0≤i,j≤3. (13)

Using equation (13), equations (12)-(12) take a condensed form.

 (ξ∗01ξ∗12ξ∗23ξ∗30)⎛⎜ ⎜ ⎜⎝1i−1−i⎞⎟ ⎟ ⎟⎠=0. (14a) (ξ∗01ξ∗12ξ∗23ξ∗30)⎛⎜ ⎜ ⎜⎝1−i−1i⎞⎟ ⎟ ⎟⎠=0. (14b) (14c) (ξ00ξ11ξ22ξ33)⎛⎜ ⎜ ⎜⎝1−11−1⎞⎟ ⎟ ⎟⎠=0. (14d) (ξ∗01ξ∗12ξ∗23ξ∗30)⎛⎜ ⎜ ⎜⎝1−11−1⎞⎟ ⎟ ⎟⎠=0. (14e) (ξ∗01ξ∗12ξ∗23ξ∗30)⎛⎜ ⎜ ⎜⎝1111⎞⎟ ⎟ ⎟⎠=0. (14f)

Equations (14a) , (14b), (14e) and (14f) collectively imply that

 (ξ01,ξ12,ξ23,ξ30)=0. (15)

Equation (14d) implies that

 (ξ00,ξ11,ξ22,ξ33) = a0(1,1,1,1)+a1(1,1,−1,−1)+a2(1,−1,−1,1), (16)

where , and are real. This is because are diagonal matrix elements of .

Equation (14b) implies that has to be of the form

 (ξ02,ξ13,ξ20,ξ31) = b0(1,1,1,1)+b1(1,i,−1,−i)+b2(1,−1,1,−1). (17)

Since is hermitian, . This implies that (using ) and (using ) and these imply that and and are real.

Thus putting the constraints imposed by equations (15), (III) and (III), tells us that in the basis is given by equation:

 K†αKα= ⎛⎜ ⎜ ⎜⎝a0+a1+a20b0+b200a0+a1−a20b0−b2b0+b20a0−a1−a200b0−b20a0−a1+a2⎞⎟ ⎟ ⎟⎠. (18)

The eigensystem of in equation (III) is given in the following table

where and are given by and and , , and are normalization factors. For to be a positive semidefinite operator it is necessary that .

Using the table above,

 K†αKα= a01 +μ0⎛⎜ ⎜ ⎜⎝cosζ0sinζ00000sinζ0−cosζ00000⎞⎟ ⎟ ⎟⎠ +μ1⎛⎜ ⎜ ⎜⎝00000cosη0sinη00000sinη0−cosη⎞⎟ ⎟ ⎟⎠. (19)

Imposing the OP condition (6) doesn’t give us any conclusion about the local (in)distinguishability of the states , , , , since the solution for (equation (III)) is not a multiple of the identity, i.e., the measurement isn’t constrained to be trivial. Hence, at this point we do not know if the states are distinguishable or not.

We now obtain the post-measurement joint states using necessary condition (6) for OP.

enables us to determine upto a left-unitary, i.e., , where is a unitary matrix. This unitary is irrelevant because physically it implies Alice performing a unitary after her measurement and we know that such a unitary transformation on Alice’s side (or Bob’s side) doesn’t alter the local distinguishability of the set of states. Hence we can assume the . Using the above table of eigenvalues and eigenvectors, we get that is given by

 Kα=√a0+μ0(cosζ2|0⟩+sinζ2|2⟩)(cosζ2⟨0|+sinζ2⟨2|)+√a0−μ0(−sinζ2|0⟩+cosζ2|2⟩)(−sinζ2⟨0|+cosζ2⟨2|)+√a0+μ1(cosη2|1⟩+sinη2|3⟩)(cosη2⟨1|+sinη2⟨3|)+√a0−μ1(−sinη2|1⟩+cosη2|3⟩)(−sinη2⟨1|+cosη2⟨3|). (20)

Using equation (20) we now give the Schmidt decomposition of the states .

 |ψ(4)α,00⟩=12√1+μ0a0|χ⟩A(cosζ2|0⟩B+sinζ2|2⟩B)+12√1−μ0a0|κ⟩A(−sinζ2|0⟩B+cosζ2|2⟩B)+12√1+μ1a0|ω⟩A(cosη2|1⟩B+sinη2|3⟩B)+12√1−μ1a0|τ⟩A(−sinη2|1⟩B+cosη2|3⟩B), (21a) |ψ(4)α,11⟩=12√1+μ0a0|χ⟩A(cosζ2|1⟩B−sinζ2|3⟩B)−12√1−μ0a0|κ⟩A(sinζ2|1⟩B+cosζ2|3⟩B)+i