LOCC distinguishability of unilaterally transformable quantum states

We consider the question of perfect local distinguishability of mutually orthogonal bipartite quantum states, with the property that every state can be specified by a unitary operator acting on the local Hilbert space of Bob. We show that if the states can be exactly discriminated by one-way LOCC where Alice goes first, then the unitary operators can also be perfectly distinguished by an orthogonal measurement on Bob's Hilbert space. We give examples of sets of N<=d maximally entangled states in $d \otimes d $ for d=4,5,6 that are not perfectly distinguishable by one-way LOCC. Interestingly for d=5,6 our examples consist of four and five states respectively. We conjecture that these states cannot be perfectly discriminated by two-way LOCC.

orthogonal basis of entangled states is locally indistinguishable [6,7,10]. The nonlocal nature of quantum information is therefore revealed when a set of orthogonal states of a composite quantum system cannot be reliably identified by LOCC. This has been particularly useful to explore quantum nonlocality and its relationship with entanglement [1,2,4,10], and has also found practical applications in quantum cryptography primitives like secret sharing, and data hiding [23][24][25][26].
The fundamental result of Walgate et al shows that it is always possible to perfectly discriminate any two orthogonal quantum states by LOCC regardless of their dimension, multipartite structure and entanglement [3]. As it turns out, quite remarkably, perfect discrimination of more than two orthogonal states is not always possible. Examples include, any three orthogonal entangled states in 2 ⊗ 2, two maximally entangled states and a product state in 2 ⊗ 2 and so on [6]. When perfect discrimination is not possible, one may distinguish the states conclusively or unambigously [18][19][20], where the unknown state is reliably identified with probability less than unity. A necessary and sufficient condition for unambigious discrimination of quantum states, not necessarily orthogonal was obtained by Chefles [20]. Recently, Bandyopadhyay and Walgate has shown that for any set of three states conclusive identification is always possible [16]. In the worst case scenario, only one member of the set, and not all, can be correctly identified, albeit with a non-zero probability.
Interestingly, the maximally entangled basis (Bell basis) in 2⊗2 [5], or a complete orthogonal entangled basis in n ⊗ m [10] are not even conclusively distnguishable, in which case we say that the sets are completely indistinguishable. Note that if an orthogonal set contains at least one product state, one can always distinguish the set conclusively. Therefore, all members of a completely indistinguishable set must necessarily be entangled.
The present work is motivated by the results on local distinguishability of orthogonal maximally entangled states [5,7,9,11], and in particular those that put an upper bound on the number of states that can be perfectly distinguished by LOCC [7,11]. For example, it was first observed in [7] that no more than d maximally entangled states in d ⊗ d can be perfectly distinguished provided the states were chosen from the Bell basis. This was soon followed by a more general result establishing this bound for any set of maximally entangled states in d ⊗ d [11].
It is therefore natural to ask whether any N orthogonal maximally entangled states in d ⊗ d can be perfectly distinguished by means of a LOCC protocol if N ≤ d. The general answer is not yet known except in dimensions 2 ⊗ 2 [3] and 3 ⊗ 3 [11]. In 2 ⊗ 2 the answer follows as a corollary of the more general result that any two orthogonal quantum states of a composite quantum system can be reliably distinguished [3]. In [11] a constructive proof was given to show that any three orthogonal maximally entangled states in 3 ⊗ 3 can be perfectly distinguished by LOCC. It is worth noting that in both [3] and [11] the maximally entangled states could be perfectly distinguished by one-way LOCC. Indeed, for almost all known sets of bipartite orthogonal states that are perfectly LOCC distinguishable, one-way protocols are sufficient. A notable exception to this can be found in [1] where it was shown that two-way LOCC is required to distinguish subsets of a locally indistinguishable orthogonal basis of 3 ⊗ 3.

II. FORMULATION OF THE PROBLEM AND RESULTS
In this work we consider the question of perfect LOCC distinguishability of bipartite orthogonal quantum states |ψ 1 , |ψ 2 , ..., |ψ N ∈ H A ⊗ H B with the property for all 1 ≤ i, j ≤ n. The above equation simply reflects the orthogonality condition for the vectors in the following way: where, U is unitary. The following result makes explicit use of the equations (6) and (7) for one-way LOCC in the directions A → B and B → A respectively.

Corollary 3. Consider a set of maximally entangled vectors
We note that the known cases in which a set of maximally entangled states can be perfectly distinguished by LOCC (these LOCC protocols are all one-way in the direction A → B [7,9,11]), the orthogonal measurements on Bob's Hilbert space make explicit use of vectors {|φ m } ∈ H B with the property φ m |U † k U l |φ m = δ kl for every m and for all k and l. Given the existing symmetry in maximally entangled states one might wonder whether there is any difference between the one-way LOCC protocols "Alice goes first" and "Bob goes first". This is an interesting question and intuitively it seems that for distinguishing maximally entangled states this should not be an issue. However we haven't been able to conclusively prove that this is the case. As noted in Corollary 3, if the states are perfectly distinguishable when Bob goes first then the orthogonality condition must hold for all k and l for some |φ ′ . Using the fact that V k = U T k the above equation can also be written as which in turn is equivalent to the condition Comparing the above condition with that of one-way LOCC in the direction A → B (as mentioned in Corollary 1) it is not clear if there is any one-to-one correspondence between the two. So we conclude that if the maximally entangled states are perfectly distinguishable by one-way LOCC in the direction A → B, then they can also be perfectly distinguished in the opposite direction provided U † k , U l = 0 for all k, l = 1, ..., N . In the latter case one can of course choose |φ ′ = |φ * .

III. ONE-WAY LOCC INDISTINGUISHABLE MAXIMALLY ENTANGLED STATES
We now give examples of one-way locally indistinguishable sets of N orthogonal maximally entangled i=0 |i ⊗ |i . These states are related to the standard maximally entangled state in the following way, where, are d × d unitary matrices for n, m = 0, 1, · · · , d − 1 .
We will now show that if the set of states {|ψ 1 , |ψ 2 , ..., |ψ n } defined by Eq. (4) Suppose that the states |ψ 1 , ..., |ψ n ∈ H ′ ⊗ H are perfectly distinguishable by one-way LOCC in the where A i is the Kraus element. Subsequent to the i th outcome of the measurement A, the reduced density matrix on H (for the input state |ψ k ) is given by where, ρ k = |ψ k ψ k |. Because a measurement now perfectly distinguishes the set of reduced density matrices {σ k,A i ∈ H : k = 1, ..., n}, they must be mutually orthogonal, that is, Noting that the states we are trying to perfectly distinguish are of the form, for k = 1, ..., n; the transformed state |ψ k,A i (unnormalized) post measurement on H ′ is given by This in turn implies that the reduced density matrices σ k,A i for all k, can be expressed in terms of σ 1,A i as, Let the spectral decomposition of the density matrix σ 1.A i be, where, 0 < λ i p ≤ 1, r p=1 λ i p = 1, and χ i p |χ i q = δ pq . Using the Eqs. (20) and (21) we can rewrite σ k,A i as, We now apply the orthogonality condition:-Tr (σ k,A i σ l,A i ) = 0, if k = l to obtain, from which it follows that every term in the summation must be identically zero. This is because each term is non-negative (note that 0 < λ i p ≤ 1) and by adding all the terms we get zero. Moreover, Eq. (23) holds for all k and l. Therefore for every p we have, from which it follows that there exist vectors {|χ p , U k |χ p ∈ H : k = 2, ..., n} forming an orthogonal set.
This is in contradiction with the fact that the unitary operators are distinguishable only in an extended tensor product space. This proves the result.
where, χ 0 Using the fact that for every i, where, U i is unitary, (25) can be rewritten as where the states {U i |χ x 1 : x = 0, 1 : i = 1, ..., N } satisfy the following orthogonality conditions if i = j. This concludes the proof. |Ψ (4) 00 . From Corollary 3, a necessary condition for these four states to be perfectly distinguishable by one-way LOCC in the direction A → B is that there must exist a vector (normalized) |φ = 3 j=0 φ j |j ∈ H B satisfying the normalization condition such that the following four vectors |φ , U 11 |φ , U 32 |φ , U 31 |φ are pairwise orthogonal. From here on we will omit the superscript in the unitaries. It is easy to verify that the six unitary operators U 11 , U 31 , U 32 , U † 11 U 32 , U † 11 U 31 , U † 32 U 31 are all distinct. We now write the orthogonality conditions: where all the exponents of ω = e is orthogonal to the following three vectors: 1, ω, ω 2 , ω 3 , 1, ω 3 , ω 2 , ω , and 1, ω 2 , 1, ω 2 . Therefore, we must have, for some λ ∈ C. We will show that the above equality cannot be satisfied except when φ i = 0 for every i and λ = 0 thereby completing the proof. To show this we need to consider two cases, namely, λ = 0 and λ = 0.
Proof of example 2: We will prove local indistinguishability in the direction A → B. A similar proof holds for B → A as well. Consider the following four maximally entangled states in 5 ⊗ 5: According to Corollary 3, a necessary condition for these four states to be perfectly distinguishable by oneway LOCC in the direction A → B, is that there must exist a vector (normalized) |φ = 4 j=0 φ j |j ∈ H B satisfying the normalization condition and such that the following four vectors |φ , U n 1 1 |φ , U n ′ 1 1 |φ , U n 2 2 |φ are pairwise orthogonal. We now write the orthogonality conditions: where all the exponents of ω = e is orthogonal to the set of following four vectors (1, 1, 1, 1, 1) , 1, ω 3 , ω, ω 4 , ω 2 , 1, ω 2 , ω 4 , ω, ω 3 , 1, ω 4 , ω 3 , ω 2 , ω ∈ C 5 . Noting that the vector 1, ω, ω 2 , ω 3 , ω 4 is orthogonal to the previous four vectors, the following identity must be valid for some λ ∈ C 5 . Proceeding as in example 1, we need to consider two cases, namely, λ = 0 and λ = 0. Proof of example 3. As in the proof of the previous example, we begin with a more general family of five orthogonal states in 6 ⊗ 6. We will prove the local indistinguishability in the direction A → B. We note that a similar proof holds in the direction B → A as well. The states are defined as follows: where, U nm = 5 j=0 e 2nπij 6 |j ⊕ 6 m n|, with n, m = 0, 1, 2, 3, 4, 5 and j ⊕ 6 m = (j + m) mod6. Also, we denote ω = e 2πi 6 .
From Corollary 3, a necessary condition that the above five states to be perfectly distinguishable by one way LOCC in the direction A → B is that there must exist a normalized vector |φ = 5 j=0 φ j |j ∈ C 6 satisfying the normalization condition and such that the following five vectors |φ , U n 1 1 |φ , U n ′ 1 1 |φ , U n 2 2 |φ and U n 3 3 |φ are pairwise orthogonal.
Case 2: Let λ = 0. This gives rise to several subcases that need to be considered individually.
and from Eq. (55) and from the normalization condition we get, Clearly the above three equations are incompatible.
On the other hand Eqs. (55) and the normalization condition give us the following two relations: The above three equations are clearly inconsistent with each other.
The remaining cases, namely when any two of the elements are zero and only one element is zero, are easily shown to be ruled out for they all give rise to conradiction with Eq. (57). This completes the proof.

VI. DISCUSSIONS AND CONCLUSIONS
We have considered in this work one way local distinguishability of a set of orthogonal states which are unilaterally transformable. That is to say, the states can be mapped onto one another by unitary operators acting on the local Hilbert spaces. We have shown that the one-way local distinguishability of such states is initimately related to the question of perfect distinguishability of the corresponding unitary operators in the local Hilbert space they act upon. In particular, if the unitary operators cannot be distinguished in their local Hilbert space but instead are perfectly distinguishable in an extended Hilbert space, then the set of orthogonal states thus generated are indistinguishable by one-way LOCC. We then apply these results to distinguish maximally entangled states by one way LOCC. are not perfectly distinguishable on the local Hilbert spaces they act upon, and instead can be perfectly distinguished in an extended tensor product space.
In the first two cases it is possible to obtain results similar to that obtained in this work with respect to the following one way LOCC in the directions A → C → B for case (a), and A → B → C for case (b).
On the other hand case (c) merits careful consideration, and it is not obvious at all how the results in this paper could be generalized to include such cases. Thus for a general multipartite system consisting of say, N subsystems, our results can be applied when the states can be mapped onto one another by applying local unitaries only one one subsystem. For more complex scenarios that involve unitaries mapping the states onto one another by acting on two or more subsystems would call for further research and beyond the scope of this paper.
Finally we would like to mention that quantum cryptography primitives like both classical and quantum data hiding, secret sharing protocols [23][24][25][26] make use of the fact that it is not possible to perfectly determine the state of a quantum system even though it was prepared in one of several orthogonal states.
In this paper several examples of locally one-way indistinguishable minimal (possibly) sets of maximally entangled states are presented with the property that they are unilaterally transformable. It is conceivable that these states with their very special properties may find applications in developing new protocols for secret sharing and data hiding.