Local distinguishability with preservation of entanglement

I consider the deterministic distinguishability of a set of orthogonal, bipartite states when only a single copy is available and the parties are restricted to local operations and classical communication, but with the additional requirement that entanglement must be preserved in the process. Several general theorems aimed at characterizing sets of states with which the parties can succeed in such a task are proven. These include (i) a maximum for the number of states when the Schmidt rank of every outcome must be at least a given minimum, (ii) an upper bound (equal to the dimension of Hilbert space if entanglement need not be preserved) for the sum over Schmidt ranks of the initial states when only one-way classical communication is allowed, and (iii) separately, a necessary and a sufficient condition on the states such that their original Schmidt ranks can always be preserved. Two additional theorems explicitly demonstrate a trade-off between the extent to which the set of states fill Hilbert space, as measured by their Schmidt ranks, and how refined the parties must make their measurements, an important factor in determining the Schmidt rank the state can retain after it has been identified. It is shown that our bound on the sum of Schmidt ranks can be exceeded if two-way communication is permitted, and this includes the case that entanglement need not be preserved, so that this sum can exceed the dimension of Hilbert space. Such questions, concerning how the various results are affected by the resources used by the parties (amount of classical communication and types of local operations), are addressed for each theorem. This subject is closely related to the problem of locally purifying an entangled state from a mixed state, which is of direct relevance to teleportation and dense coding using a mixed-state resource. In an appendix, I give an extremely simple and transparent proof of “nonlocality without entanglement,” a phenomenon originally discussed by Bennett and co-workers several years ago.

Figure 1

Representation of the set of states given in Eq. (2). Alice’s basis states are denoted along the left side of each grid, Bob’s along the top. The numbers $\left(j\right)$ inside the boxes indicate that the state $(\mid {\Psi }_j⟩)$ has the corresponding product state as a component. (a) For the bases of Eq. (2), it is easily seen that the states can be distinguished by LOCC, preserving Schmidt rank of $2$ for all outcomes. (b) When viewed in other bases, the distinguishability may be far from obvious.

Figure 2

Lattice diagram showing relationships between the various types of protocols discussed in the text. A connected path upward (downward) from one vertex, possibly passing through others, to a second vertex indicates that a sufficient (necessary) condition on the first implies the same for the second.

Figure 3

A set of states achieving the bound of theorem 1 using LOCC-P0. Each numbered box represents an $r\times r$ subspace and $N_{\max }=25$ with $\lfloor D_A∕r\rfloor =5=\lfloor D_B∕r\rfloor$ in this example.

Figure 4

The three states of Eq. (6), which can be distinguished by the SEP POVM of Eq. (7) preserving Schmidt rank of $r=2$ for all outcomes, a task that cannot be accomplished by any LOCC protocol.

Figure 5

The states of Eq. (8), having ${\sum }_j{R}_j=12$, exceeding the (one-way) bound of theorem 2, ${\sum }_j{R}_j\le D_A\lfloor D_B∕r\rfloor =10$ with $r=2$. See text for detailed two-way protocol.

Figure 6

Illustration of the procedure of cascading partitions for testing theorem 3, described in the text [the states are given in Eq. (16)]. Bob can separate out $\left\{{\rho }_5^B\right\}$ since it is orthogonal to all the others, after which Alice can divide the remaining four into $\{{\rho }_1^A,{\rho }_2^A\}$ and $\{{\rho }_3^A,{\rho }_4^A\}$. Then Bob can complete the partitioning.

Figure 7

Demonstration that theorem 3 does not provide a necessary condition. These product states can be distinguished (so $R_j=1$ is preserved in all cases), but they cannot be completely partitioned by a cascading sequence. In fact, they cannot be partitioned even once into two nonempty subsets where the reduced density operators in one subset are orthogonal to all those in the other.

Figure 8

Intuitive picture indicating how theorem 4 can be proved. The parts of this figure correspond to the two ways it can happen that ${\widehat{\rho }}_2{\widehat{\rho }}_1\ne 0$ (only selected components of $\mid {\Psi }_2⟩$ are shown). In either case, any individual measurements the parties can perform that preserve $R_j$ leave the picture essentially unchanged, which means they have not distinguished. See Appendix for a detailed proof.

Figure 9

Illustration of theorem 6. (a) For $D_A=2$ and $D_B=3$, it is not possible to preserve entanglement while distinguishing if more than one state is initially entangled. (b) Increasing $D_B$ allows higher-rank states to be fully separated from each other, and then preserving entanglement becomes possible. As seen in the right half of this diagram, however, the presence of additional states can alter this conclusion. (c) For a $3\times 3$ system with $R_1=3$, $\mid {\Psi }_2⟩$ cannot remain entangled after being identified by the measurements. (d) Again, increasing the size of ${\mathcal{H}}_B$ allows more space for the states and even with $R_1=3$ it is possible to preserve $r_2=2$.

Figure 10

$D_B$ rank-$D_A$ states that are distinguishable by one-way LOCC.

Figure 11

Examples illustrating how the bound of theorem 2 can be achieved. (a) The states of Eq. (A12) when $D_A\le D_B$, with $r=3$. Alice’s two POVM elements are each rank $3$: the first annihilates $\mid 3{⟩}_A$, the second $\mid 0{⟩}_A$. (b) The states of Eq. (A14) when $D_B<D_A$; each square box is $r\times r$. Note that states $10,11$, and $12$ are rank $(r+a)$ with $a\ne 0$.