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arXiv:1205.1031 · Physical Review A

Positive-partial-transpose-indistinguishable states via semidefinite programming

Alessandro Cosentino

Published Jan 18, 2013• 80 citations
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Abstract

We present a simple semidefinite program whose optimal value is equal to the maximum probability of perfectly distinguishing orthogonal maximally entangled states using any PPT measurement (a measurement whose operators are positive under partial transpose). When the states to be distinguished are given by the tensor product of Bell states, the semidefinite program simplifies to a linear program. In Phys. Rev. Lett. 109, 020506 (2012), Yu, Duan, and Ying exhibit a set of four maximally entangled states in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$, which is distinguishable by any PPT measurement only with probability strictly less than 1. Using semidefinite programming, we show a tight bound of $7/8$ on this probability ($3/4$ for the case of unambiguous PPT measurements). We generalize this result by demonstrating a simple construction of a set of $k$ states in ${\mathbb{C}}^{k}\ensuremath{\bigotimes}{\mathbb{C}}^{k}$ with the same property, for any $k$ that is a power of 2. By running numerical experiments, we show the local indistinguishability of certain sets of generalized Bell states in ${\mathbb{C}}^{5}\ensuremath{\bigotimes}{\mathbb{C}}^{5}$ and ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$ previously considered in the literature.

DOI
10.1103/physreva.87.012321

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A bipartite subspace $S$ is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer $k$, there is no PPT state supporting on the orthogonal complement of $S^{\otimes k}$. We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Applying this result, we demonstrate a simple criterion for verifying whether a set of bipartite quantum states is asymptotically indistinguishable by operations completely preserving positivity of partial transpose. This criterion is based on the observation that if the support of some bipartite state is strongly PPT-unextendible, then any set of bipartite quantum states which contains this state cannot be asymptotically distinguished by PPT operations. Utilizing this criterion, we further show that any entangled pure state and its orthogonal complement cannot be distinguished asymptotically by PPT operations. Meanwhile, we also investigate that the minimum dimension of strongly PPT-unextendible subspaces in a $m\otimes n$ system is $m+n-1$, which involves an extension of the result that non-positive-partial-transpose (NPT) subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87: 064302 (2013)].

•Feb 1, 2017•1 citation
We present a simple semidefinite program whose optimal value is equal to the maximum probability of perfectly distinguishing orthogonal maximally entangled states using any PPT measurement (a measurement whose operators are positive under partial transpose). When the states to be distinguished are given by the tensor product of Bell states, the semidefinite program simplifies to a linear program. In Phys. Rev. Lett. 109, 020506 (2012), Yu, Duan, and Ying exhibit a set of four maximally entangled states in C4\ensuremath⨂C4{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}C4\ensuremath⨂C4, which is distinguishable by any PPT measurement only with probability strictly less than 1. Using semidefinite programming, we show a tight bound of 7/87/87/8 on this probability (3/43/43/4 for the case of unambiguous PPT measurements). We generalize this result by demonstrating a simple construction of a set of kkk states in Ck\ensuremath⨂Ck{\mathbb{C}}^{k}\ensuremath{\bigotimes}{\mathbb{C}}^{k}Ck\ensuremath⨂Ck with the same property, for any kkk that is a power of 2. By running numerical experiments, we show the local indistinguishability of certain sets of generalized Bell states in C5\ensuremath⨂C5{\mathbb{C}}^{5}\ensuremath{\bigotimes}{\mathbb{C}}^{5}C5\ensuremath⨂C5 and C6\ensuremath⨂C6{\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}C6\ensuremath⨂C6 previously considered in the literature.
We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of k≤dk \leq dk≤d orthogonal maximally entangled states in $\complex^{d}\otimes\complex^{d}$ that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012)] gives an affirmative answer for the case k=dk = dk=d. We give, for the first time, a proof that such sets of states indeed exist even in the case k<dk < dk<d. Our result is constructive and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT). The proof uses the characterization of the PPT-distinguishability problem as a semidefinite program.
We study the distinguishability of a particular type of maximally entangled states---the ``lattice states''---using an approach of a semidefinite program. With this, we successfully construct all sets of four ququad-ququad orthogonal maximally entangled states that are locally indistinguishable and find some curious sets of six states having an interesting property of distinguishability. Also, some of the problems that arose from Cosentino and Russo [Quantum Inf. Comput. 14, 1098 (2013)] about the positive-partial-transpose distinguishability of lattice maximally entangled states can be answered.
We study the distinguishability of a particular type of maximally entangled states -- the states which are tensor products of Bell states in C4⊗C4\mathbb{C}^4\otimes\mathbb{C}^4C4⊗C4. We first prove that any three orthogonal ququad-ququad maximally entangled states can be distinguished with LOCC. Then we use a new approach of semidefinite program to construct all sets of four ququad-ququad orthogonal maximally entangled states that are PPT-indistinguishable and we find some interesting sets of six states having interesting property of distinguishability. Also, we show that our approach of the optimization problem can make some computational complex problem more tractable.
We study the distinguishability of bipartite quantum states by positive operator-valued measures with positive partial transpose (PPT POVMs). The contributions of this paper include: 1) we give a negative answer to an open problem of showing a limitation of a previous known method for detecting nondistinguishability; 2) we show that a maximally entangled state and its orthogonal complement, no matter how many copies are supplied, cannot be distinguished by the PPT POVMs, even unambiguously. This result is much stronger than the previous known ones; and 3) we study the entanglement cost of distinguishing quantum states. It is proved that √{2/3}|00〉+√{1/3}|11〉 is sufficient and necessary for distinguishing three Bell states by the PPT POVMs. An upper bound of entanglement cost of distinguishing a d ⊗ d pure state and its orthogonal complement is obtained for separable operations. Based on this bound, we are able to construct two orthogonal quantum states, which cannot be distinguished unambiguously by separable POVMs, but finite copies would make them perfectly distinguishable by local operations and classical communication. We further observe that a two-qubit maximally entangled state is always enough for distinguishing a d ⊗ d pure state and its orthogonal complement by the PPT POVMs, no matter the value of d. In sharp contrast, an entangled state with Schmidt number at least d is always needed for distinguishing such two states by separable POVMs. As an application, we show that the entanglement cost of distinguishing a d ⊗ d maximally entangled state and its orthogonal complement must be a maximally entangled state for d=2, which implies that teleportation is optimal, and in general, it could be chosen as O(logd/d).
In order to better understand the class of quantum operations that preserve the positivity of partial transpose (PPT operations) and its relation to the widely used class of local operations and classical communication (LOCC), we study the problem of distinguishing orthogonal maximally entangled states (MES) by PPT operations. Firstly, we outline a rather simple proof to show that the number of d⊗dd\otimes dd⊗d PPT distinguishable MES is at most ddd, which slightly generalizes existing results on this problem. Secondly, we construct 4 MES in 4⊗44\otimes 44⊗4 state space that cannot be distinguished by PPT operations. Before our work, it was unknown whether there exists ddd MES in d⊗dd\otimes dd⊗d state space that are locally indistinguishable. This example leads us to a novel phenomenon of Entanglement Catalysis Discrimination. Moreover, we find there exists a set of locally indistinguishable states KKK such that K⊗mK^{\otimes m}K⊗m is locally distinguishable for some finite mmm. As an interesting application, we exhibit a quantum channel with one sender and two receivers, whose one-shot zero-error local capacity is not optimal, but multi-use would enhance the capacity to achieve the full output dimension even without entangled input. Finally, we consider the entanglement cost of distinguishing three Bell states and a 2⊗22\otimes 22⊗2 entangled basis. In the former case a bipartite pure entangled state with the largest Schmidt coefficient at most 2/3 is necessary and sufficient, while in the latter case an additional Bell state, or one ebit, should be supplied.
A bipartite subspace SSS is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer kkk, there is no PPT state supporting on the orthogonal complement of S⊗kS^{\otimes k}S⊗k. We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Applying this result, we demonstrate a simple criterion for verifying whether a set of bipartite quantum states is asymptotically indistinguishable by operations completely preserving positivity of partial transpose. This criterion is based on the observation that if the support of some bipartite state is strongly PPT-unextendible, then any set of bipartite quantum states which contains this state cannot be asymptotically distinguished by PPT operations. Utilizing this criterion, we further show that any entangled pure state and its orthogonal complement cannot be distinguished asymptotically by PPT operations. Meanwhile, we also investigate that the minimum dimension of strongly PPT-unextendible subspaces in a m⊗nm\otimes nm⊗n system is m+n−1m+n-1m+n−1, which involves an extension of the result that non-positive-partial-transpose (NPT) subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87: 064302 (2013)].