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Journal of Physics A Mathematical and Theoretical

A class of separable quantum states

Yu Jing Guo, Jinchuan Hou

Published Nov 21, 2012• 16 citations
Publisher

Abstract

We present a sufficient condition for a bipartite state to be separable via the strong positive partial transposition structure introduced in Chru?ci?ski et?al (2008 Phys. Rev. A 77 022113), from which we obtain a class of separable states. In particular, we prove that any classical-quantum state is a state of our class but not vice versa. Both finite- and infinite-dimensional systems are considered.

DOI
10.1088/1751-8113/45/50/505303

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We present a sufficient condition for a bipartite state to be separable via the strong positive partial transposition structure introduced in Chru?ci?ski et?al (2008 Phys. Rev. A 77 022113), from which we obtain a class of separable states. In particular, we prove that any classical-quantum state is a state of our class but not vice versa. Both finite- and infinite-dimensional systems are considered.
We generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by Yu and Zhao [Int. J. Theor. Phys. 54, 292 (2014)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states and the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.
In this paper, we study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of(D-1) pure states, whereDis the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication. Before our work, only a class of 3 otimes 3 nonlocal separable operations was known [Bennett , Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.
The positive partial transposition (PPT) criterion and the realignment criterion constitute two of the most important criteria for detecting entanglement. The general- ized partial transposition (GPT) criterion which contains the PPT criterion and the realignment criterion as special cases, provides a necessary condition for a multipartite state to be separable. Here we extend the GPT criterion to the infinite- dimensional multipartite (bipartite) case and show that it includes multipartite (bipartite) PPT criterion and multipartite (bipartite) realignment criterion as special cases as well.
State space structure of tripartite quantum systems is analyzed. In particular, it has been shown that the set of states separable across all the three bipartitions [say, Bint(ABC){\mathcal{B}}^{int}(ABC)Bint(ABC)] is a strict subset of the set of states having positive partial transposition across the three bipartite cuts [say, Pint(ABC){\mathcal{P}}^{int}(ABC)Pint(ABC)] for all the tripartite Hilbert spaces ${\mathbb{C}}_{A}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}_{B}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}_{C}^{{d}_{3}}$ with $min{{d}_{1},{d}_{2},{d}_{3}}\ensuremath{\ge}2$. The claim is proved by constructing a state belonging to the set Pint(ABC){\mathcal{P}}^{int}(ABC)Pint(ABC) but not to Bint(ABC){\mathcal{B}}^{int}(ABC)Bint(ABC). For ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}3}$ with $d\ensuremath{\ge}3$, the construction follows from a specific type of multipartite unextendible product bases. However, such a construction is not possible for ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}3}$ since bipartite systems ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ do not allow any unextendible product bases for arbitrary nnn [Phys. Rev. Lett. 82, 5385 (1999)]. For the three-qubit system, we therefore come up with a different construction.