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arXiv:1101.5134 · Journal of Physics A Mathematical and Theoretical

Distillability and PPT entanglement of low-rank quantum states

Lin Chen, Dragomir Ž. Djoković

Published Jun 14, 2011• 46 citations
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Abstract

The bipartite quantum states ρ, with rank strictly smaller than the maximum of the ranks of the reduced states ρ A and ρ B , are distillable by local operations and classical communication (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our first main result is that this is also true for NPT states with rank equal to this maximum. (A state is PPT if the partial transpose of its density matrix is positive semidefinite, and otherwise it is NPT.) This was conjectured first in 1999 in the special case when the ranks of ρ A and ρ B are equal (see (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our second main result provides a complete solution of the separability problem for bipartite states of rank 4. Namely, we show that such a state is separable if and only if it is PPT and its range contains at least one product state. We also prove that the so-called checkerboard states are distillable if and only if they are NPT.

DOI
10.1088/1751-8113/44/28/285303

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The bipartite quantum states ρ, with rank strictly smaller than the maximum of the ranks of the reduced states ρ A and ρ B , are distillable by local operations and classical communication (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our first main result is that this is also true for NPT states with rank equal to this maximum. (A state is PPT if the partial transpose of its density matrix is positive semidefinite, and otherwise it is NPT.) This was conjectured first in 1999 in the special case when the ranks of ρ A and ρ B are equal (see (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our second main result provides a complete solution of the separability problem for bipartite states of rank 4. Namely, we show that such a state is separable if and only if it is PPT and its range contains at least one product state. We also prove that the so-called checkerboard states are distillable if and only if they are NPT.
Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.
Assume that Alice, Bob, and Charlie share a tripartite pure state$ \vert \psi _{ABC}\rangle .WeprovethatifAlicecannotdistillentanglementwitheitherBoborCharlieusing. We prove that if Alice cannot distill entanglement with either Bob or Charlie using.WeprovethatifAlicecannotdistillentanglementwitheitherBoborCharlieusing \vert \psi _{ABC}\rangle andlocaloperationswithanyoneofthefollowingconfigurationsforclassicalcommunication:and local operations with any one of the following configurations for classical communication:andlocaloperationswithanyoneofthefollowingconfigurationsforclassicalcommunication:(A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C),and, and,and(A\leftrightarrow B, A\leftrightarrow C),thenthesameisalsotruefortheothertwoconfigurations.Moreover,thishappenspreciselywhenthestateissuchthatbothitsreductionsonsystems, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems,thenthesameisalsotruefortheothertwoconfigurations.Moreover,thishappenspreciselywhenthestateissuchthatbothitsreductionsonsystemsABandandandAC$are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. In proving these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.
We consider bipartite mixed states ρ in a d ⊗ d quantum system. We say that ρ is PPT if its partial transpose 1 ⊗ T ( ρ ) is positive semidefinite, and otherwise ρ is NPT. The well-known Werner states are divided into three types: (a) the separable states (the same as the PPT states); (b) the one-distillable states (necessarily NPT); and (c) the NPT states which are not one-distillable. We give several different formulations and provide further evidence for the validity of the conjecture that Werner states of type (c) are not two-distillable.
We study entanglement distillation under operations that remain completely positive under partial-transpose conjugation, a.k.a. PPT channels, in the standard quantum Shannon theory regime of asymptotically vanishing but non-zero error. Our main results are: (a) a regularised formula for the PPT distillable entanglement in terms of a measured-relative-entropy-like quantity; and (b) two different single-letter converses, one based on operator quadratic forms and the other on Hirschman's strengthening of the Hadamard three-line theorem. As an immediate application of (b), we show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, thereby resolving an open problem in the theory of entanglement manipulation [Regula et al., NJP 21:103017, 2019]. A gap appears already for 3×33\times 33×3 Werner states: at antisymmetric weight 25/2625/2625/26, a certified upper bound of 0.621070.621070.62107 ebits lies below the (regularised) Rains bound, which equals 2526log⁡25−log⁡23≈0.64766\frac{25}{26} \log_2 5 - \log_2 3 \approx 0.647662625​log2​5−log2​3≈0.64766 ebits. On a different note, (a) implies a faithful lower bound on the PPT distillable entanglement in terms of the entanglement negativity, which provides a quantitative counterpart to the qualitatively known fact that any NPT state is PPT distillable [Eggeling et al., PRL 87:257902, 2001].
Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension $3\ifmmode\times\else\texttimes\fi{}3$, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension $3\ifmmode\times\else\texttimes\fi{}3$, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.
We show that a bipartite quantum state of rank four is distillable if its partial transpose has at least one negative eigenvalue; that is, the state is a non-positive-partial transpose (NPT). For this purpose we prove that if the partial transpose of a two-qutrit NPT state has at least two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT state of rank five which is not 1-distillable and show that it is not nnn-distillable for any given nnn when the parameter is sufficiently small. This state has the smallest rank among all 1-undistillable NPT states. We conjecture that the state is not distillable.