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arXiv:quant-ph/0204143 · Physical Review A

Asymptotic relative entropy of entanglement for orthogonally invariant states

K. Audenaert, Bart De Moor, K. G. H. Vollbrecht, Reinhard F. Werner

Published Sep 18, 2002• 83 citations
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Abstract

For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $\mathrm{O}\ensuremath{\bigotimes}\mathrm{O},$ where O is any orthogonal matrix. We show that in this case ${E}_{R}^{\ensuremath{\infty}}$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ${E}_{R}^{\ensuremath{\infty}};$ (iii) for states for which the relative entropy of entanglement ${E}_{R}$ is additive, the Rains bound is equal to ${E}_{R}.$

DOI
10.1103/physreva.66.032310

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For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ER\ensuremath∞{E}_{R}^{\ensuremath{\infty}}ER\ensuremath∞​ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form O\ensuremath⨂O,\mathrm{O}\ensuremath{\bigotimes}\mathrm{O},O\ensuremath⨂O, where O is any orthogonal matrix. We show that in this case ER\ensuremath∞{E}_{R}^{\ensuremath{\infty}}ER\ensuremath∞​ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ER\ensuremath∞;{E}_{R}^{\ensuremath{\infty}};ER\ensuremath∞​; (iii) for states for which the relative entropy of entanglement ER{E}_{R}ER​ is additive, the Rains bound is equal to ER.{E}_{R}.ER​.
Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement ERE_{R}ER​ is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of ERE_RER​ were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of ERE_RER​ based on the convex hull approximation of the set of separable states. We apply our method to calculate ERE_RER​ for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.
We first study the reversibility for a class of states under the operations which completely preserve the positivity of partial transpose (PPT) and show that the entanglement cost is equal to the distillable entanglement for a rank-two mixed state on the 4⊗4 antisymmetric subspace under PPT operations. By using a similar method in finding the irreversibility, we also find that the value of a new efficiently computable additive lower bound E η ( ρ ) for the asymptotic PPT-relative entropy of entanglement presented in [ Phys. Rev. Lett. 119 , 180506 (2017)] is equal to the regularized Rains’ bound and an upper bound E N ( ρ ) for distillable entanglement for these states. Furthermore, we find states on the symmetric subspace satisfying the relation mentioned above, generalize the antisymmetric states and symmetric states in higher dimensions, and give a specific value for distillable entanglement and entanglement cost for these states under the PPT operations.
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].
The long-standing problem of finding a closed formula for the relative entropy of entanglement (REE) for two qubits is addressed. A compact-form solution to the inverse problem, which characterizes an entangled state for a given closest separable state, is obtained. Analysis of the formula for a large class of entangled states strongly suggests that a compact analytical solution of the original problem, which corresponds to finding the closest separable state for a given entangled state, can be given only in some special cases. A few applications of the compact-form formula are given to show additivity of the REE, to relate the REE with the Rains upper bound for distillable entanglement, and to show that a Bell state does not have a unique closest separable state.
It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity of partial transpose, our results provide an explicit example of operations such that, even though the entanglement cost for an exact preparation is the same, the entanglement of distillation of a mixed state can exceed that of pure states. This means that the entanglement manipulation via a pure state can result in a larger entanglement loss than that via a mixed state.