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arXiv:1310.3530 · Physical Review A

Symmetric extension of two-qubit states

Jianxin Chen, Zhengfeng Ji, David W. Kribs, Norbert Lütkenhaus, Bei Zeng

Published Sep 17, 2014• 41 citations
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Abstract

A bipartite state ${\ensuremath{\rho}}_{AB}$ is symmetric extendible if there exists a tripartite state ${\ensuremath{\rho}}_{AB{B}^{\ensuremath{'}}}$ whose $AB$ and $A{B}^{\ensuremath{'}}$ marginal states are both identical to ${\ensuremath{\rho}}_{AB}$. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of Einstein-Podolsky-Rosen pairs, one-way distillation of secure keys, quantum marginal problems, and antidegradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state ${\ensuremath{\rho}}_{AB}$; specifically, $\mathrm{tr}({\ensuremath{\rho}}_{B}^{2})\ensuremath{\ge}\mathrm{tr}({\ensuremath{\rho}}_{AB}^{2})\ensuremath{-}4\sqrt{det{\ensuremath{\rho}}_{AB}}$. As a special case we solve the bosonic three-representability problem for the two-body reduced density matrix.

DOI
10.1103/physreva.90.032318

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A bipartite state \ensuremathρAB{\ensuremath{\rho}}_{AB}\ensuremathρAB​ is symmetric extendible if there exists a tripartite state \ensuremathρABB\ensuremath′{\ensuremath{\rho}}_{AB{B}^{\ensuremath{'}}}\ensuremathρABB\ensuremath′​ whose ABABAB and AB\ensuremath′A{B}^{\ensuremath{'}}AB\ensuremath′ marginal states are both identical to \ensuremathρAB{\ensuremath{\rho}}_{AB}\ensuremathρAB​. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of Einstein-Podolsky-Rosen pairs, one-way distillation of secure keys, quantum marginal problems, and antidegradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state \ensuremathρAB{\ensuremath{\rho}}_{AB}\ensuremathρAB​; specifically, tr(\ensuremathρB2)\ensuremath≥tr(\ensuremathρAB2)\ensuremath−4det\ensuremathρAB\mathrm{tr}({\ensuremath{\rho}}_{B}^{2})\ensuremath{\ge}\mathrm{tr}({\ensuremath{\rho}}_{AB}^{2})\ensuremath{-}4\sqrt{det{\ensuremath{\rho}}_{AB}}tr(\ensuremathρB2​)\ensuremath≥tr(\ensuremathρAB2​)\ensuremath−4det\ensuremathρAB​​. As a special case we solve the bosonic three-representability problem for the two-body reduced density matrix.
A bipartite state ${\ensuremath{\rho}}^{AB}$ has a kkk-symmetric extension if there exists a (k+1k+1k+1)-partite state ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ with marginals ${\ensuremath{\rho}}^{A{B}_{i}}={\ensuremath{\rho}}^{AB},\ensuremath{\forall}i$. The kkk-symmetric extension is called bosonic if ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ is supported on the symmetric subspace of B1B2...Bk{B}_{1}{B}_{2}...{B}_{k}B1​B2​...Bk​. Understanding the structure of symmetric and bosonic extension has various applications in the theory of quantum entanglement, quantum key distribution, and the quantum marginal problem. In particular, bosonic extension gives a tighter bound for the quantum marginal problem based on separability. In general, it is known that a ${\ensuremath{\rho}}^{AB}$ admitting symmetric extension may not have bosonic extension. In this work, we show that, when the dimension of the subsystem BBB is 2 (i.e., a qubit), ${\ensuremath{\rho}}^{AB}$ admits a kkk-symmetric extension if and only if it has a kkk-bosonic extension. Our result has an immediate application to the quantum marginal problem and indicates a special structure for qubit systems based on group representation theory.
We analyze bipartite quantum states that admit a symmetric extension. Any such state can be decomposed into a convex combination of states that allow a pure symmetric extension. A necessary condition for a state to admit a pure symmetric extension is that the spectra of the local and global density matrices are equal. This condition is also sufficient for two qubits but not for any larger systems. Using this condition, we present a conjectured necessary and sufficient condition for a two-qubit state to admit symmetric extension, which we prove in some special cases. The results from symmetric extension carry over to degradable and antidegradable channels and we use this to prove that all degradable channels with qubit output have a qubit environment.
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.
This paper provides a characterization for the set of antidegradable qubit channels. The characterization arises from the correspondence between the antidegradability of a channel and the symmetric extendibility of its Choi operator. Using an inequality derived to describe the set of bipartite qubit states which admit symmetric extension, we are able to characterize the set of all antidegradable qubit channels. Using the characterization we investigate the antidegradability of unital qubit channels and arbitrary qubit channels with respect to the dimension of the environment. We additionally provide a condition which describes qubit channels which are simultaneously degradable and antidegradable along with a classification of self-complementary qubit channels.
This thesis investigates a characterization for the set of antidegradable qubit channels. The characterization comes from a correspondence between the antidegradability of a channel, and the symmetric extendibility of the Choi operator. Using an inequality intended to describe the set of bipartite qubit states which admit symmetric extension, we demonstrate that it additionally provides a condition to characterize the set of antidegradable qubit channels. Using the characterization we investigate the antidegradability of qubit channels with respect to the rank of the Choi matrix. We also apply the antidegradability criterion to the qubit amplitude damping and qubit depolarizing channel. We end by discussing degradable and antidegradable channels in terms of their channel capacities and the applications of these results to investigating phenomena such as super-additive coherent information and superactivation.
We investigate which bipartite quantum states admit a symmetric extension and apply the results in the analysis of noise thresholds in quantum key distribution protocols with two-way postprocessing. We find that states that admit a symmetric extension can be decomposed into a convex combination of states that admit a_pure_ symmetric extension. For a state to have a pure symmetric extension, its nonzero eigenvalues must be equal to the nonzero eigenvalues of one of the subsystems. While this does not fully characterize the set of states with a pure symmetric extension in all dimensions, it is a full characterization for two qubits. A conjectured formula characterizing two-qubit states with a (mixed or pure) symmetric extension is presented. Proofs are given for some special classes. For multi-qubit Bell-diagonal states, the symmetric extension is simplified using the symmetries of the states and formulated as a semidefinite program. Quantum key distribution with two-way advantage distillation steps is then analyzed. For the commonly used repetition code advantage distillation, the failure to distill a secret key beyond the currently known noise thresholds is explained by a failure to break a symmetric extension. More general linear advantage distillation schemes are analyzed, but for those schemes analyzed, we find that they tolerate strictly less noise than repetition code advantage distillation before they fail to break a symmetric extension. The results suggest that no other linear advantage distillation scheme can reach the noise threshold of 27.6 % for the six-state protocol which is achieved by repetition code advantage distillation.