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Symmetric extendibility for a class of qudit states

Kedar S. Ranade

Published Nov 27, 2012• 3 citations
OpenAlex

Abstract

Abstract. The concept of symmetric extendibility has recently drawn attention in the context of tolerable error rates in quantum cryptography, where it can be used to decide whether quantum states shared between two parties can be purified by means of entanglement purification with one-way classical communication only. Unfortunately, at present there exists no simple general criterion to decide whether a state possesses a symmetric extension or not. In this article we derive criteria for symmetric extendibility within subclasses of all two-qudit states. Using these criteria, we can completely solve the problem for a two-parameter family of two-qudit states, which includes the isotropic states as a subclass. PACS numbers: 03.67.-a, 03.67.Dd, 03.67.HkSymmetric extendibility for a class of qudit states 2 1.

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Symmetric extendibility for a class of qudit states

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The concept of symmetric extendibility has recently drawn attention in the context of tolerable error rates in quantum cryptography, where it can be used to decide whether quantum states shared between two parties can be purified by means of entanglement purification with one-way classical communication only. Unfortunately, at present there exists no simple general criterion to decide whether a state possesses a symmetric extension or not. In this paper, we derive criteria for symmetric extendibility within subclasses of all two-qudit states. Using these criteria, we can completely solve the problem for a two-parameter family of two-qudit states, which includes the isotropic states as a subclass.

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Symmetric extendibility of quantum states has recently drawn attention in the context of quantum cryptography to judge whether quantum states shared between two distant parties can be purified by means of one-way error correction protocols. In this paper we study the symmetric extendibility in a specific class of two-qudit states, i.e., states composed of two $d$-level systems, in order to find upper bounds on tolerable error rates for a wide class of qudit-based quantum cryptographic protocols using two-way error correction. In important cases these bounds coincide with previously known lower bounds, thereby proving sharpness of these bounds in arbitrary finite-dimensional systems.

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Entanglement of qudit pairs, with single-particle Hilbert space dimension d , has important potential for quantum information processing, with applications in cryptography, algorithms, and error correction. For a pair of qudits of arbitrary even dimension d , we introduce a generalized Bell basis with definite symmetry under exchange of internal states between the two particles. We show that no complete exchange-symmetrized basis can exist for odd d . This framework extends prior work on exchange-symmetrized hyperentangled qubit bases, where d is a power of 2. For our exchange-symmetrized basis, we show that measurement devices restricted to linear evolution and local measurement (LELM) can unambiguously distinguish 2 d − 1 qudit Bell states for any even d . This achieves the upper bound in general for reliable Bell-state distinguishability via LELM and augments previously known limits for d = 2 n and d = 3 . This result is relevant to near-term realizations of quantum communication protocols.

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Abstract. The concept of symmetric extendibility has recently drawn attention in the context of tolerable error rates in quantum cryptography, where it can be used to decide whether quantum states shared between two parties can be purified by means of entanglement purification with one-way classical communication only. Unfortunately, at present there exists no simple general criterion to decide whether a state possesses a symmetric extension or not. In this article we derive criteria for symmetric extendibility within subclasses of all two-qudit states. Using these criteria, we can completely solve the problem for a two-parameter family of two-qudit states, which includes the isotropic states as a subclass. PACS numbers: 03.67.-a, 03.67.Dd, 03.67.HkSymmetric extendibility for a class of qudit states 2 1.
The concept of symmetric extendibility has recently drawn attention in the context of tolerable error rates in quantum cryptography, where it can be used to decide whether quantum states shared between two parties can be purified by means of entanglement purification with one-way classical communication only. Unfortunately, at present there exists no simple general criterion to decide whether a state possesses a symmetric extension or not. In this paper, we derive criteria for symmetric extendibility within subclasses of all two-qudit states. Using these criteria, we can completely solve the problem for a two-parameter family of two-qudit states, which includes the isotropic states as a subclass.
Symmetric extendibility of quantum states has recently drawn attention in the context of quantum cryptography to judge whether quantum states shared between two distant parties can be purified by means of one-way error correction protocols. In this paper we study the symmetric extendibility in a specific class of two-qudit states, i.e., states composed of two ddd-level systems, in order to find upper bounds on tolerable error rates for a wide class of qudit-based quantum cryptographic protocols using two-way error correction. In important cases these bounds coincide with previously known lower bounds, thereby proving sharpness of these bounds in arbitrary finite-dimensional systems.
Abstract Studies on the symmetric extendibility of quantum states have become particularly important in the context of the analysis of one-way quantum measures of entanglement, and the distillability and security of quantum protocols. In this paper we analyze composite systems containing a symmetric extendible part, with particular attention devoted to the one-way security of such systems. Further, we introduce a new one-way entanglement monotone based on the best symmetric approximation of a quantum state and the extendible number of a quantum state. We underpin these results with geometric observations about the structures of multi-party settings which posses substantial symmetric extendible components in their subspaces. The impossibility of reducing the maximal symmetric extendibility by means of the one-way local operations and classical communication method is pointed out on multiple copies. Finally, we state a conjecture linking symmetric extendibility with the one-way distillability and security of all quantum states, analyzing the behavior of a private key in the neighborhood of symmetric extendible states.
A bipartite state ${\ensuremath{\rho}}_{AB}$ is symmetric extendible if there exists a tripartite state ${\ensuremath{\rho}}_{AB{B}^{\ensuremath{'}}}$ whose ABABAB 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.
Entanglement of qudit pairs, with single-particle Hilbert space dimension d , has important potential for quantum information processing, with applications in cryptography, algorithms, and error correction. For a pair of qudits of arbitrary even dimension d , we introduce a generalized Bell basis with definite symmetry under exchange of internal states between the two particles. We show that no complete exchange-symmetrized basis can exist for odd d . This framework extends prior work on exchange-symmetrized hyperentangled qubit bases, where d is a power of 2. For our exchange-symmetrized basis, we show that measurement devices restricted to linear evolution and local measurement (LELM) can unambiguously distinguish 2 d − 1 qudit Bell states for any even d . This achieves the upper bound in general for reliable Bell-state distinguishability via LELM and augments previously known limits for d = 2 n and d = 3 . This result is relevant to near-term realizations of quantum communication protocols.