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arXiv:0906.0163 · Physical Review A

Symmetric extendibility for qudits and tolerable error rates in quantum cryptography

Kedar S. Ranade

Published Aug 3, 2009• 10 citations
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Abstract

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.

DOI
10.1103/physreva.80.022301

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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 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.
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.
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 asymptotic correctability of Bell-diagonal quantum states is generalized to elementary quantum systems of higher dimensions. Based on these results basic properties of quantum state purification protocols are investigated which are capable of purifying tensor products of Bell-diagonal states and which are based on B -steps of the Gottesman-Lo-type with the subsequent application of a Calderbank–Shor–Steane quantum code. Consequences for maximum tolerable error rates of quantum cryptographic protocols are discussed.
The concept of asymptotic correctability of Bell-diagonal quantum states is generalised to elementary quantum systems of higher dimensions. Based on these results basic properties of quantum state purification protocols are investigated which are capable of purifying tensor products of Bell-diagonal states and which are based on BBB-steps of the Gottesman-Lo-type with the subsequent application of a Calderbank-Shor-Steane quantum code. Consequences for maximum tolerable error rates of quantum cryptographic protocols are discussed.
We investigate the error tolerance of quantum cryptographic protocols using ddd-level systems. In particular, we focus on prepare-and-measure schemes that use two mutually unbiased bases and a key-distillation procedure with two-way classical communication. For arbitrary quantum channels, we obtain a sufficient condition for secret-key distillation which, in the case of isotropic quantum channels, yields an analytic expression for the maximally tolerable error rate of the cryptographic protocols under consideration. The difference between the tolerable error rate and its theoretical upper bound tends slowly to zero for sufficiently large dimensions of the information carriers.
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.