Asymptotic correctability of Bell-diagonal qudit states and lower bounds on tolerable error probabilities in quantum cryptography

Abstract

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 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.

Symmetric extendibility for qudits and tolerable error rates in quantum cryptography

“…By performing an analysis similar to that of Myhr et al, we will derive upper bounds on the tolerable error rates with two-way communication. As Myhr et al did for qubit-based protocols, we show that these upper bounds coincide with the lower bounds already known [13,14,15,16], thus proving sharpness of these bounds. In the following we will state the details of the model and our proof.…”

“…To determine whether a general state is symmetrically extendible or not is a complicated task, even for two-qubit states [11]. However, in the context of entanglement-based quantum cryptography by using arguments of the Gottesman-Lo type [5] we may concentrate on a subclass of all states, namely the (generalised) Bell-diagonal states [5,6,7,8,13,14,15] on H = C d ⊗C d where d is the dimension of a single quantum system, shared by Alice and Bob. This is achieved by a fictivemeasurement argument [5]: One can perform a measurement in the so-called Bell basis before starting the actual protocol, and such measurement does not have any measurable effect on the key.…”

“…One may ask whether similar results hold true for general finite-dimensional systems, in case we use generalised BB84 and six-state schemes. While lower bounds for such cases are known [13,14,15,16], with the exception of disentanglement bounds [17] we are not aware of a systematic study of upper bounds, which is the main purpose of this letter. By performing an analysis similar to that of Myhr et al, we will derive upper bounds on the tolerable error rates with two-way communication.…”

Symmetric extendibility for a class of qudit states

“…We now want to concentrate on an even more restricted class of states, where we can solve the problem completely, the generalised isotropic states [3]. These are Bell-diagonal states where A l0 = A l ′ 0 , A 0m = A 0m ′ and A lm = A l ′ m ′ hold for all l, m = 0.…”

“…The outline of this article is the following: in this section we shall introduce the basic concepts and notation; this includes the Hurwitz-Sylvester criterion for positivity, on which a large part of our discussion relies. In section 2 we introduce the class of U 2 -invariant two-qudit states, which are of interest in quantum cryptography [3,4]; for these states we derive a criterion (Theorem 1) in order to decide whether they are symmetrically extendible or not. We restrict our focus to the class of Bell-diagonal U 2 -invariant states, which are of even greater interest in quantum cryptography [3,4,5] in section 3 and simplify our criterion to find Theorem 2.…”