Mitsuru Hamada

Information rates achievable with algebraic codes on quantum discrete memoryless channels

The highest information rate at which quantum error-correction schemes work reliably on a channel is called the quantum capacity. Here this is proven to be lower-bounded by the limit of coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. The quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension. The codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, the bound proven here is actually the highest possible rate at which symplectic stabilizer codes work reliably

Lower bounds on the quantum capacity and highest error exponent of general memoryless channels

Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the authors earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.

An exponential lower bound on the highest fidelity achievable by quantum error-correcting codes

On a quantum memoryless channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity of the channel.

Conjugate Codes for Secure and Reliable Information Transmission

A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum code. It is known that conjugate code pairs are applicable to (quantum) cryptography. In this work, polynomially constructible and efficiently decodable conjugate code pairs are presented. The constructed pairs achieve the highest known achievable rate on additive channels.

Teleportation and entanglement distillation in the presence of correlation among bipartite mixed states

The teleportation channel associated with an arbitrary bipartite state denotes the map that represents the change suffered by a teleported state when the bipartite state is used instead of the ideal maximally entangled state for teleportation. This work presents and proves an explicit expression of the teleportation channel for teleportation using Weyls projective unitary representation of (Z/dZ){sup 2n} for integers d{>=}2, n{>=}1, which has been known for n=1. This formula allows any correlation among the n bipartite mixed states, and an application shows the existence of reliable schemes for distillation of entanglement from a sequence of mixed states with correlation.

Reliability of Calderbank-Shor-Steane codes and security of quantum key distribution

This paper describes the security of quantum key distribution (QKD) to share a random secret string of digits between two parties based on the principle of quantum mechanics. A Colderbank-Shor-Steane quantum code had been implicitly used in the BB84 protocol, which makes fidelity of this code to unity and the mutual information between the shared key and the data obtained approaches zero. In the BB84 protocol, the sender transmits digits from which encoded into an orthogonal basis or the conjugate basis. These CSS codes achieve a higher rate called as Shannon rate.

The minimum number of rotations about two axes for constructing an arbitrarily fixed rotation.

For any pair of three-dimensional real unit vectors m^ and n^ with |m^Tn^|<1 and any rotation U, let Nm^,n^(U) denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about either m^ or n^. This work gives the number Nm^,n^(U) as a function of U. Here, a rotation means an element D of the special orthogonal group SO(3) or an element of the special unitary group SU(2) that corresponds to D. Decompositions of U attaining the minimum number Nm^,n^(U) are also given explicitly.

Algebraic and quantum theoretical approach to coding on wiretap channels

Algebraic codes designed on a criterion that has arisen from quantum cryptography are analyzed. The primary purpose of this analysis is to demonstrate the constructibility of codes for wiretap channels. Specifically, it is shown that reliable conjugate code pairs, or Calderbank-Shor-Steane (CSS) quantum error-correcting codes, can immediately be converted into reliable and secure codes for wiretap channels (Wyner, 1975). Here, a pair of linear codes is called a conjugate code pair if either of the pair contains the dual of the other; A code is called reliable if it has exponentially small decoding error probability, and it is called secure if the mutual information between the transmitted confidential message and the leaked data is exponentially small. It is argued that conjugate code pairs are applicable to wiretap channels whether they are classical or quantum theoretical. In particular, this implies that recently obtained conjugate code pairs that are constructible with polynomial complexity achieve positive rates for wiretap channels.

Efficient quotient codes decodable in polynomial time for quantum error correction and cryptography

Quantum error-correcting codes that achieve high rates in the Shannon theoretic sense and that are efficiently decodable are presented.

Disjointness of random sequence sets with respect to distinct probability measures

It is shown that the set of deterministic random sequences (of symbols from a finite alphabet) with respect to a computable probability measure /spl mu/, in Martin-Lofs (1966) sense, and the set of deterministic random sequences with respect to another computable probability measure /spl nu/ are disjoint if /spl mu/ and /spl nu/ are different and the measures are either i.i.d. or homogeneous finite-order irreducible Markov measures.