Tomographic Quantum Cryptography

We present a protocol for quantum cryptography in which the data obtained for mismatched bases are used in full for the purpose of quantum state tomography. Eavesdropping on the quantum channel is seriously impeded by requiring that the outcome of the tomography is consistent with unbiased noise in the channel. We study the incoherent eavesdropping attacks that are still permissible and establish under which conditions a secure cryptographic key can be generated. The whole analysis is carried out for channels that transmit quantum systems of any finite dimension.Comment: REVTeX4, 9 pages, 3 figures, 1 tabl

Quantum Key Distribution using Multilevel Encoding: Security Analysis

We present security proofs for a protocol for Quantum Key Distribution (QKD) based on encoding in finite high-dimensional Hilbert spaces. This protocol is an extension of Bennett's and Brassard's basic protocol from two bases, two state encoding to a multi bases, multi state encoding. We analyze the mutual information between the legitimate parties and the eavesdropper, and the error rate, as function of the dimension of the Hilbert space, while considering optimal incoherent and coherent eavesdropping attacks. We obtain the upper limit for the legitimate party error rate to ensure unconditional security when the eavesdropper uses incoherent and coherent eavesdropping strategies. We have also consider realistic noise caused by detector's noise.Comment: 8 pages, 3 figures, REVTe

Tomographic Quantum Cryptography: Equivalence of Quantum and Classical Key Distillation

The security of a cryptographic key that is generated by communication through a noisy quantum channel relies on the ability to distill a shorter secure key sequence from a longer insecure one. For an important class of protocols, which exploit tomographically complete measurements on entangled pairs of any dimension, we show that the noise threshold for classical advantage distillation is identical with the threshold for quantum entanglement distillation. As a consequence, the two distillation procedures are equivalent: neither offers a security advantage over the other.Comment: 4 pages, 1 figur

Encoding a qubit in an oscillator

Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.Comment: 22 pages, 8 figures, REVTeX, title change (qudit -> qubit) requested by Phys. Rev. A, minor correction

Unitary quantum gates, perfect entanglers and unistochastic maps

Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3 \pi \approx 0.85. We establish explicit conditions under which a given one-qubit bistochastic map is unistochastic, so it can be obtained by partial trace over a one--qubit environment initially prepared in the maximally mixed state.Comment: 14 pages including 6 figures in eps, version 4, title changed according to a suggestion of the editor

Extremal spacings between eigenphases of random unitary matrices and their tensor products

Extremal spacings between eigenvalues of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N = 4. We study ensembles of tensor product of k random unitary matrices of size n which describe independent evolution of a composite quantum system consisting of k subsystems. In the asymptotic case, as the total dimension N = n^k becomes large, the nearest neighbor distribution P(s) becomes Poissonian, but statistics of extreme spacings P(s_min) and P(s_max) reveal certain deviations from the Poissonian behavior