QIC Abstracts

 Vol.5 No.3 May 15, 2005
Research and Review Articles:
Cryptanalysis of a Practical Quantum Key Distribution with  Polarization-Entangled Photons  (pp181-186)
         Th. Beth, J. Muller-Quade and R. Steinwandt
Recently, a quantum key exchange protocol has been described\cite{PFLM04}, which served as basis for securing an actual bank transaction by means of quantum cryptography \cite{ZVS04}. The authentication scheme used to this aim has been proposed by Peev et al. \cite{PML04}. Here we show, that this authentication is insecure in the sense that an attacker can provoke a situation where initiator and responder of a key exchange end up with different keys. Moreover, it may happen that an attacker can decrypt a part of the plaintext protected with the derived encryption key.

Commutative version of the local Hamiltonian problem and common eigenspace problem  (pp187-215)
         S. Bravyi and M. Vyalyi
We study the complexity of a problem Common Eigenspace --- verifying consistency of eigenvalue equations for composite quantum systems. The input of the problem is a family of pairwise commuting Hermitian operators H_1,\ldots,H_r on a Hilbert space (\CC^d)^{\otimes n} and a string of real numbers \lambda=(\lambda_1,\ldots,\lambda_r). The problem is to determine whether the common eigenspace specified by equalities H_a|\psi\ra=\lambda_a|\psi\ra, a=1,\ldots,r has a positive dimension. We consider two cases: (i) all operators H_a are k-local; (ii) all operators H_a are factorized. It can be easily shown that both problems belong to the class \QMA --- quantum analogue of \NP, and that some \NP-complete problems can be reduced to either (i) or (ii). A non-trivial question is whether the problems (i) or (ii) belong to \NP? We show that the answer is positive for some special values of k and d. Also we prove that the problem (ii) can be reduced to its special case, such that all operators H_a are factorized projectors and all \lambda_a=0.

Lagrangian representation for fermionic linear optics  (pp216-238)
         S. Bravyi
Notions of a Gaussian state and a Gaussian linear map are generalized to the case of anticommuting (Grassmann) variables. Conditions under which a Gaussian map is trace preserving and (or) completely positive are formulated. For any Gaussian map an explicit formula relating correlation matrices of input and output states is presented. This formalism allows to develop the Lagrangian representation for fermionic linear optics (FLO). It covers both unitary operations and the single-mode projectors associated with FLO measurements. Using the Lagrangian representation we reduce a classical simulation of FLO to a computation of Gaussian integrals over Grassmann variables. Explicit formulas describing evolution of a quantum state under FLO operations are put forward.

Communicating continuous quantum variables between different Lorentz frames  (pp239-246)
         P. Kok, T.C. Ralph and G.J. Milburn
We show how to communicate Heisenberg-limited continuous (quantum) variables between Alice and Bob in the case where they occupy two inertial reference frames that differ by an unknown Lorentz boost. There are two effects that need to be overcome: the Doppler shift and the absence of synchronized clocks. Furthermore, we show how Alice and Bob can share Doppler-invariant entanglement, and we demonstrate that the protocol is robust under photon loss.

Deterministic Local Conversion of Incomparable States by Collective LOCC (pp247-257)
         I. Chattopadhyay and D. Sarkar
Incomparability of pure bipartite entangled states under deterministic LOCC is a very strange phenomena. We find two possible ways of getting our desired pure entangled state which is incomparable with the given input state, by collective LOCC with certainty. The first one is by providing some pure entanglement through the lower dimensional maximally-entangled states or using further less amount of entanglement and the next one is by collective operation on two pairs which are individually incomparable. It is quite surprising that we are able to achieve maximally entangled states of any Schmidt rank from a finite number of 2x2 pure entangled states only by deterministic LOCC. We provide general theory for the case of 3x3 system of incomparable states by the above processes where incomparability seems to be the most hardest one.

Classical and quantum fingerprinting with shared randomness and one-sided error (pp258-271)
         R.T. Horn, A.J. Scott, J. Walgate, R. Cleve, A.I. Lvovsky and B.C. Sanders
Within the simultaneous message passing model of communication complexity, under a public-coin assumption, we derive the minimum achievable worst-case error probability of a classical fingerprinting protocol with one-sided error. We then present entanglement-assisted quantum fingerprinting protocols attaining worst-case error probabilities that breach this bound.

Webcorner updates (pp272-272)
         P. Kok

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