QIC Abstracts

 Vol.4 No.6&7  December 31, 2004
Quantum Information and Underlying Mathematics
Editorial note ( pp409-410)
Non-stabilizer quantum codes from Abelian subgroups of the error group (pp411-436)
         V. Arvind, P.P. Kurur and K.R. Parthasarathy
This paper is motivated by the computer-generated nonadditive code described in Rains et al\cite{rains97nonadditive}. We describe a theory of non-stabilizer codes of which the nonadditive code of Rains et al is an example. Furthermore, we give a general strategy of constructing good nonstabilizer codes from good stabilizer codes and give some explicit constructions and asymptotically good nonstabilizer codes. Like in the case of stabilizer codes, we can design fairly efficient encoding and decoding procedures.

Instrumental processes, entropies, information in quantum continual measurements (pp437-449)
         A. Barchielli and G. Lupieri
In this paper we will give a short presentation of the quantum L\'evy-Khinchin formula and of the formulation of quantum continual measurements based on stochastic differential equations, matters which we had the pleasure to work on in collaboration with Prof.\ Holevo. Then we will begin the study of various entropies and relative entropies, which seem to be promising quantities for measuring the information content of the continual measurement under consideration and for analysing its asymptotic behaviour.

Optical demonstrations of statistical decision theory for quantum systems (pp450-459)
         S.M. Barnett
The work of Holevo and other pioneers of quantum information theory has given us limits on the performance of communication systems. Only recently, however, have we been able to perform laboratory demonstrations approaching the ideal quantum limit. This article presents some of the known limits and bounds together with the results of our experiments based on optical polarisation.

A resource-based view of quantum information (pp460-466)
         C.H. Bennett
We survey progress in understanding quantum information in terms of equivalences, reducibilities, and asymptotically achievable rates for transformations among nonlocal resources such as classical communication, entanglement, and particular quantum states or channels. In some areas, eg source coding, there are straightforward parallels to classical information theory; in others eg entanglement-assisted communication, new effects and tradeoffs appear that are beginning to be fairly well understood, or the remaining uncertainty has become focussed on a few simple open questions, such as conjectured additivity of the Holevo capacity. In still other areas, e.g. the role of the back communication and the classification of tripartite entanglement, much remains unknown, and it appears unlikely that an adequate description exists in terms of a finite number of resources.

On the quantumness of a Hilbert space (pp467-478)
         C.A. Fuchs
We derive an exact expression for the quantumness of a Hilbert space (defined in C.A. Fuchs and M. Sasaki, Quant. Info. Comp. {\bf 3}, 377 (2003)), and show that in composite Hilbert spaces the signal states must contain at least some entangled states in order to achieve such a sensitivity. Furthermore, we establish that the accessible fidelity for symmetric informationally complete signal ensembles is equal to the quantumness. Though spelling the most trouble for an eavesdropper because of this, it turns out that the accessible fidelity is nevertheless easy for her to achieve in this case: Any measurement consisting of rank-one POVM elements is an optimal measurement, and the simple procedure of reproducing the projector associated with the measurement outcome is an optimal output strategy.

Statistical estimation of a quantum operation (pp479-488)
         A. Fujiwara
This paper reviews some recent developments in quantum channel estimation theory, putting emphasis on an active interplay between noncommutative statistics and information geometry.

Classical capacity of free-space optical communication (pp489-499)
         V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J.H. Shapiro, B.J. Yen and H.P. Yuen
The classical-information capacity of lossy bosonic channels is studied, with emphasis on the far-field free space channel.

Comments on multiplicativity of maximal p-norms when p=2 (pp500-512)
         C. King and M.B. Ruskai
We consider the maximal p-norm associated with a completely positive map and the question of its multiplicativity under tensor products. We give a condition under which this multiplicativity holds when p = 2, and we describe some maps which satisfy our condition. This class includes maps for which multiplicativity is known to fail for large p. Our work raises some questions of independent interest in matrix theory; these are discussed in two appendices.

On consistency of quantum theory and macroscopic objectivity (pp513-522)
         L. Lanz, B. Vacchini and O. Melsheimer
We argue that the consistency problem between quantum theory and macroscopic objectivity must be placed inside a quantum description of macroscopic non-equilibrium systems. Resorting to thermodynamic concepts inside quantum field theory seems to be necessary.

Toward implementation of coding for quantum sources and channels (pp526-536)
         M. Sasaki
We review our experiment on quantum source and channel codings, the most fundamental operations in quantum info-communications. For both codings, entangling letter states is essential. Our model is based on the polarization-location coding, and a quasi-single photon linear optics implementation to entangle the polarization and location degrees of freedom. Using single-photon events in a subset of possible cases, we simulate quantum coding-decoding operations for nonorthogonal states under the quasi-pure state condition. In the quantum channel coding, we double the spatial bandwidth (number of optical paths), and demonstrate the information more than double can be transmitted. In the quantum source coding, we halve the spatial bandwidth to compress the data and decompress the original data with the high fidelity approaching the theoretical limit.

The classical capacity achievable by a quantum channel assisted by limited entanglement (pp537-545)
         P.W. Shor
We give the trade-off curve showing the capacity of a quantum channel as a function of the amount of entanglement used by the sender and receiver for transmitting information. The endpoints of this curve are given by the Holevo-Schumacher-Westmoreland capacity formula and the entanglement-assisted capacity, which is the maximum over all input density matrices of the quantum mutual information. The proof we give is based on the Holevo-Schumacher-Westmoreland formula, and also gives a new and simpler proof for the entanglement-assisted capacity formula.

The uncertainty relation for joint measurement of position and momentum (pp546-562)
         R.F. Werner
We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (\Delta P)(\Delta Q)\geq C\hbar, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision \Delta Q of the position measurement, and the perturbation \Delta P of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.

Quantum and classical message protect identification via quantum channels  (pp564-578)
         A. Winter
We discuss concepts of message identification in the sense of Ahlswede and Dueck via general quantum channels, extending investigations for classical channels, initial work for classical--quantum (cq) channels and ``quantum fingerprinting''. We show that the identification capacity of a discrete memoryless quantum channel for classical information can be larger than that for transmission; this is in contrast to all previously considered models, where it turns out to equal the common randomness capacity (equals transmission capacity in our case): in particular, for a noiseless qubit, we show the identification capacity to be 2, while transmission and common randomness capacity are 1. Then we turn to a natural concept of identification of quantum messages (i.e. a notion of ``fingerprint'' for quantum states). This is much closer to quantum information transmission than its classical counterpart (for one thing, the code length grows only exponentially, compared to double exponentially for classical identification). Indeed, we show how the problem exhibits a nice connection to visible quantum coding. Astonishingly, for the noiseless qubit channel this capacity turns out to be 2: in other words, one can compress two qubits into one and this is optimal. In general however, we conjecture quantum identification capacity to be different from classical identification capacity.

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