Showing papers by "Andreas Winter published in 2001"

The capacity of the quantum multiple-access channel

Abstract: We define classical quantum multiway channels for transmission of classical information, after the previous work by Allahverdyan and Saakian (see Quantum Computing and Quantum Communications (Lecture Notes in Computer Science). Berlin, Germany: Springer-Verlag, vol.1509, 1999). Bounds on the capacity region are derived in a uniform way, which are analogous to the classically known ones, simply replacing Shannon (1961) entropy with von Neumann (1955) entropy. For the single receiver case (multiple-access channel) the elect capacity region is determined. These results are applied to the case of noisy channels, with arbitrary input signal states. A second issue of this work is the presentation of a calculus of quantum information quantities, based on the algebraic formulation of quantum theory.

"Extrinsic" and "intrinsic" data in quantum measurements: asymptotic convex decomposition of positive operator valued measures

Abstract: We study the problem of separating the data produced by a given quantum measurement (on states from a memoryless source which is unknown except for its average state), described by a positive operator valued measure (POVM), into a "meaningful" (intrinsic) and a "not meaningful" (extrinsic) part. We are able to give an asymptotically tight separation of this form, with the "intrinsic" data quantfied by the Holevo mutual information of a certain state ensemble associated to the POVM and the source, in a model that can be viewed as the asymptotic version of the convex decomposition of POVMs into extremal ones. This result is applied to a similar separation therorem for quantum instruments and quantum operations, in their Kraus form. Finally we comment on links to related subjects: we stress the difference between data and information (in particular by pointing out that information typically is strictly less than data), derive the Holevo bound from our main result, and look at its classical case: we show that this includes the solution to the problem of extrinsic/intrinsic data separation with a known source, then compare with the well-known notion of sufficient statistics. The result on decomposition of quantum operations is used to exhibit a new aspect of the concept of entropy exchange of an open dynamics. An appendix collects several estimates for mixed state fidelity and trace norm distance, that seem to be new, in particular a construction of canonical purification of mixed states that turns out to be valuable to analyze their fidelity.

On the reversible extraction of classical information from a quantum source

Abstract: Consider a source ϵ of pure quantum states with von Neumann entropy S . By the quantum source coding theorem, arbitrarily long strings of signals may be encoded asymptotically into S qubits per signal (the Schumacher limit) in such a way that entire strings may be recovered with arbitrarily high fidelity. Suppose that classical storage is free while quantum storage is expensive and suppose that the states of ϵ do not fall into two or more orthogonal subspaces. We show that if ϵ can be compressed with arbitrarily high fidelity into A qubits per signal plus any amount of auxiliary classical storage, then A must still be at least as large as the Schumacher limit S of ϵ. Thus no part of the quantum information content of ϵ can be faithfully replaced by classical information. If the states do fall into orthogonal subspaces, then A may be less than S , but only by an amount not exceeding the amount of classical information specifying the subspace for a signal from the source.

Compression of quantum-measurement operations

Abstract: We generalize recent work of Massar and Popescu dealing with the amount of classical data that is produced by a quantum measurement on a quantum state ensemble. In the previous work it was shown how spurious randomness generally contained in the outcomes can be eliminated without decreasing the amount of knowledge, to achieve an amount of data equal to the von Neumann entropy of the ensemble. Here we extend this result by giving a more refined description of what constitute equivalent measurements (that is measurements which provide the same knowledge about the quantum state) and also by considering incomplete measurements. In particular we show that one can always associate to a POVM with elements a_j, an equivalent POVM acting on many independent copies of the system which produces an amount of data asymptotically equal to the entropy defect of an ensemble canonically associated to the ensemble average state and the initial measurement (a_j). In the case where the measurement is not maximally refined this amount of data is strictly less than the von Neumann entropy, as obtained in the previous work. We also show that this is the best achievable, i.e. it is impossible to devise a measurement equivalent to the initial measurement (a_j) that produces less data. We discuss the interpretation of these results. In particular we show how they can be used to provide a precise and model independent measure of the amount of knowledge that is obtained about a quantum state by a quantum measurement. We also discuss in detail the relation between our results and Holevo's bound, at the same time providing a new proof of this fundamental inequality.

Quantum Finite State Transducers

Abstract: We introduce quantum finite state transducers (qfst), and study the class of relations which they compute. It turns out that they share many features with probabilistic finite state transducers, especially regarding undecidability of emptiness (at least for low probability of success). However, like their 'little brothers', the quantum finite automata, the power of qfst is incomparable to that of their probabilistic counterpart. This we show by discussing a number of characteristic examples.

On the fidelity of two pure states

Abstract: The fidelity of two pure states (also known as transition probability) is a symmetric function of two operators, and well founded operationally as an event probability in a certain preparation-test pair. Motivated by the idea that the fidelity is the continuous quantum extension of the combinatorial equality function, we enquire whether there exists a symmetric operational way of obtaining the fidelity. It is shown that this is impossible. Finally, we discuss the optimal universal approximation by a quantum operation.

Scalable programmable quantum gates and a new aspect of the additivity problem for the classical capacity of quantum channels

Abstract: We consider two apparently separated problems: in the first part of the paper we study the concept of a scalable (approximate) programmable quantum gate (SPQG). These are special (approximate) programmable quantum gates, with nice properties that could have implications on the theory of universal computation. Unfortunately, as we prove, such objects do not exist in the domain of usual quantum theory. In the second part the problem of noisy dense coding (and generalizations) is addressed. We observe that the additivity problem for the classical capacity obtained is of apparently greater generality than for the usual quantum channel (completely positive maps): i.e., the latter occurs as a special case of the former, but, as we shall argue with the help of the non-existence result of the first part, the former cannot be reduced to an instance of the latter. We conclude by suggesting that the additivity problem for the classical capacity of quantum channels, as posed until now, may conceptually not be in its appropriate generality.