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How much is a quantum controller controlled by the controlled system?

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

We consider unitary transformations on a bipartite system A × B. To what extent entails the ability to transmit information from A to B the ability to transfer information in the converse direction? We prove a dimension-dependent lower bound on the classical channel capacity C(A ← B) in terms of the capacity C(A → B) for the case that the bipartite unitary operation consists of controlled local unitaries on B conditioned on basis states on A. If the local operations are given by the regular representation of a finite group G we have C(A → B) = log |G| and C(A ← B) = log N where N is the sum over the degrees of all inequivalent representations. Hence the information deficit C(A → B) − C(A ← B) between the forward and the backward capacity depends on the “non-abelianness” of the control group. For regular representations, the ratio between backward and forward capacities cannot be smaller than 1/2. The symmetric group S n reaches this bound asymptotically. However, for the general case (without group structure) all bounds must depend on the dimensions since it is known that the ratio can tend to zero. Our results can be interpreted as statements on the strength of the inevitable backaction of a quantum system on its controller.

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Authors and Affiliations

  1. School of Computer Science and Electrical Engineering, University of Central Florida, Orlando, FL, 32816, USA

    Dominik Janzing

  2. Department of Computer Science and Engineering, University of Washington, Seattle, WA, 98195, USA

    Thomas Decker

Authors
  1. Dominik Janzing
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  2. Thomas Decker
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Correspondence to Dominik Janzing.

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Janzing, D., Decker, T. How much is a quantum controller controlled by the controlled system?. AAECC 19, 241–258 (2008). https://doi.org/10.1007/s00200-008-0076-y

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  • DOI: https://doi.org/10.1007/s00200-008-0076-y

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