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

TL;DR

This paper investigates the relationship between the capacity of quantum systems to transmit information forward and backward, revealing how the control system's structure influences the backaction and information flow in bipartite quantum operations.

Contribution

It establishes a dimension-dependent lower bound on backward capacity in terms of forward capacity for controlled unitaries, linking backaction strength to the control group's structure.

Findings

Backward capacity is bounded by the control group's properties.

Regular representations of finite groups achieve the minimal capacity ratio asymptotically.

Without group structure, the capacity ratio can approach zero.

Abstract

We consider unitary transformations on a bipartite system A x 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. This can be interpreted as a statement on the strength of the inevitable backaction of a quantum system on its controller. 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.