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arXiv:quant-ph/0210071

Optimal approximate reversal of quantum operations on a single qubit

Ye Yeo

Published Oct 10, 2002• 0 citations
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Abstract

We demonstrate how insights gained from reformulating the problem of quantum teleportation into one of reversing quantum operations, and designing optimum completely positive maps for teleportation, can enable one to explore optimal approximate reversal of quantum operations on a single qubit. In particular, we show that the optimal approximate reversal of a generalized depolarizing channel can be achieved using only unitary transformations. We also show that for a quantum channel, which reveals some information about the input state, extremal completely positive maps and not unitary transformations yield optimal approximate reversal.

DOI
10.48550/arxiv.quant-ph/0210071

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In general quantum operations, or quantum channels cannot be inverted by physical operations, i.e., by completely positive trace-preserving maps. An arbitrary state passing through a quantum channel loses its fidelity with the input. Given a quantum channel E\mathcal{E}E, we discuss the concept of its quasi-inverse as a completely positive trace-preserving map Eqi{\mathcal{E}}^{qi}Eqi which when composed with E\mathcal{E}E increases its average input-output fidelity in an optimal way. The channel Eqi{\mathcal{E}}^{qi}Eqi comes as close as possible to the inverse of a quantum channel. We give a complete classification of such maps for qubit channels and provide quite a few illustrative examples.
The Bayesian inverse of a quantum channel E\mathcal{E}E is a channel F\mathcal{F}F in the reverse direction of E\mathcal{E}E that yields time-symmetric correlations for sequential measurements performed on open quantum systems. Such an operational form of time-reversal symmetry for open quantum systems is quite remarkable, as the dynamics of open quantum systems are inherently irreversible due to system-environment interactions. Similar to the Petz map, a Bayesian inverse F\mathcal{F}F is defined with respect to a fiducial reference state ρρρ for the channel E\mathcal{E}E. However, Bayesian inverses do not always exist, and it is often a non-trivial task to determine the set of states ρρρ for which a Bayesian inverse of E\mathcal{E}E exists. In this work, we solve the general problem of quantum Bayesian inversion for unital channels acting on a single qubit. Our analysis is streamlined by demonstrating that finding a Bayesian inverse for a unital qubit channel may be reduced to finding a Bayesian inverse of a Pauli channel, which is simply a mixture of unitary channels associated with the Pauli matrices. As such, we provide a complete description of when operational time-reversal symmetry is attainable for sequential measurements of a single qubit in the presence of unital noise.
Quantum operations provide a general description of the state changes allowed by quantum mechanics. Simple necessary and sufficient conditions for an ideal quantum operation to be reversible by a unitary operation are derived in this paper. These results generalize recent work on reversible measurements by Mabuchi and Zoller [Phys. Rev. Lett. 76, 3108 (1996)]. Quantum teleportation can be understood as a special case of the problem of reversing quantum operations. We characterize completely teleportation schemes of the type proposed by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)].
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