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arXiv:quant-ph/9608001 · Physical Review A

Reversible quantum operations and their application to teleportation

Michael A. Nielsen, Carlton M. Caves

Published Apr 1, 1997• 109 citations
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Abstract

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)].

DOI
10.1103/physreva.55.2547

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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)].
A scheme for teleporting an unknown quantum state of many particles is proposed. The scheme operates essentially by prearranging the sharing of an Einstein-Podolsky-Rosen-correlated pair of particles every time. We show that after performing a series of Bell-state measurements and single-particle unitary transformations, the unknown state of many particles, which was destroyed at one place, can be reconstructed at another place. Our scheme is actually obtained by generalizing an earlier scheme of Bennett et al. [ Phys. Rev. Lett. 70 (1993) 1895 ; 76 (1996) 722 ] known as quantum teleportation to the multiparticle case.
We introduce a separability criterion based on the positive map $\ensuremath{\Gamma}:\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}(\mathrm{Tr}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho},$ where $\ensuremath{\rho}$ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of $\ensuremath{\Gamma}$ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, $\ensuremath{\Gamma}$ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\ifmmode\times\else\texttimes\fi{}2$ and $2\ifmmode\times\else\texttimes\fi{}3$ systems. Finally, a simple connection between this map for two qubits and complex conjugation in the ``magic'' basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed.