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arXiv:quant-ph/0105124 · Physical Review A

Extremal equation for optimal completely positive maps

Jaromı́r Fiurášek

Published Nov 16, 2001• 42 citations
arXivHTMLPDF

Abstract

We derive an extremal equation for an optimal completely positive map, which most closely approximates a given transformation between pure quantum states. Moreover, we also obtain an upper bound on the maximal mean fidelity that can be attained by the optimal approximate transformation. The developed formalism is applied to a universal-NOT gate, quantum cloning machines, quantum entanglers, and a qubit \ensuremath{\vartheta} shifter.

DOI
10.1103/physreva.64.062310

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We derive an extremal equation for an optimal completely positive map, which most closely approximates a given transformation between pure quantum states. Moreover, we also obtain an upper bound on the maximal mean fidelity that can be attained by the optimal approximate transformation. The developed formalism is applied to a universal-NOT gate, quantum cloning machines, quantum entanglers, and a qubit \ensuremath{\vartheta} shifter.
Recently, a lot of attention has been devoted to finding physically realizable operations that realize as closely as possible certain desired transformations between quantum states, e.g., quantum cloning, teleportation, quantum gates, etc. Mathematically, this problem boils down to finding a completely positive trace-preserving (CPTP) linear map that maximizes the (mean) fidelity between the map itself and the desired transformation. In this communication, we want to draw attention to the fact that this problem belongs to the class of so-called semidefinite programming (SDP) problems. As SDP problems are convex, it immediately follows that they do not suffer from local optima. Furthermore, this implies that the numerical optimization of the CPTP map can, and should, be done using methods from the well-established SDP field, as these methods exploit convexity and are guaranteed to converge to the real solution. Finally, we show how the duality inherent to convex and SDP problems can be exploited to prove analytically the optimality of a proposed solution. We give an example of how to apply this proof method by proving the optimality of Hardy and Song's optimal qubit $\ensuremath{\theta}$ shifter (e-print quant-ph/0102100).
We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.
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.
We present the universal cloning transformation of states in arbitrary-dimensional Hilbert spaces. This unitary transformation attains the optimal fidelity of cloning as specified by Werner [Phys. Rev. A 58, 1827 (1998)]. With this cloning transformation, pure as well as impure states can be optimally copied, and the quality of the copies does not depend on the state being copied. We discuss the properties of quantum clones. In particular, we show that in the limit of high dimension the fidelity of clones does not converge to zero but attains the limit 1/21/21/2. We also show that our cloning transformation is most suitable for cloning of entanglement.
We study the problem of universal quantum cloning – taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a ddd-dimensional unknown pure state with its NNN copies given as input, which was first obtained by Hayashi in terms of an infinite set of covariant positive operator valued measures (POVM's) and by Bru\ss{} and Macchiavello establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we construct optimal estimators with finite POVMs using exact quadratures on a hypersphere. These finite optimal estimators are not generally universal, where universality means the fidelity is independent of input states. However, any optimal estimator with finite POVM for M(>N)M(>N)M(>N) copies is universal if it is used for NNN copies as input.