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arXiv:1901.08848 · Physical Review A

Comment on “Optimal convex approximations of quantum states”

Xiao-Bin Liang, Bo Li, Shao-Ming Fei

Published Jan 22, 2019• 4 citations
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Abstract

In a recent paper, M. F. Sacchi [Phys. Rev. A 96, 042325 (2017)] addressed the general problem of approximating an unavailable quantum state by the convex mixing of different available states. For the case of qubit mixed states, we show that the analytical solutions in some cases are invalid. In this Comment, we present complete analytical solutions for the optimal convex approximation. Our solutions can be viewed as correcting and supplementing the results in the aforementioned paper.

DOI
10.1103/physreva.99.016301

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In a recent paper, M. F. Sacchi [Phys. Rev. A 96, 042325 (2017)] addressed the general problem of approximating an unavailable quantum state by the convex mixing of different available states. For the case of qubit mixed states, we show that the analytical solutions in some cases are invalid. In this Comment, we present complete analytical solutions for the optimal convex approximation. Our solutions can be viewed as correcting and supplementing the results in the aforementioned paper.
We consider the problem of optimally approximating an unavailable quantum state $\ensuremath{\rho}$ by the convex mixing of states drawn from a set of available states ${{\ensuremath{\nu}}_{i}}$. The problem is recast to look for the least distinguishable state from $\ensuremath{\rho}$ among the convex set ${\ensuremath{\sum}}_{i}{p}_{i}{\ensuremath{\nu}}_{i}$, and the corresponding optimal weights pi{{p}_{i}}pi​ provide the optimal convex mixing. We present the complete solution for the optimal convex approximation of a qubit mixed state when the set of available states comprises the three bases of the Pauli matrices.
We address the problem of optimally approximating the action of a desired and unavailable quantum channel $\mathrm{\ensuremath{\Phi}}$ having at our disposal a single use of a given set of other channels ${{\mathrm{\ensuremath{\Psi}}}_{i}}$. The problem is recast to look for the least distinguishable channel from $\mathrm{\ensuremath{\Phi}}$ among the convex set ${\ensuremath{\sum}}_{i}{p}_{i}{\mathrm{\ensuremath{\Psi}}}_{i}$, and the corresponding optimal weights pi{{p}_{i}}pi​ provide the optimal convex mixing of the available channels ${{\mathrm{\ensuremath{\Psi}}}_{i}}$. For single-qubit channels we study specifically cases where the available convex set corresponds to covariant channels or to Pauli channels, and the desired target map is an arbitrary unitary transformation or a generalized damping channel.
Abstract Approximating a quantum state by the convex mixing of some given states has strong experimental significance and provides alternative understandings of quantum resource theory. It is essentially a complex optimal problem which, up to now, has only partially solved for qubit states. Here, the most general case is focused on that the approximation of a d‐dimensional objective quantum state by the given state set consisting of any number of (mixed‐) states. The problem is thoroughly solved with a closed solution of the minimal distance in the sense of l2 norm between the objective state and the set. In particular, the minimal number of states in the given set is presented to achieve the optimal distance. The validity of this closed solution is further verified numerically by several randomly generated quantum states.
We investigate the optimal convex approximation of the quantum state with respect to a set of available states. By isometric transformation, we have presented the general mathematical model and its solutions together with a triple uncertainty equality relation. Meanwhile, we show a concise inequality criterion for decomposing qubit mixed states. The new results include previous ones as special cases. Our model and method may be applied to solve similar problems in high-dimensional and multipartite scenarios.
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Abstract We investigate the optimal convex approximation, optimally approximating a desired and unavailable qubit state by the convex mixing of a given set of available states. When the available states are the eigenvectors of three Pauli matrices, we present the complete exact solution for the optimal convex approximation of an arbitrary qubit state based on the fidelity distance. By the comparison of optimal states based on fidelity and trace norm, the advantages and disadvantages of the optimal convex approximation are identified. Several examples are provided to support this. We also analyze the geometrical properties of the target states which can be completely represented by a set of available states. Using the feature of convex combination, we derive the maximum range of completely represented states and clearly illustrate that any qubit state can be optimally prepared by at most three available states in known useable states.