arXiv:2105.07336 · Europhysics Letters (EPL)

Optimal convex approximation of qubit states and geometry of completely represented states

Hongyi Zhou, Ting Gao, Fengli Yan

Published Oct 1, 2021• 3 citations

Abstract

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.

DOI: 10.1209/0295-5075/ac3362

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arXiv:2105.07336 · Europhysics Letters (EPL)

Optimal convex approximation of qubit states and geometry of completely represented states

Hongyi Zhou, Ting Gao, Fengli Yan

Published Oct 1, 2021• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.1209/0295-5075/ac3362

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Complete optimal convex approximations of qubit states under

B_2

B 2 distance

We consider the optimal approximation of arbitrary qubit states with respect to an available states consisting the eigenstates of two of three Pauli matrices, the

B_2

-distance of an arbitrary target state. Both the analytical formulae of the

B_2

-distance and the corresponding complete optimal decompositions are obtained. The trade-off relations for both the sum and the squared sum of the

B_2

-distances have been analytically and numerically investigated.

{{p}_{i}}

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.

Optimal convex approximation of quantum channels based on

α

-affinity

α

-affinity measure. We construct a channel distance metric induced by the α-affinity and the ChoiJamiolkowski isomorphism, which satisfies the required properties of a well-defined channel distance. Subsequently, we present an optimization framework for the convex approximation of quantum channels, and derive analytical solutions for the optimal convex approximation of single-qubit unitary channels over both the SU(2)-covariant and Pauli channel families, obtaining closed-form expressions for the optimal parameters and the minimal approximation distance. This framework is further applied to the amplitude-damping channel, yielding the explicit form of its optimal approximation and the associated minimal α-affinity distance. In contrast to conventional approaches based on the diamond norm, our framework provides a systematic and analytically tractable approach to quantum channel approximation under realistic constraints.