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Chinese Physics B

Optimal convex approximations of qubit states based on l 1-norm of coherence

Li‐qiang Zhang, Yan-Dong 彦东 Du 杜, Chang‐shui Yu

Published Jul 3, 2025• 0 citations
Publisher

Abstract

Abstract Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory, offering critical guidance for experimental implementations. In this paper, we embark on an in-depth exploration of the use of a quantum state prepared by the convex combination of given qubit states to optimally approximate the l 1-norm of coherence of the target quantum state, striving to make the prepared state and the target state as similar as possible. Here, we present the analytical solution for the optimal distance for any N given quantum states. We find that the optimal approximation problem for any N > 4 quantum states can be transformed into an optimal approximation problem for no more than four quantum states, which not only significantly streamlines the problem but also proves advantageous for laboratories in terms of material conservation. Ultimately, a one-to-one comparison between the analytical and numerical solutions verifies the effectiveness of our approach.

DOI
10.1088/1674-1056/adeb5d

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Abstract Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory, offering critical guidance for experimental implementations. In this paper, we embark on an in-depth exploration of the use of a quantum state prepared by the convex combination of given qubit states to optimally approximate the l 1-norm of coherence of the target quantum state, striving to make the prepared state and the target state as similar as possible. Here, we present the analytical solution for the optimal distance for any N given quantum states. We find that the optimal approximation problem for any N > 4 quantum states can be transformed into an optimal approximation problem for no more than four quantum states, which not only significantly streamlines the problem but also proves advantageous for laboratories in terms of material conservation. Ultimately, a one-to-one comparison between the analytical and numerical solutions verifies the effectiveness of our approach.
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.
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.
Optimal convex approximation of quantum channels based on ααα-affinity
Determining the minimal distance between a target channel and a convex hull of predefined set of implementable channels is a fundamental problem in quantum resource theory, and provides key guidance for experimental implementations. In this work, we develop a unified analytical framework for optimal convex approximation of quantum channels based on the quantum ααα-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.
Abstract The preparation of quantum states lies at the foundation in the quantum information processing. The convex mixing of some existing quantum states is one of the effective candidate. In this paper, we mainly study how a target quantum state can be optimally prepared by not more than three given pure states. The analytic optimal distance based on the fidelity is found. We also show that the preparation with more than four states can be essentially converted to the case with not more than four states, which can be similarly solved as the case with three states. The validity is illustrated by the comparison of our analytical and numerical results.
Abstract Utilizing the l 1 norm as the quantum state distance metric, we consider the Pauli distance of arbitrary target quantum states, that is, the optimal convex approximation of any qubit mixed states relative to the available state of pairs of eigenstates in three Pauli matrices. We get the complete analytical expressions for B 1 -distance, B 2 -distance and B 3 -distance. We determine the parameter range for the optimal approximation of each B 2 -distance in eight distinct scenarios. Furthermore, some trade-off relations about B 2 -distances are also studied.
In the quantum resource theory， finding the optimal quantum state within the convex combination of given quantum states that is closest to the target quantum state is not only a practical approach for quantifying quantum properties but also holds guiding significance for experimental operations. Common distance metrics for quantum states include trace distance， L1 <?fx-imagestate width="3.64066648" height="3.72533321"?> <?fx-imagestate width="3.64066648" height="3.72533321"?> -norm， and fidelity， through which the optimal quantum state that is most indistinguishable from the target quantum state can be identified. In this paper， based on specific quantum coherence information tasks， we use the Csk <?fx-imagestate width="4.57200003" height="3.72533321"?> <?fx-imagestate width="4.57200003" height="3.72533321"?> -coherence metric and Csku <?fx-imagestate width="4.57200003" height="3.97933316"?> <?fx-imagestate width="4.57200003" height="3.97933316"?> -coherence metric to analyze and compare the approximation effects of optimal quantum states based on these three state distances， and clarify the applicable range of each distance metric. Furthermore， we find that the optimal quantum state approximation based on fidelity can always preserve more skew-information coherence， which gives it an advantage in quantum information tasks that consume skew-information coherence. This also indirectly verifies the correlation between the fidelity distance metric and skew-information coherence to some extent. The choice of state distance also depends on different quantum information tasks， which consume different quantum resources. In this paper， we consider two quantum resources： skew-information coherence Csk <?fx-imagestate width="4.57200003" height="3.72533321"?> <?fx-imagestate width="4.57200003" height="3.72533321"?> and average skew-information coherence Csku <?fx-imagestate width="4.57200003" height="3.97933316"?> <?fx-imagestate width="4.57200003" height="3.97933316"?> . When the given quantum state is in the B1' <?fx-imagestate width="3.72533321" height="3.97933316"?> <?fx-imagestate width="3.72533321" height="3.97933316"?> set， no state distance metric can preserve the skew-information coherence Csk <?fx-imagestate width="4.57200003" height="3.72533321"?> <?fx-imagestate width="4.57200003" height="3.72533321"?> ， but all of them retain a certain amount of average skew-information coherence Csku <?fx-imagestate width="4.57200003" height="3.97933316"?> <?fx-imagestate width="4.57200003" height="3.97933316"?> . Among them， the optimal approximation based on fidelity exhibits the best preservation effect. From the B2' <?fx-imagestate width="3.72533321" height="3.97933316"?> <?fx-imagestate width="3.72533321" height="3.97933316"?> distance， it is found that when the range of the analytical solution is x+z>1 <?fx-imagestate width="15.40933418" height="2.96333337"?> <?fx-imagestate width="15.40933418" height="2.96333337"?> and x+z2+y2>1 <?fx-imagestate width="27.94000053" height="4.82600021"?> <?fx-imagestate width="27.94000053" height="4.82600021"?> where x， y， and z refer to the Bloch vector of the target quantum state ρ <?fx-imagestate width="2.11666679" height="3.64066648"?>