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"?>
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"?>
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.
The resource theoretic measure of quantum coherence is basis dependent, and the amount of coherence contained in a state is different in different bases. We obtained analytical solutions for the maximum coherence by optimizing the reference basis and highlighted the essential role of the mutually unbiased bases (MUBs) on attaining the maximum coherence. Apart from the relative entropy of coherence, we showed that the MUBs are optimal for the robustness of coherence, the coherence weight, and the modified skew information measure of coherence for any state. Moreover, the MUBs are optimal for all the faithful coherence measures if the state is pure or is of the single qubit. We also highlighted an upper bound for the l1 norm of coherence and compared it with the other bounds as well as the maximum one attainable by optimizing the reference basis.
Quantifying coherence is a key task in both quantum-mechanical theory and practical applications. Here, a reliable quantum coherence measure is presented by utilizing the quantum skew information of the state of interest subject to a certain broken observable. This coherence measure is proven to fulfill all the criteria (especially the strong monotonicity) recently introduced in the resource theories of quantum coherence. The coherence measure has an analytic expression and an obvious operational meaning related to quantum metrology. In terms of this coherence measure, the distribution of the quantum coherence, i.e., how the quantum coherence is distributed among the multiple parties, is studied and a corresponding polygamy relation is proposed. As a further application, it is found that the coherence measure forms the natural upper bounds for quantum correlations prepared by incoherent operations. The experimental measurements of our coherence measure as well as the relative-entropy coherence and lp-norm coherence are studied finally.
In the quantitative theory of quantum coherence, the amount of coherence is defined as the distance between the given state to the closest incoherent state. The set of incoherent states is conventionally defined as any state with a diagonal density matrix. One of the objections to this formulation is that the incoherent states are intrinsically basis-dependent, which makes the amount of coherence also a basis-dependent quantity. Basis-independent measures have recently been proposed where the incoherent state is taken as the maximally mixed state. We show that this is the only possible choice of reference incoherent state, without modifying the original definition of coherence. We find a relation between the two formulations by defining a contribution to the coherence due to the basis choice. The hierarchical relationship between quantum coherence and the various quantum correlations is explored in detail. Finally, we illustrate some operational uses of the basis-independent quantum coherence in quantum information theory tasks.
In terms of the metric-adjusted skew information (an important and versatile class of quantum Fisher information), which generalizes the seminal Wigner-Yanase skew information arising naturally from the study of quantum measurement, we propose a family of coherence measures of states relative to quantum channels, and reveal their basic properties such as unitary covariance, convexity, and monotonicity. Furthermore, we evaluate these coherence measures of states relative to several prototypical quantum channels, and make a comparative study for this family of coherence measures with relative entropy of coherence. This provides a general approach to coherence of states relative to quantum channels, which also captures decoherence on the states caused by quantum channels and asymmetry of states relative to quantum channels.
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.