Quantum Generalizations of Multivariate Classical Fidelities

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The authors introduce several quantum generalizations of multivariate classical fidelities, such as the average pairwise z-fidelity, multivariate semi-definite programming fidelity, secrecy-based multivariate fidelity, and multivariate log-Euclidean fidelity. These multivariate quantum fidelities satisfy desirable properties like reduction to classical multivariate fidelities for commuting states, data processing inequality, symmetry, faithfulness, orthogonality, direct-sum property, and joint concavity.

Samenvatting

The paper begins by reviewing classical and quantum bivariate fidelities, including the Uhlmann and Holevo fidelities. It then introduces several multivariate classical fidelities, such as the Matusita multivariate fidelity and the average pairwise fidelity, and establishes relationships between them.
The main contribution is the introduction of quantum generalizations of these multivariate classical fidelities. The authors propose three main variants that reduce to the average pairwise fidelity for commuting states:

Average pairwise z-fidelities: These generalize the classical average pairwise fidelity using the z-fidelity for z ≥ 1/2. The Uhlmann and Holevo average pairwise fidelities are special cases.

Multivariate semi-definite programming (SDP) fidelity: This is obtained by extending the SDP formulation of the Uhlmann fidelity to multiple states.

Secrecy-based multivariate fidelity: This is inspired by an existing secrecy measure and also reduces to the average pairwise fidelity for commuting states.

The authors show that all three of these variants satisfy the desired properties for a multivariate fidelity, including reduction to classical multivariate fidelities, data processing inequality, symmetry, faithfulness, orthogonality, direct-sum property, and joint concavity. They also establish uniform continuity bounds for some of these fidelities.
Additionally, the authors introduce the multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. They show that it satisfies most of the desired properties and has an operational interpretation in terms of quantum hypothesis testing.
Finally, the authors define maximal and minimal extensions of multivariate classical fidelities and analyze their properties.

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Multivariate Fidelities

by Theshani Nur... om arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16101.pdf

Diepere vragen

How do the proposed multivariate quantum fidelities compare in terms of computational complexity and operational interpretations

The proposed multivariate quantum fidelities vary in terms of computational complexity and operational interpretations.

Average Pairwise z-Fidelities: These fidelities involve calculating the z-fidelity for all pairs of quantum states in the tuple and then averaging them. The computational complexity increases with the number of states in the tuple, making it more computationally intensive for larger tuples. Operationally, these fidelities can be interpreted as the average similarity between pairs of quantum states in the tuple.

Multivariate Semi-Definite Programming (SDP) Fidelity: This fidelity involves solving a semi-definite programming problem, which can be computationally demanding. However, it provides a powerful tool for quantifying the similarity between multiple quantum states. Operationally, it can be interpreted as a measure of how well a set of quantum states can be distinguished.

Secrecy-Based Multivariate Fidelity: Inspired by a secrecy measure, this fidelity quantifies the average fidelity between each state in the tuple and a fixed state, maximizing over all possible fixed states. This measure can provide insights into the security of quantum communication protocols.

Multivariate Log-Euclidean Fidelity: This fidelity is a quantum generalization of the Matusita multivariate fidelity and is defined through log-Euclidean divergences. It offers a unique perspective on the similarity of multiple quantum states and can be operationally interpreted in terms of quantum hypothesis testing.

In summary, the computational complexity and operational interpretations of the multivariate quantum fidelities vary based on the specific formulation and the number of states involved in the analysis.

Can the multivariate quantum fidelities be used to define novel quantum information-theoretic quantities, such as multivariate quantum mutual information or multivariate quantum channel capacities

The multivariate quantum fidelities have the potential to define novel quantum information-theoretic quantities and metrics.

Multivariate Quantum Mutual Information: By extending the concept of mutual information to multiple quantum states, the multivariate quantum fidelities could be used to define a multivariate quantum mutual information measure. This measure would capture the total amount of shared information among multiple quantum systems.

Multivariate Quantum Channel Capacities: The multivariate quantum fidelities can also be utilized to define multivariate quantum channel capacities. These capacities would quantify the maximum rates at which information can be reliably transmitted through a quantum channel when multiple quantum states are involved.

By leveraging the properties and operational interpretations of the multivariate quantum fidelities, novel information-theoretic quantities can be defined to enhance our understanding of quantum systems and their interactions.

Are there any applications or use cases where the multivariate quantum fidelities could provide insights that are not captured by bivariate fidelities or other existing multivariate measures

The multivariate quantum fidelities offer unique insights and applications that are not captured by bivariate fidelities or other existing multivariate measures.

Quantum State Comparison: The multivariate fidelities provide a comprehensive way to compare the similarity of multiple quantum states simultaneously. This can be particularly useful in scenarios where the relationships between multiple quantum systems need to be analyzed.

Quantum Hypothesis Testing: The operational interpretations of the multivariate fidelities in terms of quantum hypothesis testing offer a practical application in distinguishing between different quantum states or scenarios. This can be valuable in quantum communication and cryptography protocols.

Security Analysis: The secrecy-based multivariate fidelity can be applied in security analysis to quantify the level of secrecy or privacy in quantum communication systems involving multiple states. It provides a measure of how well the states can be kept secret from eavesdroppers.

Overall, the multivariate quantum fidelities open up new avenues for exploring the relationships and properties of multiple quantum states, leading to a deeper understanding of quantum information processing and communication.