Fidelity and Entanglement of Random Bipartite Pure States: Insights and Applications

TL;DR

This paper explores the relationship between fidelity and entanglement in random bipartite pure states, revealing consistent average fidelities across dimensions and providing analytical tools for benchmarking quantum devices.

Contribution

It introduces a detailed analysis of fidelity-entanglement distributions in random bipartite states and extends findings to higher dimensions, offering new insights for quantum benchmarking.

Findings

Average fidelity for qubits is 1/4 within a narrow entanglement range.

Average fidelity for qudits is 1/d^2, independent of entanglement.

Analytical fidelity distribution functions aid in quantum device benchmarking.

Abstract

We investigate the fidelity of Haar random bipartite pure states from a fixed reference quantum state and their bipartite entanglement. By plotting the fidelity and entanglement on perpendicular axes, we observe that the resulting plots exhibit non-uniform distributions. The distribution depends on the entanglement of the fixed reference quantum state used to quantify the fidelity of the random pure bipartite states. We find that the average fidelity of typical random pure bipartite qubits within a narrow entanglement range with respect to a randomly chosen fixed bipartite qubit is $\frac{1}{4}$. Extending our study to higher dimensional bipartite qudits, we find that the average fidelity of typical random pure bipartite qudits with respect to a randomly chosen fixed bipartite qudit remains constant within a narrow entanglement range. The values of these constants are \(\frac{1}{d^2}\), with d being the dimension of the local Hilbert space of the bipartite qudit system, suggesting a consistent relationship between entanglement and fidelity across different dimensions. The probability distribution functions of fidelity with respect to a product state are analytically studied and used as a reference for the benchmarking of distributed quantum computing devices.