Issue 13, 2022

Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

Abstract

Using IBM's publicly accessible quantum computers, we have analyzed the entropies of Schrödinger's cat states, which have the form Ψ = (1/2)1/2 [|0 0 0⋯0〉 + |1 1 1⋯1〉]. We have obtained the average Shannon entropy SSo of the distribution over measurement outcomes from 75 runs of 8192 shots, for each of the numbers of entangled qubits, on each of the quantum computers tested. For the distribution over N fault-free measurements on pure cat states, SSo would approach one as N → ∞, independent of the number of qubits; but we have found that SSo varies nearly linearly with the number of qubits n. The slope of SSoversus the number of qubits differs among computers with the same quantum volumes. We have developed a two-parameter model that reproduces the near-linear dependence of the entropy on the number of qubits, based on the probabilities of observing the output 0 when a qubit is set to |0〉 and 1 when it is set to |1〉. The slope increases as the error rate increases. The slope provides a sensitive measure of the accuracy of a quantum computer, so it serves as a quickly determinable index of performance. We have used tomographic methods with error mitigation as described in the qiskit documentation to find the density matrix ρ and evaluate the von Neumann entropies of the cat states. From the reduced density matrices for individual qubits, we have calculated the entanglement entropies. The reduced density matrices represent mixed states with approximately 50/50 probabilities for states |0〉 and |1〉. The entanglement entropies are very close to one.

Graphical abstract: Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

Supplementary files

Article information

Article type
Paper
Submitted
17 nov. 2021
Accepted
17 févr. 2022
First published
01 mars 2022

Phys. Chem. Chem. Phys., 2022,24, 7666-7681

Author version available

Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

N. D. Jansen, M. Loucks, S. Gilbert, C. Fleming-Dittenber, J. Egbert and K. L. C. Hunt, Phys. Chem. Chem. Phys., 2022, 24, 7666 DOI: 10.1039/D1CP05255A

To request permission to reproduce material from this article, please go to the Copyright Clearance Center request page.

If you are an author contributing to an RSC publication, you do not need to request permission provided correct acknowledgement is given.

If you are the author of this article, you do not need to request permission to reproduce figures and diagrams provided correct acknowledgement is given. If you want to reproduce the whole article in a third-party publication (excluding your thesis/dissertation for which permission is not required) please go to the Copyright Clearance Center request page.

Read more about how to correctly acknowledge RSC content.

Social activity

Spotlight

Advertisements