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 S_So 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, S_So would approach one as N → ∞, independent of the number of qubits; but we have found that S_So varies nearly linearly with the number of qubits n. The slope of S_Soversus 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.