Unambiguous Examining the Orthogonality of Multiple Quantum States

TL;DR

This paper proves that it is impossible to unambiguously determine whether multiple unknown quantum states are orthogonal or to measure their orthogonality degree with a threshold, highlighting fundamental limits in quantum state comparison.

Contribution

The paper establishes a stronger no-go theorem showing the impossibility of unambiguous orthogonality testing among multiple unknown quantum states.

Findings

Unambiguous measurement cannot confirm orthogonality with non-zero probability.

Extended orthogonality comparison with thresholds is also impossible unambiguously.

The results are stronger than existing no-go theorems in quantum state comparison.

Abstract

In this article, we study an opposite problem of universal quantum state comparison, that is unambiguous determining whether multiple unknown quantum states from a Hilbert space are orthogonal or not. We show that no unambiguous quantum measurement can accomplish this task with a non-zero probability. Moreover, we extend the problem to a more general case, that is to compare how orthogonal multiple unknown quantum states are with a threshold, and it turns out that given any threshold this extended task is also impossible by any unambiguous quantum measurement except for a trivial case. It will be seen that the impossibility revealed in our problem is stronger than that in the universal quantum state comparison problem and distinct from those in the existing "no-go" theorems.