Maximally entangled states are not always useful for single shot distinguishability of unitaries

TL;DR

The paper investigates how entanglement affects single-shot discrimination of finite sets of unitary channels. Using Hellström-type analyses and convex-hull criteria, it shows that maximally entangled probes are not universally optimal: for two unitaries they are equivalent to product probes, but in $d\ge 3$ there exist pairs perfectly distinguishable with non-maximally entangled inputs while all maximally entangled probes fail or are equivalent depending on the trace of $U_1^\dagger U_2$. It provides two constructive families of unitaries illustrating when non-maximally entangled or product inputs outperform maximally entangled ones and derives necessary and sufficient conditions for such advantages, highlighting a nuanced role of entanglement beyond the maximally entangled paradigm. The results offer guidance for probe design in quantum channel discrimination and prompt further study of input-state optimality for broader channel classes.

Abstract

There have been many instances where the maximally entangled state as a probe acts better than the product and the non-maximally entangled states in the task of distinguishing quantum channels. We provide a proof that for single-shot discrimination of two unitary channels, entangled and product states are operationally equivalent. But we identify pairs of unitaries that are perfectly distinguishable using a non-maximally entangled state, but not with a maximally entangled one. This contrast becomes more pronounced when the number of unitaries exceeds two. In every dimension $\geqslant 3$, we show that there exists a class of unitaries that are indistinguishable under maximally entangled probes, yet perfectly distinguishable using product or non-maximally entangled inputs. Another interesting set of unitaries in every dimension $\geqslant 3$ has been presented where only non-maximally entangled state acts as the successful probe, while product states and maximally entangled states cannot.

Figures (1)

• Figure 1: An example is depicted, where the position of the eigenvalues of $U_1^\dagger U_2$ has been shown here in the complex plane and $U_1$ and $U_2$ are $3$-dimensional unitaries. $\mathcal{D}_P=\frac{1}{2}(1+\sqrt{1-|r_1|^2})$ and $\mathcal{D}_{ME}=\frac{1}{2}(1+\sqrt{1-|r_2|^2})$. In general, $\mathcal{D}_P\geqslant\mathcal{D}_{ME}$.

Theorems & Definitions (16)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof