The Independence of Distinguishability and the Dimension of the System

TL;DR

This paper proves that the distinguishability of quantum states under LOCC, PPT, and SEP operations is independent of the system dimension, showing indistinguishability is an intrinsic property of the states.

Contribution

It establishes that indistinguishability of quantum states remains unchanged when viewed in larger systems, revealing a fundamental property of quantum states.

Findings

Indistinguishability is independent of system dimension.

Maximal numbers of LOCC$_{1}$ distinguishable states are determined.

Constructs LOCC indistinguishable product bases in general systems.

Abstract

The are substantial studies on distinguishabilities, especially local distinguishability, of quantum states. It is shown that a necessary condition of a local distinguishable state set is the total Schmidt rank not larger than the system dimension. However, if we view states in a larger system, the restriction will be invalid. Hence, a nature problem is that can indistinguishable states become distinguishable by viewing them in a larger system without employing extra resources. In this paper, we consider this problem for (perfect or unambiguous) LOCC$_{1}$, PPT and SEP distinguishabilities. We demonstrate that if a set of states is indistinguishable in $\otimes _{k=1}^{K} C^{d _{k}}$, then it is indistinguishable even being viewed in $\otimes _{k=1}^{K} C^{d _{k}+h _{k}}$, where $K, d _{k}\geqslant2, h _{k}\geqslant0$ are integers. This shows that such distinguishabilities are properties of states themselves and independent of the dimension of quantum system. Our result gives the maximal numbers of LOCC$_{1}$ distinguishable states and can be employed to construct a LOCC indistinguishable product basis in general systems. Our result is suitable for general states in general systems. For further discussions, we define the local-global indistinguishable property and present a conjecture.