Ancilla-Assisted Discrimination of Quantum Gates

TL;DR

This paper investigates the limits of ancilla-assisted quantum gate discrimination, establishing bounds on the number of gates that can be perfectly distinguished with an ancillary system of given dimension.

Contribution

It provides bounds on the number of quantum gates that can be discriminated with ancilla assistance and estimates the minimal ancilla dimension needed for discrimination.

Findings

Any rd+1 quantum gates cannot be perfectly discriminated with r-dimensional ancilla.

There exist rd quantum gates that can be perfectly discriminated with r-dimensional ancilla.

The order O(r) of discriminable gates with r-dimensional ancilla is proven to be optimal.

Abstract

The intrinsic idea of superdense coding is to find as many gates as possible such that they can be perfectly discriminated. In this paper, we consider a new scheme of discrimination of quantum gates, called ancilla-assisted discrimination, in which a set of quantum gates on a $d-$dimensional system are perfectly discriminated with assistance from an $r-$dimensional ancilla system. The main contribution of the present paper is two-fold: (1) The number of quantum gates that can be discriminated in this scheme is evaluated. We prove that any $rd+1$ quantum gates cannot be perfectly discriminated with assistance from the ancilla, and there exist $rd$ quantum gates which can be perfectly discriminated with assistance from the ancilla. (2) The dimensionality of the minimal ancilla system is estimated. We prove that there exists a constant positive number $c$ such that for any $k\leq cr$ quantum gates, if they are $d$-assisted discriminable, then they are also $r$-assisted discriminable, and there are $c^{\prime}r\textrm{}(c^{\prime}>c)$ different quantum gates which can be discriminated with a $d-$dimensional ancilla, but they cannot be discriminated if the ancilla is reduced to an $r-$dimensional system. Thus, the order $O(r)$ of the number of quantum gates that can be discriminated with assistance from an $r-$dimensional ancilla is optimal. The results reported in this paper represent a preliminary step toward understanding the role ancilla system plays in discrimination of quantum gates as well as the power and limit of superdense coding.