How long can it take for a quantum channel to forget everything?

TL;DR

This paper explores the properties of quantum channels that completely erase information after a finite number of applications, establishing bounds on the number of steps needed based on the system's dimension.

Contribution

It provides a bound on the order of forgetfulness for quantum channels and constructs explicit examples, also extending results to memory channels with additional inputs and outputs.

Findings

Bound on the forgetfulness order: k ≤ d^2 - 1.

Explicit construction scheme for such channels.

Representation of memory channels with bounded memory depth.

Abstract

We investigate quantum channels, which after a finite number $k$ of repeated applications erase all input information, i.e., channels whose $k$-th power (but no smaller power) is a completely depolarizing channel. We show that on a system with Hilbert space dimension $d$, the order is bounded by $k\leq d^2-1$, and give an explicit construction scheme for such channels. We also consider strictly forgetful memory channels, i.e., channels with an additional input and output in every step, which after exactly $k$ steps retain no information about the initial memory state. We establish an explicit representation for such channels showing that the same bound applies for the memory depth $k$ in terms of the memory dimension $d$.