Information-preserving structures: A general framework for quantum zero-error information

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system’s ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., “noiseless,” “unitarily correctible,” etc.) and introduce two natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational criteria, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel’s information-preserving structures.

Figure 1

Image of the Bloch sphere under a dephasing channel.

Figure 2

Action of a “squashing” channel on a classical trit, and a code (thick red line) that is not preserved even though all $1$-norm distances between code states remain fixed.

Figure 3

The block-diagonal shape of the matrices (states) comprising an IPS.

Figure 4

Image of the Bloch sphere under the nonphysical, non-CP “pancake map.”

Figure 5

An input-output graph for a classical channel on four states. Vertices on the left represent input states, those on the right represent output states, and edges represent possible mappings.