Quantum predictive filtering

How can relevant information be extracted from a quantum information source? In many situations, only some part of the total information content produced by an information source is useful. Can one then find an efficient encoding, in the sense of retaining the largest fraction of relevant information? This paper offers one possible solution by giving a generalization of a classical method designed to retain as much relevant information as possible in a lossy data compression. A key feature of the method is to introduce a second information source to define relevance. We quantify the advantage a quantum encoding has over the best classical encoding in general, and we demonstrate using examples that a substantial quantum advantage is possible. We show analytically, however, that if the relevant information is purely classical, then a classical encoding is optimal.

Figure 1

We consider a data source $X$ and define “relevance” by introducing a quantum channel, ${\mathcal{R}}^{X\to Y}$, to a second information source, $Y$. Information from $X$ is encoded in a memory $M$, and only information about $Y$ is deemed important. Relevant information is quantified by an information measure, $I_{\text{pred}}$, quantifying the correlations between $M$ and $Y$. We refer to $I_{\text{pred}}$ as “predictive power.” The correlations between $M$ and $X$, quantified by an information measure $I_{\text{mem}}$, we refer to as “memory.” The latter quantifies the total encoded information. An encoding is considered more efficient if it has greater $I_{\text{pred}}$ at the same $I_{\text{mem}}$.

Figure 2

An illustration of the causal structure of three related coding protocols. In each case the input is a purification ${\psi }_{XR}$ of the initial data ${\rho }_X$. The protocols differ only by local maps applied to either the data, $X$, or the purification reference, $R$. (a) The “relevance coding protocol” considered in this paper. The data are encoded in a memory by an encoding map, ${\mathcal{E}}^{X\to M}$, while a “relevance channel” ${\mathcal{R}}^{R\to Y}$ is applied to the reference to define relevant information. (b) The conventional protocol for quantum rate-distortion coding. No “relevance channel” is used, and all of the information in the initial data is considered equally important. A decoding map, ${\mathcal{D}}^{M\to Z}$, needs to be introduced to measure the fidelity of the encoded and original data. (c) A generalized rate-distortion coding protocol, of which the two protocols in (a) and (b) are special cases.

Figure 3

The information plane is spanned by $I_{\text{mem}}$ on the horizontal and $I_{\text{pred}}$ on the vertical axis. It can be divided into three regions: the infeasible, the quantum feasible, and the classically feasible region. These regions can be mapped out using the quantum and classical upper bounds on $I_{\text{mem}}$ and the optimal curves found using the quantum information bottleneck method. The optimal quantum curve generally lies above the optimal classical curve, and we quantify the gap by the two positive measures ${\delta }_{\text{pred}}$ and ${\delta }_{\text{mem}}$, as illustrated in the figure. The circles indicate the $\alpha \to 0$ limits of the optimal quantum and classical solutions.

Figure 4

Optimal curve in the information plane for the classical even process. Classical and quantum solutions coincide exactly. Dots show numerical results for a memory of dimension $d_M=3$ (blue) and $d_M=2$ (gray) for comparison. (Inset) Magnification of the region where the solutions for $d_M=3$ and $d_M=2$ bifurcate.

Figure 5

Characteristics of optimal encodings for a qubit phase damping channel. (a)–(c) Quantum solutions in pink, classical in blue. (a) Optimal solutions depicted in the information plane. (b) Conditional entropy $S\left[R\right|M]$ and memory entropy $S\left[M\right]$. (c) Purity of the optimal solution ${\rho }_{MR}^{\text{opt}}$. (d) Quantum correlations in the optimal solutions, quantified by concurrence (green) and quantum discord (purple), for the input (dark) and output (bright) systems, $MR$ and $MY$, respectively.

Figure 6

Optimal encodings for an amplitude damping channel with redundant initial data. (a) The optimal quantum and classical curves in the information plane. (b) The conditional entropy $S\left[X\right|M]$ and the memory entropy $S\left[M\right]$. (c) The purity of the optimal solution ${\rho }_{MR}^{\text{opt}}$. (d) Quantum correlations in the optimal solutions, quantified by concurrence and quantum discord, for the input and output systems, $M{R}_1$ and $MY$, respectively, where $R_1$ refers to the first (relevant) data qubit.

Figure 7

Predictive quantum advantage for the amplitude damping channel, as quantified by ${\delta }_{\text{pred}}\left[I_{\text{mem}}\right]$ (top) and ${\delta }_{\text{mem}}\left[I_{\text{pred}}\right]$ (bottom).