Quantum state merging and negative information

TL;DR

The paper introduces quantum state merging as a fundamental primitive and proves that the optimal entanglement cost equals the quantum conditional entropy $S(A|B)$ while the optimal classical communication cost equals $I(A:R)$. It provides a rigorous one-shot framework, proves achievability and optimality, and derives a range of applications including distributed compression, quantum side-information coding, multipartite entanglement of assistance, and the capacity region of the quantum multiple access channel. An operational proof of strong subadditivity is given via state merging, linking core quantum information quantities to network tasks. Overall, the work unifies several quantum information processing tasks under the state merging paradigm and clarifies the roles of coherent information and mutual information in quantum communication.

Abstract

We consider a quantum state shared between many distant locations, and define a quantum information processing primitive, state merging, that optimally merges the state into one location. As announced in [Horodecki, Oppenheim, Winter, Nature 436, 673 (2005)], the optimal entanglement cost of this task is the conditional entropy if classical communication is free. Since this quantity can be negative, and the state merging rate measures partial quantum information, we find that quantum information can be negative. The classical communication rate also has a minimum rate: a certain quantum mutual information. State merging enabled one to solve a number of open problems: distributed quantum data compression, quantum coding with side information at the decoder and sender, multi-party entanglement of assistance, and the capacity of the quantum multiple access channel. It also provides an operational proof of strong subadditivity. Here, we give precise definitions and prove these results rigorously.

Figures (3)

• Figure 1: A graphical representation of the building blocks of classical information theory. The total information of the source producing pairs of random variables $X,Y$ is $H(XY)$, while the information contained in just variable $X$ ($Y$) is $H(X)$ ($H(Y)$). The information common to both variables is the mutual information $I(X:Y)$, while the partial informations are $H(X|Y)$ and $H(Y|X)$. In the quantum case, the quantum mutual information $I(A:B)$ can be greater than the total information $S(AB)$, which can be also greater than the local informations $S(A)$ and $S(B)$. To compensate, the partial informations $S(A|B)$ and $S(B|A)$ can be negative.
• Figure 2: The rate region for distributed compression by two parties with individual rates $R_A$ and $R_B$. The total rate $R_{AB}$ is bounded by $S(AB)$. The top left diagram shows the rate region of a source with positive conditional entropies; the top right and bottom left diagrams show the purely quantum case of sources where $S(B|A)<0$ or $S(A|B)<0$. It is even possible that both $S(B|A)$ and $S(A|B)$ are negative, as shown in the bottom right diagram, but observe that the rate-sum $S(AB)$ has to be positive.
• Figure 3: The rate region for the multiple-access channel for two parties with individual rates $R_A$ and $R_B$. The total rate $R_{AB}$ is bounded by $I(AB\rangle C)$. The top left diagram shows the rate region when both rates are positive; the top right and bottom left diagrams show the case where $I(B\rangle C) < 0$ or $I(A\rangle C)<0$. I.e. here, Bob (Alice) can invest entanglement so that the other party can send at a rate $I(A\rangle BC) \geq R_A \geq I(A\rangle C)$ ($I(B\rangle AC) \geq R_B \geq I(B\rangle C)$). In the bottom right diagram, both parties may have the option of achieving the higher rate by having the other party invest entanglement.

Theorems & Definitions (16)

• Definition 1: State merging
• Theorem 2: Quantum State Merging
• Proposition 3: Merging condition
• Proposition 4: One-shot merging
• Remark 5
• Lemma 6
• Remark 7
• Theorem 8
• Definition 9
• Proposition 10: Random measurement gives covering