Quantum information can be negative

Abstract

Given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the "partial information" one system needs conditioned on it's prior information. It turns out to be given by an extremely simple formula, the conditional entropy. In the classical case, partial information must always be positive, but we find that in the quantum world this physical quantity can be negative. If the partial information is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, the sender and receiver instead gain the corresponding potential for future quantum communication. We introduce a primitive "quantum state merging" which optimally transfers partial information. We show how it enables a systematic understanding of quantum network theory, and discuss several important applications including distributed compression, multiple access channels and multipartite assisted entanglement distillation (localizable entanglement). Negative channel capacities also receive a natural interpretation.

Figures (2)

• Figure 1: Diagrammatic representation of the process of state merging. Initially the state $| \psi \rangle$ is shared between the three systems R(eference), A(lice) and B(ob). After the communication Alice's system is in a pure state, while Bob holds not only his but also her initial share. Note that the reference's state $\rho_R$ has not changed, as indicated by the curve separating R from AB. The protocol for state merging is as follows: Let Alice and Bob have a large number $n$ of the state $\rho_{AB}$. To begin, we note that we only need to describe the protocol for negative $S(A|B)$, as otherwise Alice and Bob can share $nS(A|B)$ EPR pairs (by sending this number of quantum bits) and create a state $| \psi \rangle_{AA'BB'R}$ with $S(AA'|BB')<0$. This is because adding an EPR pair reduces the conditional entropy by one unit. However, $S(A|B)<0$ is equivalently expressed as $S(A) > S(AB) = S(R)$, and it is knownshor-capLloyd-capigor-cap that measurement in a uniformly random basis on Alice's $n$ systems projects Bob and R into a state $| \varphi \rangle_{BR}$ whose reduction to R is very close to $\rho_R$. But this means that Bob can, by a local operation, transform $| \varphi \rangle_{BR}$ to $| \psi \rangle_{ABR}$. Finally, by coarse-graining the random measurement, Alice essentially projects onto a good quantum codeshor-capLloyd-capigor-cap of rate $-S(A|B)$; this still results in Bob obtaining the full state $\rho_{AB}$, but now, just under $-nS(A|B)$ EPR pairs are also created. These codes can also be obtained by an alternative constructionDevetakWinter-hash.
• 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. If one party compresses at a rate $S(B)$, then the other party can over-compress at a rate $S(A|B)$, by merging her state with the state which will end up with the decoder. Time-sharing gives the full rate region, since the bounds evidently cannot be improved. Analogously, for $m$ parties $A_i$, and all subsets $T\subseteq\{1,2,\ldots,m\}$ holding a combined state with entropy $S(T)$, the rate sums $R_T = \sum_{i\in T} R_{A_i}$ have to obey $R_T > S(T|\bar{T})$ with $\bar{T} = \{1,2,\ldots,m\}\setminus T$ the complement of set $T$.