Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality

We consider the classical correlations that two observers can extract by measurements on a bipartite quantum state and we discuss how they are related to the quantum mutual information of the state. We show with several examples how complementarity gives rise to a gap between the quantum and the classical correlations and we relate our quantitative finding to the so-called classical correlation locked in a quantum state. We derive upper bounds for the sum of classical correlation obtained by measurements in different mutually unbiased bases and we show that the complementarity gap is also present in the deterministic quantum computation with one quantum bit.

Figure 1

Four quantum information processing protocols involving shared bipartite quantum states: (a) Extracting classical measurement records by local measurements with the aim to maximize the mutual information between the records. (b) Merging Alice’s part of the state to Bob while preserving any correlations with a (possibly fictitious) reservoir $R$. The merging uses local operations and classical communication, and it may consume $\left(K>L\right)$ or, actually, produce $\left(L>K\right)$ maximally entangled states between Alice and Bob. (c) Interconverting between $M$ copies of a quantum state and $K$ maximally entangled qubit pairs, $|{\Psi }^{-}⟩=\frac{1}{\sqrt{2}}(|0_A{1}_B⟩-|1_A{0}_B⟩\phantom{)}$, while allowing classical communication. (d) Extracting classical measurement records while allowing classical communication. This may increase the mutual information by a larger amount than what is transmitted through the classical channel, an effect referred to as “locking” of classical information in a quantum state (see Sec. 3B).

Figure 2

Plot of $Q$ for Werner states with $d=2$, 3, and 10 (from below).

Figure 3

The DQC1 model.

Figure 4

Plot of $Q$ of the DQC1 model for $10+1$ qubits as a function of $\alpha$.