Local versus nonlocal information in quantum-information theory: Formalism and phenomena

In spite of many results in quantum information theory, the complex nature of compound systems is far from clear. In general the information is a mixture of local and nonlocal (“quantum”) information. It is important from both pragmatic and theoretical points of view to know the relationships between the two components. To make this point more clear, we develop and investigate the quantum-information processing paradigm in which parties sharing a multipartite state distill local information. The amount of information which is lost because the parties must use a classical communication channel is the deficit. This scheme can be viewed as complementary to the notion of distilling entanglement. After reviewing the paradigm in detail, we show that the upper bound for the deficit is given by the relative entropy distance to so-called pseudoclassically correlated states; the lower bound is the relative entropy of entanglement. This implies, in particular, that any entangled state is informationally nonlocal—i.e., has nonzero deficit. We also apply the paradigm to defining the thermodynamical cost of erasing entanglement. We show the cost is bounded from below by relative entropy of entanglement. We demonstrate the existence of several other nonlocal phenomena which can be found using the paradigm of local information. For example, we prove the existence of a form of nonlocality without entanglement and with distinguishability. We analyze the deficit for several classes of multipartite pure states and obtain that in contrast to the GHZ state, the Aharonov state is extremely nonlocal. We also show that there do not exist states for which the deficit is strictly equal to the whole informational content (bound local information). We discuss the relation of the paradigm with measures of classical correlations introduced earlier. It is also proved that in the one-way scenario, the deficit is additive for Bell diagonal states. We then discuss complementary features of information in distributed quantum systems. Finally we discuss the physical and theoretical meaning of the results and pose many open questions.

Figure 1

Drawing work from a single heat bath using knowledge about the position of the molecule (the Szilard engine). In the first stage the molecule is known to be on the right-hand side. Next, a piston is inserted, and the molecule pushes it out, thus performing $kT\text{bits}$ of work. After this stage, the position of the molecule is unknown, and we cannot use it to perform more work.

Figure 2

CLOCC protocol of concentration of information to local form is a series of actions aiming to reach the set of pseudoclassically correlated states. The solid lines denote reversible actions: sending dephased qubits or local unitary transformations. The dotted lines denote dephasings. The goal is to make the total entropy increase $\Delta S=\Delta S_1+\Delta S_2+\cdots$ minimal. Then the deficit is given by $\Delta =\Delta S$, because once the state is pseudoclassically correlated, its full information content can be localized.

Figure 3

Plot of $I_l^x$ versus $x^2$ for measurement in basis (61) for the $W$ state. The optimal basis for maximizing $I_l^x$ is for Alice to dephase (or measure) with $x^2=1∕3$ or $2∕3$. The basis ∣±⟩ $(x^2=1∕2)$ is not optimal.

Figure 4

Plot of the function $I_l^{xy}$ [Eq. (62)] for the three-qubit state in Eq. (60) for the case when $a=e=0$, $b=0.1$, in the $(c,r)$ plane. Here $x=r$, $y=\sqrt{1-r^2}$, and $r>0$. The value of localizable information $I_l$ for a given $c$ is the supremum of $I_l^{xy}$ for that value of $c$.

Figure 5

Zero-way deficit is plotted versus mutual information for 100 000 random two-qubit states. The upper line stands for ${\Delta }^{\stackrel{̸}{0}}=I_M$. The lower line denotes isotropic states of Eq. (66). Two regimes are evident: in the first regime, there are states for which the deficit is almost equal to mutual information. In the second region, the deficit tends to half the mutual information.