Squashed Entanglement - Motivation For Definition of CMI Entanglement

Motivation For Definition of CMI Entanglement

CMI entanglement has its roots in classical (non-quantum) information theory, as we explain next.

Given any two random variables, classical information theory defines the mutual information, a measure of correlations, as

.

Eq.(2)

For three random variables, it defines the CMI as

.

Eq.(3)

It can be shown that .

Now suppose is the density matrix for a tripartite system . We will represent the partial trace of with respect to one or two of its subsystems by with the symbol for the traced system erased. For example, . One can define a quantum analogue of Eq.(2) by

,

Eq.(4)

and a quantum analogue of Eq.(3) by

.

Eq.(5)

It can be shown that . This inequality is often called the strong-subadditivity property of quantum entropy.

Consider three random variables with probability distribution, which we will abbreviate as . For those special of the form

,

Eq.(6)

it can be shown that . Probability distributions of the form Eq.(6) are in fact described by the Bayesian network shown in Fig.1.

One can define a classical CMI entanglement by

,

Eq.(7)

where is the set of all probability distributions in three random variables, such that for all . Because, given a probability distribution, one can always extend it to a probability distribution that satisfies Eq.(6), it follows that the classical CMI entanglement, is zero for all . The fact that always vanishes is an important motivation for the definition of . We want a measure of quantum entanglement that vanishes in the classical regime.

Suppose for is a set of non-negative numbers that add up to one, and for is an orthonormal basis for the Hilbert space associated with a quantum system . Suppose and, for are density matrices for the systems and, respectively. It can be shown that the following density matrix

Eq.(8)

satisfies . Eq.(8) is the quantum counterpart of Eq.(6). Tracing the density matrix of Eq.(8) over, we get, which is a separable state. Therefore, given by Eq.(1) vanishes for all separable states.

When is a pure state, one gets . This agrees with the definition of entanglement of formation for pure states, as given in Ben96.

Next suppose for are some states in the Hilbert space associated with a quantum system . Let be the set of density matrices defined previously for Eq.(1). Define to be the set of all density matrices that are elements of and have the special form . It can be shown that if we replace in Eq.(1) the set by its proper subset, then Eq.(1) reduces to the definition of entanglement of formation for mixed states, as given in Ben96. and represent different degrees of knowledge as to how was created. represents total ignorance.

Since CMI entanglement reduces to entanglement of formation if one minimizes over instead of, one expects that CMI entanglement inherits many desirable properties from entanglement of formation.