Squashed entanglement
Thus, CMI entanglement is defined as an extremum of a functional
, the quantum Conditional Mutual Information (CMI), below.
, in agreement with the definition of entanglement of formation for pure states.
is the Von Neumann entropy of density matrix
CMI entanglement has its roots in classical (non-quantum) information theory, as we explain next.
, classical information theory defines the mutual information, a measure of correlations, as For three random variables
This inequality is often called the strong-subadditivity property of quantum entropy.
(6)[citation needed], it follows that the classical CMI entanglement,
We want a measure of quantum entanglement that vanishes in the classical regime.
is an orthonormal basis for the Hilbert space associated with a quantum system
This agrees with the definition of entanglement of formation for pure states, as given in Ben96.
are some states in the Hilbert space associated with a quantum system
be the set of density matrices defined previously for Eq.(1).
(1) reduces to the definition of entanglement of formation for mixed states, as given in Ben96.
(3), first entered information theory lore, shortly after Shannon's seminal 1948 paper and at least as early as 1954 in McG54.
However, it appears that Cerf and Adami did not realize the relation of CMI to entanglement or the possibility of obtaining a measure of quantum entanglement based on CMI; this can be inferred, for example, from a later paper, Cer97, where they try to use
The first paper to explicitly point out a connection between CMI and quantum entanglement appears to be Tuc99.
(1) of CMI entanglement was first given by Tucci in a series of 6 papers.
In Tuc00b, he pointed out the classical probability motivation of Eq.
(1), and its connection to the definitions of entanglement of formation for pure and mixed states.
In Tuc01a, he presented an algorithm and computer program, based on the Arimoto-Blahut method of information theory, for calculating CMI entanglement numerically.
In Tuc01b, he calculated CMI entanglement analytically, for a mixed state of two qubits.
In Hay03, Hayden, Jozsa, Petz and Winter explored the connection between quantum CMI and separability.
It was not however, until Chr03, that it was shown that CMI entanglement is in fact an entanglement measure, i.e. that it does not increase under Local Operations and Classical Communication (LOCC).
The proof adapted Ben96 arguments about entanglement of formation.
In Chr03, they also proved many other interesting inequalities concerning CMI entanglement, including that it was additive, and explored its connection to other measures of entanglement.
In Chr05, Christandl and Winter calculated analytically the CMI entanglement of some interesting states.
In Ali03, Alicki and Fannes proved the continuity of CMI entanglement.
In BCY10, Brandao, Christandl and Yard showed that CMI entanglement is zero if and only if the state is separable.
In Hua14, Huang proved that computing squashed entanglement is NP-hard.
