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Wikipedia / Squashed entanglement

Squashed entanglement, also called CMI entanglement (CMI can be pronounced "see me"), is an information theoretic measure of quantum entanglement is the density matrix is defined by

. Thus, CMI entanglement is defined as an extremum of a functional, the quantum Conditional Mutual Information (CMI), below. A more general version of Eq.(1) replaces the "min" (minimum) in Eq.(1) by an "inf" (infimum, in agreement with the definition of entanglement of formation is the Von Neumann entropy.

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

, it defines the CMI as

.

. We will represent the partial trace. One can define a quantum analogue of Eq.(2) by

and a quantum analogue of Eq.(3) by

. This inequality is often called the strong-subadditivity property of quantum entropy.

of the form

. 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

that satisfies Eq.(6). We want a measure of quantum entanglement that vanishes in the classical regime.

is an orthonormal basis for the Hilbert space, respectively. It can be shown that the following density matrix

, which is a separable state given by Eq.(1) vanishes for all separable states.

. This agrees with the definition of entanglement of formation for pure states, as given in Ben96.

, then Eq.(1) reduces to the definition of entanglement of formation for mixed states, as given in Ben96 represents total ignorance.

, one expects that CMI entanglement inherits many desirable properties from entanglement of formation.

History

was first proved by Lieb and Ruskai in LR73.

Classical CMI, given by Eq.(3), first entered information theory lore, shortly after Shannon's seminal 1948 paper and at least as early as 1954 in McG54. The quantum CMI, given by Eq.(5), was first defined by Cerf and Adami in Cer96. 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 instead of CMI to understand entanglement. The first paper to explicitly point out a connection between CMI and quantum entanglement appears to be Tuc99.

The final definition Eq.(1) of CMI entanglement was first given by Tucci in a series of 6 papers. (See, for example, Eq.(8) of Tuc02 and Eq.(42) of Tuc01a). 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. The name squashed entanglement first appeared in Chr03. 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.

References

  • Ali03

  • BCY10

  • Ben96

  • Cer96

  • Cer97

  • Chr03

  • Chr05

  • Chr06 Cambridge PhD thesis.

  • Hay03

  • Hua14

  • LR73 Elliott H. Lieb, Mary Beth Ruskai, "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", Journal of Mathematical Physics 14 (1973) 1938–1941.

  • McG54 W.J. McGill, "Multivariate Information Transmission", IRE Trans. Info. Theory 4 (1954) 93–111.

  • Tuc99

  • Tuc00a

  • Tuc00b

  • Tuc01a

  • Tuc01b

  • Tuc02