Squashed entanglement
Encyclopedia
Squashed entanglement, also called CMI entanglement (CMI can be pronounced "see me"), is an information theoretic measure of quantum entanglement for a bipartite quantum system. If 
is the density matrix
of a system
composed of two subsystems 
and 
, then the CMI entanglement 
of system 
is defined by
where
is the set of all density matrices 
for a tripartite system 
such that 
. Thus, CMI entanglement is defined as an extremum of a functional 
of 
. We define 
, 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
). When
is a pure state,
, in agreement with the definition of entanglement of formation for pure states. Here 
is the Von Neumann entropy
of density matrix
.
, as we explain next.
Given any two random variables
, classical information theory defines the mutual information
, a measure of correlations, as
For three random variables
, it defines the CMI as
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
and a quantum analogue of Eq.(3) by
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
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
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
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.
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, where they try to use
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.
is the density matrixDensity matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
of a system
composed of two subsystems and , then the CMI entanglement of system is defined bywhere
is the set of all density matrices for a tripartite system such that . Thus, CMI entanglement is defined as an extremum of a functional of . We define , the quantum Conditional Mutual Information (CMI), below. A more general version of Eq.(1) replaces the ``min" (minimum) in Eq.(1) by an ``inf" (infimumInfimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
). When
is a pure state,, in agreement with the definition of entanglement of formation for pure states. Here is the Von Neumann entropyVon Neumann entropy
In quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics....
of density matrix
.Motivation for definition of CMI entanglement
CMI entanglement has its roots in classical (non-quantum) information theoryInformation theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
, as we explain next.
Given any two random variables
, classical information theory defines the mutual informationMutual information
In probability theory and information theory, the mutual information of two random variables is a quantity that measures the mutual dependence of the two random variables...
, a measure of correlations, as
For three random variables
, it defines the CMI asIt can be shown that
.Now suppose
is the density matrix for a tripartite system . We will represent the partial tracePartial trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function...
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) byand a quantum analogue of Eq.(3) by
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 formit can be shown that
. Probability distributions of the form Eq.(6) are in fact described by the Bayesian networkBayesian network
A Bayesian network, Bayes network, belief network or directed acyclic graphical model is a probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph . For example, a Bayesian network could represent the probabilistic...
shown in Fig.1.
One can define a classical CMI entanglement by
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 matrixsatisfies
. Eq.(8) is the quantum counterpart of Eq.(6). Tracing the density matrix of Eq.(8) over , we get , which is a separable stateSeparable states
In quantum mechanics, separable quantum states are states without quantum entanglement.- Separable pure states :For simplicity, the following assumes all relevant state spaces are finite dimensional...
. Therefore,
given by Eq.(1) vanishes for all separable states.When
is a pure state, one gets. Thisagrees 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.History
The important inequality was first proved by Lieb and Ruskai in LR73.Classical CMI, given by Eq.(3), first entered information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
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, where they try to use
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
Rate distortion theory
Rate–distortion theory is a major branch of information theory which provides the theoretical foundations for lossy data compression; it addresses the problem of determining the minimal amount of entropy R that should be communicated over a channel, so that the source can be approximately...
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
Separability
In mathematics and in physics, separability may refer to properties of:* Separable differential equations** Separable ordinary differential equation, a class of equations that can be separated into a pair of integrals...
.
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.