# Squashed entanglement

**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 $\varrho_{A, B}$ is the density matrix of a system (*A*, *B*) composed of two subsystems *A* and *B*, then the CMI Entanglement *E**C**M**I* of system (*A*, *B*) is defined by

$Eq.(1)\qquad E_{CMI}(\varrho_{A, B}) = \frac{1}{2}\min_{\varrho_{A,B,\Lambda}\in K}S(A:B | \Lambda)$,

where *K* is the set of all density matrices $\varrho_{A, B, \Lambda}$ for a tripartite system (*A*, *B*, Λ) such that $\varrho_{A, B}=tr_\Lambda (\varrho_{A, B, \Lambda})$. Thus, CMI Entanglement is defined as an extremum of a functional *S*(*A* : *B*∣Λ) of $\varrho_{A, B, \Lambda}$. We define *S*(*A* : *B*∣Λ), 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 $\varrho_{A, B}$ is a pure state, $E_{CMI}(\varrho_{A, B})=S(\varrho_{A})=S(\varrho_{B})$, in agreement with the definition of Entanglement of Formation for pure states.

### 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 *A*, *B*, Classical Information Theory defines the Mutual Information, a measure of correlations, as

*E**q*.(2) *H*(*A* : *B*) = *H*(*A*) + *H*(*B*) − *H*(*A*, *B*) .

For three random variables *A*, *B*, *C*, it defines the CMI as $Eq.(3)\qquad\begin{matrix}
H(A : B | \Lambda)&=& H(A|\Lambda)+ H(B | \Lambda)- H(A, B | \Lambda)\\
&=&H(A, \Lambda)+H(B, \Lambda)-H(\Lambda)-H(A, B, \Lambda)
\end{matrix}$.

It can be shown that *H*(*A* : *B*∣Λ) ≥ 0.

Now suppose $\varrho_{A, B, \Lambda}$ is the density matrix for a tripartite system (*A*, *B*, Λ). We will represent the partial trace of $\varrho_{A, B, \Lambda}$ with respect to one or two of its subsystems by $\varrho_{A, B, \Lambda}$ with the symbol for the traced system erased. For example, $\varrho_{A, B}= trace_\Lambda(\varrho_{A, B, \Lambda})$. One can define a quantum analogue of Eq.(2) by

$Eq.(4)\qquad S(A:B) = S(\varrho_{A}) + S(\varrho_{B}) -S(\varrho_{A, B}) \,$,

and a quantum analogue of Eq.(3) by

$Eq.(5)\qquad S(A:B|\Lambda) = S(\varrho_{A, \Lambda}) +S(\varrho_{B, \Lambda}) -S(\varrho_\Lambda) -S(\varrho_{A, B, \Lambda}) \,$.

It can be shown that *S*(*A* : *B*∣Λ) ≥ 0. This inequality is often called the strong sub-additivity property of quantum entropy.

Consider three random variables *A*, *B*, Λ with probability distribution *P**A*, *B*, Λ(*a*, *b*, *λ*), which we will abbreviate as *P*(*a*, *b*, *λ*). For those special *P*(*a*, *b*, *λ*) of the form

*E**q*.(6) *P*(*a*, *b*, *λ*) = *P*(*a*∣*λ*)*P*(*b*∣*λ*)*P*(*λ*) ,

it can be shown that *H*(*A* : *B*∣Λ) = 0. 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

*E**q*.(7) *E**C**M**I*(*P**A*, *B*) = min*P**A*, *B*, Λ ∈ *K**H*(*A* : *B*∣Λ),

where *K* is the set of all probability distributions *P**A*, *B*, Λ in three random variables *A*, *B*, Λ, such that ∑*λ**P**A*, *B*, Λ(*a*, *b*, *λ*) = *P**A*, *B*(*a*, *b*) for all *a*, *b*. Because, given a probability distribution *P**A*, *B*, one can always extend it to a probability distribution *P**A*, *B*, Λ that satisfies Eq.(6), it follows that the classical CMI Entanglement, *E**C**M**I*(*P**A*, *B*), is zero for all *P**A*, *B*. The fact that *E**C**M**I*(*P**A*, *B*) always vanishes is an important motivation for the definition of $E_{CMI}( \varrho_{A,B})$. We want a measure of quantum entanglement that vanishes in the classical regime.

Suppose *w**λ* for *λ* = 1, 2, ..., *d**i**m*(Λ) is a set of non-negative numbers that add up to one, and ∣*λ* > for *λ* = 1, 2, ..., *d**i**m*(Λ) is an orthonormal basis for the Hilbert space associated with a quantum system Λ. Suppose $\varrho_A^\lambda$ and $\varrho_B^\lambda$, for *λ* = 1, 2, ..., *d**i**m*(Λ) are density matrices for the systems *A* and *B*, respectively. It can be shown that the following density matrix

$Eq.(8)\qquad \varrho_{A, B, \Lambda}=\sum_\lambda \varrho_A^\lambda \varrho_B^\lambda w_\lambda |\lambda><\lambda| \,$

satisfies *S*(*A* : *B*∣Λ) = 0. Eq.(8) is the quantum counterpart of Eq.(6). Tracing the density matrix of Eq.(8) over Λ, we get $\varrho_{A,B} = \sum_\lambda \varrho_A^\lambda \varrho_B^\lambda w_\lambda \,$, which is a separable state. Therefore, $E_{CMI}(\varrho_{A, B})$ given by Eq.(1) vanishes for all separable states.

When $\varrho_{A, B}$ is a pure state, one gets $E_{CMI}(\varrho_{A, B})=S(\varrho_{A})=S(\varrho_{B})$. This agrees with the definition of Entanglement of Formation for pure states, as given in **Ben96**.

Next suppose ∣*ψ**A*, *B**λ* > for *λ* = 1, 2, ..., *d**i**m*(Λ) are some states in the Hilbert space associated with a quantum system (*A*, *B*). Let *K* be the set of density matrices defined previously for Eq.(1). Define *K**o* to be the set of all density matrices $\varrho_{A, B, \Lambda}$ that are elements of *K* and have the special form $\varrho_{A, B, \Lambda} = \sum_\lambda|\psi_{A,B}^\lambda><\psi_{A,B}^\lambda| w_\lambda |\lambda> <\lambda|\,$. It can be shown that if we replace in Eq.(1) the set *K* by its proper subset *K**o*, then Eq.(1) reduces to the definition of Entanglement of Formation for mixed states, as given in **Ben96**. *K* and *K**o* represent different degrees of knowledge as to how $\varrho_{A, B, \Lambda}$ was created. *K* represents total ignorance.

### History

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 *S*(*A*∣*B*) 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.

Since CMI Entanglement reduces to Entanglement of Formation if one minimizes over *K**o* instead of *K*, one expects that CMI Entanglement inherits many desirable properties from Entanglement of Formation. As first shown in **Ben96**, Entanglement of Formation does not increase under LOCC (Local Operations and Classical Communication). In **Chr03**, Christandl and Winter showed that CMI Entanglement also does not increase under LOCC, by adapting **Ben96** arguments about Entanglement of Formation. In **Chr03**, they also proved many other interesting inequalities concerning CMI Entanglement, 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.

### References

**Ali03**R. Alicki, M. Fannes, ``Continuity of quantum mutual information", quant-ph/0312081**Ben96**C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, ``Mixed State Entanglement and Quantum Error Correction", quant-ph/9604024**Cer96**N. J. Cerf, C. Adami, ``Quantum Mechanics of Measurement", quant-ph/9605002**Cer97**N.J. Cerf, C. Adami, R.M. Gingrich, ``Quantum conditional operator and a criterion for separability", quant-ph/9710001**Chr03**M. Christandl, A. Winter, ``Squashed Entanglement - An Additive Entanglement Measure", quant-ph/0308088**Chr05**M. Christandl, A. Winter, ``Uncertainty, Monogamy, and Locking of Quantum Correlations", quant-ph/0501090**Chr06**M. Christandl, Ph.D. Thesis, quant-ph/0604183**Hay03**P. Hayden, R. Jozsa, D. Petz, A. Winter, ``Structure of states which satisfy strong subadditivity of quantum entropy with equality" quant-ph/0304007**McG54**W.J. McGill, ``Multivariate Information Transmission", IRE Trans. Info. Theory**4**(1954) 93-111.**Tuc99**R.R. Tucci, ``Quantum Entanglement and Conditional Information Transmission", quant-ph/9909041**Tuc00a**R.R. Tucci,``Separability of Density Matrices and Conditional Information Transmission", quant-ph/0005119**Tuc00b**R.R. Tucci, ``Entanglement of Formation and Conditional Information Transmission", quant-ph/0010041**Tuc01a**R.R. Tucci, ``Relaxation Method For Calculating Quantum Entanglement", quant-ph/0101123**Tuc01b**R.R. Tucci, ``Entanglement of Bell Mixtures of Two Qubits", quant-ph/0103040**Tuc02**R.R. Tucci, ``Entanglement of Distillation and Conditional Mutual Information", quant-ph/0202144