# Strong Subadditivity of Quantum Entropy

Strong subadditivity of entropy (SSA) was long known and appreciated in classical probability theory and information theory. Its extension to quantum mechanical entropy (the von Neumann entropy) was conjectured by D.W. Robinson and D. Ruelle  in 1966 and O. E. Lanford III and D. W. Robinson  in 1968 and proved in 1973 by E.H. Lieb and M.B. Ruskai. It is a basic theorem in modern quantum information theory.

SSA concerns the relation between the entropies of various subsystems of a larger system consisting of three subsystems (or of one system with three degrees of freedom). The proof of this relation in the classical case is quite easy but the quantum case is difficult because of the non-commutativity of the density matrices describing the subsystems.

Some useful references here are.

## Definitions

### Density matrix

A density matrix is a Hermitian, positive semi-definite matrix of trace one. It describes a quantum system in a mixed state. Density matrices on a tensor product are denoted by superscripts, e.g., $\rho ^{12}$ is a density matrix on ${\mathcal {H}}^{12}$ .

### Entropy

$S(\rho ):=-{\rm {Tr}}(\rho \log \rho )$ .

### Relative entropy

$S(\rho ||\sigma )={\rm {Tr}}(\rho \log \rho -\rho \log \sigma )\geq 0$ .

### Joint concavity

$g(\lambda A_{1}+(1-\lambda )A_{2},\lambda B_{1}+(1-\lambda )B_{2})\geq \lambda g(A_{1},B_{1})+(1-\lambda )g(A_{2},B_{2}).$ $S(\rho ^{12})\leq S(\rho ^{1})+S(\rho ^{2})$ This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies $S(\rho ^{12}|\rho ^{1})=S(\rho ^{12})-S(\rho ^{1})$ and $S(\rho ^{12}|\rho ^{2})=S(\rho ^{12})-S(\rho ^{2})$ are both non-negative. In the quantum case, however, both can be negative, e.g. $S(\rho ^{12})$ can be zero while $S(\rho ^{1})=S(\rho ^{12})>0$ . Nevertheless, the subadditivity upper bound on $S(\rho ^{12})$ continues to hold. The closest thing one has to $S(\rho ^{12})-S(\rho ^{1})\geq 0$ is the Araki–Lieb triangle inequality 

$S(\rho ^{12})\geq |S(\rho ^{1})-S(\rho ^{2})|$ which is derived in  from subadditivity by a mathematical technique known as 'purification'.

Suppose that the Hilbert space of the system is a tensor product of three spaces: ${\mathcal {H}}={\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}\otimes {\mathcal {H}}^{3}.$ . Physically, these three spaces can be interpreted as the space of three different systems, or else as three parts or three degrees of freedom of one physical system.

### Statement

$S(\rho ^{123})+S(\rho ^{2})\leq S(\rho ^{12})+S(\rho ^{23})$ ,

Equivalently, the statement can be recast in terms of conditional entropies to show that for tripartite state $\rho ^{ABC}$ ,

$S(A\mid BC)\leq S(A\mid B)$ .

This can also be restated in terms of quantum mutual information,

$I(A:BC)\geq I(A:B)$ .

These statements run parallel to classical intuition, except that quantum conditional entropies can be negative, and quantum mutual informations can exceed the classical bound of the marginal entropy.

The strong subadditivity inequality was improved in the following way by Carlen and Lieb 

$S(\rho ^{12})+S(\rho ^{23})-S(\rho ^{123})-S(\rho ^{2})\geq 2\max\{S(\rho ^{1})-S(\rho ^{12}),S(\rho ^{2})-S(\rho ^{12}),0\}$ ,

As mentioned above, SSA was first proved by E.H.Lieb and M.B.Ruskai in, using Lieb's theorem that was proved in. The extension from a Hilbert space setting to a von Neumann algebra setting, where states are not given by density matrices, was done by Narnhofer and Thirring .

The theorem can also be obtained by proving numerous equivalent statements, some of which are summarized below.

## Wigner–Yanase–Dyson conjecture

E. P. Wigner and M. M. Yanase  proposed a different definition of entropy, which was generalized by F.J. Dyson.

### The Wigner–Yanase–Dyson p-skew information

$I_{p}(\rho ,K)={\frac {1}{2}}{\rm {Tr}}[\rho ^{p},K^{*}][\rho ^{1-p},K],$ ### Concavity of p-skew information

It was conjectured by E. P. Wigner and M. M. Yanase in  that $p$ - skew information is concave as a function of a density matrix $\rho$ for a fixed $0\leq p\leq 1$ .

This theorem is an essential part of the proof of SSA in.

In their paper  E. P. Wigner and M. M. Yanase also conjectured the subadditivity of $p$ -skew information for $p={\tfrac {1}{2}}$ , which was disproved by Hansen by giving a counterexample.

## First two statements equivalent to SSA

It was pointed out in  that the first statement below is equivalent to SSA and A. Ulhmann in  showed the equivalence between the second statement below and SSA.

Both of these statements were proved directly in.

## Monotonicity of quantum relative entropy

The relative entropy decreases monotonically under certain operations on density matrices, the most important and basic of which is the following. Consider the map $T$ from ${\mathcal {B}}({\mathcal {H}}^{12})\rightarrow {\mathcal {B}}({\mathcal {H}}^{12})$ given by $T=1_{{\mathcal {H}}^{1}}\otimes Tr_{{\mathcal {H}}^{2}}$ . Then

which is called Monotonicity of quantum relative entropy under partial trace.

To see how this follows from the joint convexity of relative entropy, observe that $T$ can be written in Uhlmann's representation as

$T(\rho ^{12})=N^{-1}\sum _{j=1}^{N}(1_{{\mathcal {H}}^{1}}\otimes U_{j})\rho ^{12}(1_{{\mathcal {H}}^{1}}\otimes U_{j}^{*}),$ for some finite $N$ and some collection of unitary matrices on ${\mathcal {H}}^{2}$ (alternatively, integrate over Haar measure). Since the trace (and hence the relative entropy) is unitarily invariant, inequality (Template:EquationNote) now follows from (Template:EquationNote). This theorem is due to Lindblad  and Uhlmann, whose proof is the one given here.

$S(\rho ^{12}||\rho ^{1}\otimes \rho ^{2})\leq S(\rho ^{123}||\rho ^{1}\otimes \rho ^{23}).$ Therefore,

$S(\rho ^{123}||\rho ^{1}\otimes \rho ^{23})-S(\rho ^{12}||\rho ^{1}\otimes \rho ^{2})=S(\rho ^{12})+S(\rho ^{23})-S(\rho ^{123})-S(\rho ^{2})\geq 0,$ which is SSA. Thus, the monotonicity of quantum relative entropy (which follows from (Template:EquationNote) implies SSA.

Owing to the Stinespring factorization theorem, equation (Template:EquationNote) is valid not only for partial traces but also when $T$ is a quantum operation, i.e., a completely positive, trace preserving map. In this general case the inequality is called Monotonicity of quantum relative entropy.

## Relationship among inequalities

All of the above important inequalities are equivalent to each other, and can also be proved directly. The following are equivalent:

• Monotonicity of quantum relative entropy (MONO);
• Monotonicity of quantum relative entropy under partial trace (MPT);
• Joint convexity of quantum relative entropy (JC);

The following implications show the equivalence between these inequalities.

$\rho _{12}\mapsto S(\rho _{1})-S(\rho _{12})$ is convex. In  it was observed that this convexity yields MPT;

$S(\rho _{4})+S(\rho _{2})\leq S(\rho _{12})+S(\rho _{14}).$ Moreover, if $\rho _{124}$ is pure, then $S(\rho _{2})=S(\rho _{14})$ and $S(\rho _{4})=S(\rho _{12})$ , so the equality holds in the above inequality. Since the extreme points of the convex set of density matrices are pure states, SSA follows from JC;

See, for a discussion.

## The case of equality

### Equality in monotonicity of quantum relative entropy inequality

In, D. Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel:

$S(T\rho ||T\sigma )=S(\rho ||\sigma ),$ if and only if there exists a quantum operator ${\hat {T}}$ such that

${\hat {T}}T\sigma =\sigma ,$ and ${\hat {T}}T\rho =\rho .$ Moreover, ${\hat {T}}$ can be given explicitly by the formula

${\hat {T}}\omega =\sigma ^{1/2}T^{*}{\Bigl (}(T\sigma )^{-1/2}\omega (T\sigma )^{-1/2}{\Bigr )}\sigma ^{1/2},$ D. Petz also gave another condition  when the equality holds in Monotonicity of quantum relative entropy: the first statement in Theorem below. Differentiating it at $t=0$ we have the second condition. Moreover, M.B. Ruskai gave another proof of the second statement.

$S(T\rho ||T\sigma )=S(\rho ||\sigma ),$ if and only if the following equivalent conditions are satisfied:

### Equality in strong subadditivity inequality

P. Hayden, R. Jozsa, D. Petz and A. Winter described the states for which the equality holds in SSA,.

A state $\rho ^{ABC}$ on a Hilbert space ${\mathcal {H}}^{A}\otimes {\mathcal {H}}^{B}\otimes {\mathcal {H}}^{C}$ satisfies strong subadditivity with equality if and only if there is a decomposition of second system as

${\mathcal {H}}^{B}=\bigoplus _{j}{\mathcal {H}}^{B_{j}^{L}}\otimes {\mathcal {H}}^{B_{j}^{R}}$ into a direct sum of tensor products, such that

$\rho ^{ABC}=\bigoplus _{j}q_{j}\rho ^{AB_{j}^{L}}\otimes \rho ^{B_{j}^{R}C},$ ## Operator extension of strong subadditivity

In his paper  I. Kim studied an operator extension of strong subadditivity, proving the following inequality:

$Tr_{12}{\Bigl (}\rho ^{123}(-\log(\rho ^{12})-\log(\rho ^{23})+\log(\rho ^{2})+\log(\rho ^{123})){\Bigr )}\geq 0.$ The proof of this inequality is based on Effros's theorem, for which particular functions and operators are chosen to derive the inequality above. M. B. Ruskai describes this work in details in  and discusses how to prove a large class of new matrix inequalities in the tri-partite and bi-partite cases by taking a partial trace over all but one of the spaces.