# 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 [1] in 1966 and O. E. Lanford III and D. W. Robinson [2] in 1968 and proved in 1973 by E.H. Lieb and M.B. Ruskai.[3] 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.[4][5][6]

## Definitions

We will use the following notation throughout: A Hilbert space is denoted by ${\displaystyle {\mathcal {H}}}$, and ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$ denotes the bounded linear operators on ${\displaystyle {\mathcal {H}}}$. Tensor products are denoted by superscripts, e.g., ${\displaystyle {\mathcal {H}}^{12}={\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}}$. The trace is denoted by ${\displaystyle {\rm {Tr}}}$.

### 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., ${\displaystyle \rho ^{12}}$ is a density matrix on ${\displaystyle {\mathcal {H}}^{12}}$.

### Entropy

The von Neumann quantum entropy of a density matrix ${\displaystyle \rho }$ is

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

### Relative entropy

Umegaki's[7] quantum relative entropy of two density matrices ${\displaystyle \rho }$ and ${\displaystyle \sigma }$ is

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

### Joint concavity

A function ${\displaystyle g}$ of two variables is said to be jointly concave if for any ${\displaystyle 0\leq \lambda \leq 1}$ the following holds

${\displaystyle g(\lambda A_{1}+(1-\lambda )A_{2},\lambda B_{1}+(1-\lambda )B_{2})\geq \lambda g(A_{1},B_{1})+(1-\lambda )g(A_{2},B_{2}).}$

Ordinary subadditivity [8] concerns only two spaces ${\displaystyle {\mathcal {H}}^{12}}$ and a density matrix ${\displaystyle \rho ^{12}}$. It states that

${\displaystyle S(\rho ^{12})\leq S(\rho ^{1})+S(\rho ^{2})}$

This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies ${\displaystyle S(\rho ^{12}|\rho ^{1})=S(\rho ^{12})-S(\rho ^{1})}$ and ${\displaystyle S(\rho ^{12}|\rho ^{2})=S(\rho ^{12})-S(\rho ^{2})}$ are both non-negative. In the quantum case, however, both can be negative, e.g. ${\displaystyle S(\rho ^{12})}$ can be zero while ${\displaystyle S(\rho ^{1})=S(\rho ^{12})>0}$. Nevertheless, the subadditivity upper bound on ${\displaystyle S(\rho ^{12})}$ continues to hold. The closest thing one has to ${\displaystyle S(\rho ^{12})-S(\rho ^{1})\geq 0}$ is the Araki–Lieb triangle inequality [8]

${\displaystyle S(\rho ^{12})\geq |S(\rho ^{1})-S(\rho ^{2})|}$

which is derived in [8] from subadditivity by a mathematical technique known as 'purification'.

Suppose that the Hilbert space of the system is a tensor product of three spaces: ${\displaystyle {\mathcal {H}}={\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}\otimes {\mathcal {H}}^{3}.}$. 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

For any tri-partite state ${\displaystyle \rho ^{123}}$ the following holds

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

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

${\displaystyle S(A\mid BC)\leq S(A\mid B)}$.

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

${\displaystyle I(A:BC)\geq I(A:B)}$.

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 [9]

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

with the optimal constant ${\displaystyle 2}$.

As mentioned above, SSA was first proved by E.H.Lieb and M.B.Ruskai in,[3] using Lieb's theorem that was proved in.[10] 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 .[11]

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 [12] proposed a different definition of entropy, which was generalized by F.J. Dyson.

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

The Wigner–Yanase–Dyson ${\displaystyle p}$-skew information of a density matrix ${\displaystyle \rho }$. with respect to an operator ${\displaystyle K}$ is

${\displaystyle I_{p}(\rho ,K)={\frac {1}{2}}{\rm {Tr}}[\rho ^{p},K^{*}][\rho ^{1-p},K],}$

### Concavity of p-skew information

It was conjectured by E. P. Wigner and M. M. Yanase in [13] that ${\displaystyle p}$- skew information is concave as a function of a density matrix ${\displaystyle \rho }$ for a fixed ${\displaystyle 0\leq p\leq 1}$.

Since the term ${\displaystyle -{\tfrac {1}{2}}{\rm {Tr}}\rho KK^{*}}$ is concave (it is linear), the conjecture reduces to the problem of concavity of ${\displaystyle Tr\rho ^{p}K^{*}\rho ^{1-p}K}$. As noted in,[10] this conjecture (for all ${\displaystyle 0\leq p\leq 1}$) implies SSA, and was proved for ${\displaystyle p={\tfrac {1}{2}}}$ in,[13] and for all ${\displaystyle 0\leq p\leq 1}$ in [10] in the following more general form: The function of two matrix variables Template:NumBlk is jointly concave in ${\displaystyle A}$ and ${\displaystyle B,}$ when ${\displaystyle 0\leq r\leq 1}$ and ${\displaystyle p+r\leq 1}$.

This theorem is an essential part of the proof of SSA in.[3]

In their paper [13] E. P. Wigner and M. M. Yanase also conjectured the subadditivity of ${\displaystyle p}$-skew information for ${\displaystyle p={\tfrac {1}{2}}}$, which was disproved by Hansen[14] by giving a counterexample.

## First two statements equivalent to SSA

It was pointed out in [8] that the first statement below is equivalent to SSA and A. Ulhmann in [15] showed the equivalence between the second statement below and SSA.

Both of these statements were proved directly in.[3]

## Joint convexity of relative entropy

As noted by Lindblad [16] and Uhlmann ,[17] if, in equation (Template:EquationNote), one takes ${\displaystyle K=1}$ and ${\displaystyle r=1-p,A=\rho }$ and ${\displaystyle B=\sigma }$ and differentiates in ${\displaystyle p}$ at ${\displaystyle p=0}$ one obtains the Joint convexity of relative entropy : i.e., if ${\displaystyle \rho =\sum _{k}\lambda _{k}\rho _{k}}$, and ${\displaystyle \sigma =\sum _{k}\lambda _{k}\sigma _{k}}$, then Template:NumBlk where ${\displaystyle \lambda _{k}\geq 0}$ with ${\displaystyle \sum _{k}\lambda _{k}=1}$.

## 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 ${\displaystyle T}$ from ${\displaystyle {\mathcal {B}}({\mathcal {H}}^{12})\rightarrow {\mathcal {B}}({\mathcal {H}}^{12})}$ given by ${\displaystyle T=1_{{\mathcal {H}}^{1}}\otimes Tr_{{\mathcal {H}}^{2}}}$ . 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 ${\displaystyle T}$ can be written in Uhlmann's representation as

${\displaystyle T(\rho ^{12})=N^{-1}\sum _{j=1}^{N}(1_{{\mathcal {H}}^{1}}\otimes U_{j})\rho ^{12}(1_{{\mathcal {H}}^{1}}\otimes U_{j}^{*}),}$

for some finite ${\displaystyle N}$ and some collection of unitary matrices on ${\displaystyle {\mathcal {H}}^{2}}$ (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 [16] and Uhlmann,[15] whose proof is the one given here.

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

Therefore,

${\displaystyle S(\rho ^{123}||\rho ^{1}\otimes \rho ^{23})-S(\rho ^{12}||\rho ^{1}\otimes \rho ^{2})=S(\rho ^{12})+S(\rho ^{23})-S(\rho ^{123})-S(\rho ^{2})\geq 0,}$

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 ${\displaystyle T}$ 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.

${\displaystyle \rho _{12}\mapsto S(\rho _{1})-S(\rho _{12})}$ is convex. In [3] it was observed that this convexity yields MPT;

${\displaystyle S(\rho _{4})+S(\rho _{2})\leq S(\rho _{12})+S(\rho _{14}).}$

Moreover, if ${\displaystyle \rho _{124}}$ is pure, then ${\displaystyle S(\rho _{2})=S(\rho _{14})}$ and ${\displaystyle S(\rho _{4})=S(\rho _{12})}$, 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,[19][20] for a discussion.

## The case of equality

### Equality in monotonicity of quantum relative entropy inequality

In,[21][22] D. Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel:

${\displaystyle S(T\rho ||T\sigma )=S(\rho ||\sigma ),}$

if and only if there exists a quantum operator ${\displaystyle {\hat {T}}}$ such that

${\displaystyle {\hat {T}}T\sigma =\sigma ,}$ and ${\displaystyle {\hat {T}}T\rho =\rho .}$

Moreover, ${\displaystyle {\hat {T}}}$ can be given explicitly by the formula

${\displaystyle {\hat {T}}\omega =\sigma ^{1/2}T^{*}{\Bigl (}(T\sigma )^{-1/2}\omega (T\sigma )^{-1/2}{\Bigr )}\sigma ^{1/2},}$

D. Petz also gave another condition [21] when the equality holds in Monotonicity of quantum relative entropy: the first statement in Theorem below. Differentiating it at ${\displaystyle t=0}$ we have the second condition. Moreover, M.B. Ruskai gave another proof of the second statement.

${\displaystyle S(T\rho ||T\sigma )=S(\rho ||\sigma ),}$

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,.[23]

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

${\displaystyle {\mathcal {H}}^{B}=\bigoplus _{j}{\mathcal {H}}^{B_{j}^{L}}\otimes {\mathcal {H}}^{B_{j}^{R}}}$

into a direct sum of tensor products, such that

${\displaystyle \rho ^{ABC}=\bigoplus _{j}q_{j}\rho ^{AB_{j}^{L}}\otimes \rho ^{B_{j}^{R}C},}$

## Operator extension of strong subadditivity

In his paper [24] I. Kim studied an operator extension of strong subadditivity, proving the following inequality:

${\displaystyle Tr_{12}{\Bigl (}\rho ^{123}(-\log(\rho ^{12})-\log(\rho ^{23})+\log(\rho ^{2})+\log(\rho ^{123})){\Bigr )}\geq 0.}$

The proof of this inequality is based on Effros's theorem,[25] for which particular functions and operators are chosen to derive the inequality above. M. B. Ruskai describes this work in details in [26] 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.

## References

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2. O. Lanford III, D. W. Robinson, Jour. Mathematical Physics, 9, 1120 (1968)
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14. F. Hansen, The Wigner-Yanase Entropy is Not Subadditive, J. Stat. Phys. 126, 643–648 (2007).
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23. P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of States which Satisfy Strong Subadditivity of Quantum Entropy with Equality, Comm. Math. Phys. 246, 359–374 (2003).
24. I. Kim, Operator Extension of Strong Subadditivity of Entropy, arXiv:1210.5190 (2012).
25. E. G. Eﬀros. A Matrix Convexity Approach to Some Celebrated Quantum Inequalities. Proc. Natl. Acad. Sci. USA 106(4), 1006–1008 (2009).
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