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'''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 <ref>D. W. Robinson and D. Ruelle, Mean Entropy of States in Classical Statistical Mechanis, Communications in Mathematical Physics 5, 288 (1967)</ref> in 1966 and O. E. Lanford III and D. W. Robinson <ref>O. Lanford III, D. W. Robinson, Jour. Mathematical
Physics, 9, 1120 (1968)</ref> in 1968 and proved
in 1973 by E.H. Lieb and M.B. Ruskai.<ref name="LR73_2">E. H. Lieb, M. B. Ruskai, Proof of the Strong Subadditivity of Quantum Mechanichal Entropy, J. Math. Phys. 14, 1938–1941 (1973).</ref> It is a basic theorem in modern [[quantum information theory]].

SSA concerns the relation between the [[von Neumann entropy|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 matrix|density matrices]] describing
the subsystems.

Some useful references here are.<ref>M. Nielsen, I. Chuang Quantum Computation and Quantum Information, Cambr. U. Press, (2000)</ref><ref>M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer (1993)</ref><ref name="C09">E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).</ref>

==Definitions==

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

===Density matrix===
A [[density matrix]] is a [[Hermitian matrix|Hermitian]], [[Positive semi-definite matrix|positive semi-definite]] matrix of [[Trace class|trace]] one. It describes a [[quantum system]] in a [[Quantum state|mixed state]]. Density matrices on a tensor product are denoted by superscripts, e.g.,
<math>\rho^{12}</math> is a density matrix on <math>\mathcal{H}^{12}</math>.

===Entropy===
The von Neumann [[Von Neumann entropy|quantum entropy]] of a density matrix <math>\rho</math> is
:<math>S(\rho):=-{\rm Tr}(\rho\log \rho)</math>.

===Relative entropy===
Umegaki's<ref>H. Umegaki, Conditional Expectation in an Operator Algebra. IV. Entropy and Information, Kodai Math. Sem. Rep. 14, 59–85, (1962)</ref> [[quantum relative entropy]] of two density matrices <math>\rho</math> and <math>\sigma</math> is
:<math>S(\rho||\sigma)={\rm Tr}(\rho\log\rho-\rho\log\sigma)\geq 0 </math>.

===Joint concavity===
A function <math>g</math> of two variables is said to be ''' jointly concave''' if for any <math> 0\leq \lambda\leq 1</math> the following holds
:<math>
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).
</math>

==Subadditivity of entropy==

Ordinary subadditivity <ref name="AL70">H. Araki, E. H. Lieb, Entropy Inequalities, Commun. Math. Phys. 18, 160–170 (1970).</ref> concerns only two spaces <math>\mathcal{H}^{12}</math> and a density matrix <math>\rho^{12}</math>. It states that
:<math> S(\rho^{12}) \leq S(\rho^1) +S(\rho^2) </math>
This inequality is true, of course, in classical probability theory, but the latter also contains the
theorem that the [[Conditional entropy|conditional entropies]] <math> S(\rho^{12} | \rho^1)= S(\rho^{12} )-S(\rho^1)</math> and <math> S(\rho^{12} | \rho^2)=S(\rho^{12} ) -S(\rho^2)</math> are both non-negative. In the quantum case, however, both can be negative,
e.g. <math> S(\rho^{12}) </math>
can be zero while <math> S(\rho^1) = S(\rho^{12}) >0</math>. Nevertheless, the subadditivity upper bound on <math> S(\rho^{12}) </math> continues to hold. The closest thing one has
to <math> S(\rho^{12})- S(\rho^1)\geq 0 </math> is the Araki–Lieb triangle inequality <ref name="AL70" />
: <math> S(\rho^{12}) \geq |S(\rho^1) -S(\rho^2)| </math>
which is derived in <ref name="AL70" /> from subadditivity by a mathematical technique known as 'purification'.

==Strong subadditivity (SSA)==

Suppose that the Hilbert space of the system is a [[tensor product]] of three spaces: <math>\mathcal{H}=\mathcal{H}^1\otimes \mathcal{H}^2\otimes \mathcal{H}^3.</math>. 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.

Given a density matrix <math>\rho^{123}</math> on <math>\mathcal{H}</math>, we define a density matrix <math>\rho^{12}</math> on <math>\mathcal{H}^1\otimes \mathcal{H}^2</math> as a [[partial trace]]:
<math>\rho^{12}={\rm Tr}_{\mathcal{H}^3} \rho^{123}</math>. Similarly, we can define density matrices: <math>\rho^{23}</math>, <math>\rho^{13}</math>, <math>\rho^1</math>, <math>\rho^2</math>, <math>\rho^3</math>.

===Statement===
For any tri-partite state <math>\rho^{123}</math> the following holds
:<math>S(\rho^{123})+S(\rho^2)\leq S(\rho^{12})+S(\rho^{23})</math>,
where <math> S(\rho^{12})=-{\rm Tr}_{\mathcal{H}^{12}} \rho^{12} \log \rho^{12}</math>, for example.

Equivalently, the statement can be recast in terms of [[Conditional quantum entropy|conditional entropies]] to show that for tripartite state <math>\rho^{ABC}</math>,
:<math>S(A\mid BC)\leq S(A\mid B)</math>.
This can also be restated in terms of [[quantum mutual information]],
:<math>I(A:BC)\geq I(A:B)</math>.
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 <ref>Eric A. Carlen, Elliott H. Lieb, Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy, Letters in Mathematical Physics, v.101, 1, 1-11, (2012)</ref>
:<math>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 \} </math>,
with the optimal constant <math>2</math>.

As mentioned above, SSA was first proved by E.H.Lieb and M.B.Ruskai in,<ref name="LR73_2" /> using Lieb's theorem that was proved in.<ref name="L73">E. H. Lieb, Convex Trace Function and Proof of Wigner–Yanase–Dyson Conjecture, Adv. Math. 11, 267–288 (1973).</ref>
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
.<ref>H. Narnhofer, W.Thirring, From Relative Entropy to Entropy, Fizika 17, 258–262, (1985)</ref>

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 <ref name="WY63">E. P. Wigner, M. M. Yanase, Information Content of Distributions, Proc. Nat. Acad. Sci. USA 49, 910–918 (1963).</ref> proposed a different definition of entropy, which was generalized by F.J. Dyson.

===The Wigner–Yanase–Dyson ''p''-skew information===
'''The Wigner–Yanase–Dyson <math>p</math>-skew information''' of a density matrix <math>\rho</math>. with respect to an operator <math>K</math> is
:<math> I_p(\rho, K)=\frac{1}{2}{\rm Tr}[\rho^p, K^*][\rho^{1-p}, K],</math>
where <math>[A,B]=AB-BA</math> is a commutator, <math> K^* </math> is the
adjoint of <math>K</math> and <math>0\leq p\leq 1</math> is fixed.

===Concavity of ''p''-skew information===
It was conjectured by E. P. Wigner and M. M. Yanase in <ref name="WY64">E. P. Wigner, M. M. Yanase, On the Positive Semi-Definite Nature of a Certain Matrix Expression, Can. J. Math. 16, 397–406, (1964).</ref> that <math>p</math>- skew information is concave as a function of a density matrix <math>\rho</math> for a fixed <math>0\leq p\leq 1</math>.

Since the term <math>-\tfrac{1}{2}{\rm Tr}\rho KK^*</math> is concave (it is linear), the conjecture reduces to the problem of concavity of <math>Tr\rho^p K^*\rho^{1-p}K</math>. As noted in,<ref name="L73" /> this conjecture (for all <math> 0 \leq p \leq 1</math>) implies SSA, and was proved
for <math> p= \tfrac{1}{2}</math> in,<ref name= "WY64" /> and for all <math> 0\leq p \leq 1 </math> in <ref name="L73" />
in the following more general form: The function of
two matrix variables
{{NumBlk|:|<math> A, B \mapsto {\rm Tr} A^{r}K^*B^pK </math> |{{EquationRef|1}}}}
is jointly concave in <math> A</math> and <math> B,</math>
when <math>0\leq r\leq 1</math> and <math>p+r \leq 1</math>.

This theorem is an essential part of the proof of SSA in.<ref name="LR73_2" />

In their paper <ref name="WY64"/> E. P. Wigner and M. M. Yanase also conjectured the subadditivity of <math>p</math>-skew information for <math>p=\tfrac{1}{2}</math>, which was disproved by Hansen<ref>F. Hansen, The Wigner-Yanase Entropy is Not Subadditive, J. Stat. Phys.
126, 643–648 (2007).</ref> by giving a counterexample.

==First two statements equivalent to SSA==

It was pointed out in <ref name="AL70" /> that the first statement below is equivalent to SSA and A. Ulhmann in <ref name="U73">A. Ulhmann, Endlich Dimensionale Dichtmatrizen, II, Wiss. Z. Karl-Marx-University Leipzig 22 Jg. H. 2., 139 (1973).</ref> showed the equivalence between the second statement below and SSA.
* <math> S(\rho^1)+S(\rho^3)-S(\rho^{12})-S(\rho^{23})\leq 0.</math> Note that the conditional entropies <math>S(\rho^{12}|\rho^1)</math> and <math>S(\rho^{23}|\rho^3)</math> do not have to be both non-negative.
* The map <math> \rho^{12}\mapsto S(\rho^1)-S(\rho^{12}) </math> is convex.

Both of these statements were proved directly in.<ref name="LR73_2" />

==Joint convexity of relative entropy==

As noted by Lindblad <ref name="Ldb74">G. Lindblad, Expectations and Entropy Inequalities for Finite Quantum Systems, Commun. Math. Phys. 39, 111–119 (1974).</ref> and Uhlmann
,<ref name="U77">A. Ulhmann, Relative Entropy and the Wigner–Yanase–Dyson–Lieb Concavity in an Interpolation Theory, Comm. Math. Phys,54, 21–32, (1977).</ref> if, in equation ({{EquationNote|1}}), one takes <math> K=1</math> and <math> r=1-p, A=\rho</math> and
<math>B=\sigma</math> and differentiates in <math> p</math> at <math>p=0</math> one
obtains the '''Joint convexity of relative entropy ''':
i.e., if <math>\rho=\sum_k\lambda_k\rho_k</math>, and <math>\sigma=\sum_k\lambda_k\sigma_k</math>, then
{{NumBlk|:| <math> S\Bigl(\sum_k \lambda_k\rho_k||\sum_k\lambda_k \sigma_k \Bigr)\leq \sum_k\lambda_k S(\rho_k||\sigma_k),</math> |{{EquationRef|2}}}}
where <math>\lambda_k\geq 0</math> with <math>\sum_k\lambda_k=1</math>.

==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 <math>T</math> from <math> \mathcal{B}(\mathcal{H}^{12})
\rightarrow \mathcal{B}(\mathcal{H}^{12})</math> given by
<math>T=1_{\mathcal{H}^1}\otimes Tr_{\mathcal{H}^2}</math> . Then

{{NumBlk|:| <math> S(T\rho||T\sigma)\leq S(\rho||\sigma), </math> |{{EquationRef|3}}}}

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
<math> T</math> can be written in Uhlmann's representation as
:<math> 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^*), </math>
for some finite <math> N</math> and some collection of unitary matrices on <math> \mathcal{H}^2 </math> (alternatively, integrate over [[Haar measure]]). Since the trace (and hence the relative entropy) is unitarily invariant,
inequality ({{EquationNote|3}}) now follows from ({{EquationNote|2}}). This theorem is due to Lindblad <ref name="Ldb74" />
and Uhlmann,<ref name="U73" /> whose proof is the one given here.

SSA is obtained from ({{EquationNote|3}})
with <math> \mathcal{H}^1 </math> replaced by <math> \mathcal{H}^{12} </math> and
<math> \mathcal{H}^2 </math> replaced <math> \mathcal{H}^3 </math>. Take <math> \rho = \rho^{123}, \sigma = \rho^1\otimes \rho^{23}, T= 1_{\mathcal{H}^{12}}\otimes Tr_{\mathcal{H}^3}</math>.
Then ({{EquationNote|3}}) becomes
:<math> S(\rho^{12}||\rho^1\otimes \rho^2)\leq S(\rho^{123}||\rho^1\otimes\rho^{23}).</math>

Therefore,
:<math>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, </math>
which is SSA. Thus,
the monotonicity of quantum relative entropy (which follows from ({{EquationNote|1}}) implies SSA.

Owing to the [[Stinespring factorization theorem]], equation ({{EquationNote|3}}) is valid not only for partial traces
but also when <math>T</math> 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);
* Strong subadditivity (SSA);
* Joint convexity of quantum relative entropy (JC);

The following implications show the equivalence between these inequalities.
* MONO <math>\Rightarrow </math> MPT: follows since the MPT is a particular case of MONO;

* MPT <math>\Rightarrow </math> MONO: was showed by Lindblad,<ref name="Ldb75">G. Lindblad, Completely Positive Maps and Entropy Inequalities, Commun. Math. Phys. 40, 147–151 (1975).</ref> using a representation of stochastic maps as a partial trace over an auxiliary system;

* MPT <math>\Rightarrow</math> SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";

* SSA <math>\Rightarrow</math> MPT: by choosing <math>\rho_{123}</math> to be block diagonal, one can show that SSA implies that the map
<math>\rho_{12}\mapsto S(\rho_1)-S(\rho_{12})</math> is convex. In <ref name="LR73_2" /> it was observed that this convexity yields MPT;

* MPT <math>\Rightarrow</math> JC: as it was mentioned above, by choosing <math>\rho_{12}</math> (and similarly, <math>\sigma_{12}</math>) to be block diagonal matrix with blocks <math>\lambda_k\rho_k</math> (and <math>\lambda_k\sigma_k</math>), the partial trace is a sum over blocks so that <math>\rho:=\rho_2=\sum_k\lambda_k\rho_k</math>, so from MPT one can obtain JC;

* JC <math>\Rightarrow</math> SSA: using the 'purification process', Araki and Lieb,<ref name="AL70" /><ref name="L75">E. H. Lieb, Some Convexity and Subadditivity Properties of Entropy, Bull. AMS 81, 1–13 (1975).</ref> observed that one could obtain new useful inequalities from the known ones. By purifying <math>\rho_{123}</math> to <math>\rho_{1234}</math> it can be shown that SSA is equivalent to
:<math> S(\rho_4)+S(\rho_2)\leq S(\rho_{12})+S(\rho_{14}). </math>
Moreover, if <math>\rho_{124}</math> is pure, then <math>S(\rho_2)=S(\rho_{14})</math> and <math>S(\rho_4)=S(\rho_{12})</math>, 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,<ref name="L75" /><ref name="R02">M. B. Ruskai, Inequalities for Quantum Entropy: A Review with Conditions
for Equality, J. Math. Phys. 43, 4358–4375 (2002); erratum 46, 019901 (2005)</ref> for a discussion.

==The case of equality==

===Equality in monotonicity of quantum relative entropy inequality===
In,<ref name="P86_1">D. Petz, Sufficient Subalgebras and the Relative Entropy of States of a von Neumann Algebra, Commun. Math.Phys. 105, 123–131 (1986).</ref><ref name="P86_2">D. Petz, Sufficiency of Channels over von Neumann Algebras, Quart. J. Math. Oxford 35, 475–483 (1986).</ref> D. Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel:

For all states <math>\rho</math> and <math>\sigma</math> on a Hilbert space <math>\mathcal{H}</math> and all quantum operators <math>T: \mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{K})</math>,
:<math> S(T\rho||T\sigma)= S(\rho||\sigma), </math>
if and only if there exists a quantum operator <math>\hat{T}</math> such that
:<math> \hat{T}T\sigma=\sigma,</math> and <math>\hat{T}T\rho=\rho.</math>
Moreover, <math>\hat{T}</math> can be given explicitly by the formula
:<math> \hat{T}\omega=\sigma^{1/2}T^*\Bigl((T\sigma)^{-1/2}\omega(T\sigma)^{-1/2} \Bigr)\sigma^{1/2}, </math>
where <math>T^*</math> is the [[Hermitian adjoint|adjoint map]] of <math>T</math>.

D. Petz also gave another condition <ref name="P86_1" /> when the equality holds in Monotonicity of quantum relative entropy: the first statement in Theorem below. Differentiating it at <math>t=0</math> we have the second condition. Moreover, M.B. Ruskai gave another proof of the second statement.

For all states <math>\rho</math> and <math>\sigma</math> on <math>\mathcal{H}</math> and all quantum operators <math>T: \mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{K})</math>,
:<math> S(T\rho||T\sigma)= S(\rho||\sigma),</math>
if and only if the following equivalent conditions are satisfied:
* <math> T^*(T(\rho)^{it}T(\sigma)^{it})=\rho^{it}\sigma^{-it}</math> for all real <math>t</math>.
*<math> \log\rho-\log\sigma=T^*\Bigl(\log T(\rho)-\log T(\sigma) \Bigr).</math>
where <math>T^*</math> is the adjoint map of <math>T</math>.

===Equality in strong subadditivity inequality===
P. Hayden, R. Jozsa, D. Petz and A. Winter described the states for which the equality holds in SSA,.<ref name="HJPW03">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).</ref>

A state <math>\rho^{ABC}</math> on a Hilbert space <math>\mathcal{H}^A\otimes\mathcal{H}^B\otimes\mathcal{H}^C</math> satisfies strong subadditivity with equality if and only if there is a decomposition of second system as
:<math> \mathcal{H}^B=\bigoplus_j \mathcal{H}^{B^L_j}\otimes \mathcal{H}^{B^R_j} </math>
into a direct sum of tensor products, such that
:<math> \rho^{ABC}=\bigoplus_j q_j\rho^{AB^L_j}\otimes\rho^{B^R_jC},</math>
with states <math>\rho^{AB^L_j}</math> on <math>\mathcal{H}^A\otimes\mathcal{H}^{B^L_j}</math> and <math>\rho^{B^R_jC}</math> on <math>\mathcal{H}^{B^R_j}\otimes\mathcal{H}^C</math>, and a probability distribution <math>\{q_j\}</math>.

==Operator extension of strong subadditivity==
In his paper <ref name="K12">I. Kim, Operator Extension of Strong Subadditivity of Entropy, arXiv:1210.5190 (2012).</ref> I. Kim studied an operator extension of strong subadditivity, proving the following inequality:

For a tri-partite state (density matrix) <math>\rho^{123}</math> on <math>\mathcal{H}^1\otimes \mathcal{H}^2\otimes\mathcal{H}^3</math>,
:<math> Tr_{12}\Bigl(\rho^{123}(-\log(\rho^{12})-\log(\rho^{23})+\log(\rho^2)+\log(\rho^{123}))\Bigr) \geq 0.</math>

The proof of this inequality is based on [[Trace inequalities#Effros's theorem|Effros's theorem]],<ref>E. G. Eﬀros. A Matrix Convexity Approach to Some Celebrated
Quantum Inequalities. Proc. Natl. Acad. Sci. USA 106(4), 1006–1008 (2009).</ref> for which particular functions and operators are chosen to derive the inequality above. M. B. Ruskai describes this work in details in <ref name="R12">M. B. Ruskai, Remarks on Kim’s Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions, arXiv:1211.0049 (2012).</ref> 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.

==See also==
* [[Von Neumann entropy]]
* [[Conditional quantum entropy]]
* [[Quantum mutual information]]

==References==
{{reflist}}



[[Category:Quantum mechanical entropy]]
[[Category:Quantum mechanics]]

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