Strong subadditivity of quantum entropy

We use the following notation throughout the following: A Hilbert space is denoted by $\backslash mathcal\{H\}$, and $\backslash mathcal\{B\}(\backslash mathcal\{H\})$ denotes the bounded linear operators on $\backslash mathcal\{H\}$.

Tensor products are denoted by superscripts, e.g., $\backslash mathcal\{H\}^\{12\}=\backslash mathcal\{H\}^1\backslash otimes\; \backslash mathcal\{H\}^2$. The trace

is denoted by $\{\backslash rm\; Tr\}$.

A density matrix is a Hermitian, positive semi-definite matrix of trace one. It allows for the description of a quantum system in a mixed state. Density matrices on a tensor product are denoted by superscripts, e.g.,

$\backslash rho^\{12\}$ is a density matrix on $\backslash mathcal\{H\}^\{12\}$.

The von Neumann quantum entropy of a density matrix $\backslash rho$ is

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

Umegaki's{{cite journal | last=Umegaki | first=Hisaharu | title=Conditional expectation in an operator algebra. IV. Entropy and information | journal=Kodai Mathematical Seminar Reports | publisher=Tokyo Institute of Technology, Department of Mathematics | volume=14 | issue=2 | year=1962 | issn=0023-2599 | doi=10.2996/kmj/1138844604 | pages=59–85|doi-access=free}} quantum relative entropy of two density matrices $\backslash rho$ and $\backslash sigma$ is

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

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

:$$

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 {{cite journal|last1=Araki|first1=Huzihiro|last2=Lieb|first2=Elliott H.|title=Entropy inequalities|journal=Communications in Mathematical Physics|volume=18|issue=2|year=1970|pages=160–170|issn=0010-3616|doi=10.1007/BF01646092|bibcode=1970CMaPh..18..160A|s2cid=189832417}} concerns only two spaces $\backslash mathcal\{H\}^\{12\}$ and a density matrix $\backslash rho^\{12\}$. It states that

:$S(\backslash rho^\{12\})\; \backslash leq\; S(\backslash rho^1)\; +S(\backslash rho^2)$

This inequality is true, of course, in classical probability theory, but the latter also contains the

theorem that the conditional entropies $S(\backslash rho^\{12\}\; |\; \backslash rho^1)=\; S(\backslash rho^\{12\}\; )-S(\backslash rho^1)$ and $S(\backslash rho^\{12\}\; |\; \backslash rho^2)=S(\backslash rho^\{12\}\; )\; -S(\backslash rho^2)$ are both non-negative. In the quantum case, however, both can be negative,

e.g. $S(\backslash rho^\{12\})$

can be zero while $S(\backslash rho^1)\; =\; S(\backslash rho^\{2\})\; >0$. Nevertheless, the subadditivity upper bound on $S(\backslash rho^\{12\})$ continues to hold. The closest thing one has

to $S(\backslash rho^\{12\})-\; S(\backslash rho^1)\backslash geq\; 0$ is the Araki–Lieb triangle inequality

: $S(\backslash rho^\{12\})\; \backslash geq\; |S(\backslash rho^1)\; -S(\backslash rho^2)|$

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: $\backslash mathcal\{H\}=\backslash mathcal\{H\}^1\backslash otimes\; \backslash mathcal\{H\}^2\backslash otimes\; \backslash mathcal\{H\}^3.$. 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.

Given a density matrix $\backslash rho^\{123\}$ on $\backslash mathcal\{H\}$, we define a density matrix $\backslash rho^\{12\}$ on $\backslash mathcal\{H\}^1\backslash otimes\; \backslash mathcal\{H\}^2$ as a partial trace:

$\backslash rho^\{12\}=\{\backslash rm\; Tr\}\_\{\backslash mathcal\{H\}^3\}\; \backslash rho^\{123\}$. Similarly, we can define density matrices: $\backslash rho^\{23\}$, $\backslash rho^\{13\}$, $\backslash rho^1$, $\backslash rho^2$, $\backslash rho^3$.

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

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

where $S(\backslash rho^\{12\})=-\{\backslash rm\; Tr\}\_\{\backslash mathcal\{H\}^\{12\}\}\; \backslash rho^\{12\}\; \backslash log\; \backslash rho^\{12\}$, for example.

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

:$S(A\backslash mid\; BC)\backslash leq\; S(A\backslash mid\; B)$.

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

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

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 {{cite journal | doi = 10.1007/s11005-012-0565-6 | volume=101 | title=Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy | year=2012 | journal=Letters in Mathematical Physics | pages=1–11 | last1 = Carlen | first1 = Eric A. | last2 = Lieb | first2 = Elliott H.| issue=1 | arxiv = 1203.4719| bibcode=2012LMaPh.101....1C | s2cid=119317605 }}

:$S(\backslash rho^\{12\})+S(\backslash rho^\{23\})-S(\backslash rho^\{123\})-S(\backslash rho^2)\; \backslash geq\; 2\backslash max\backslash \{S(\backslash rho^1)-S(\backslash rho^\{13\}),S(\backslash rho^3)-S(\backslash rho^\{13\}),\; 0\; \backslash \}$,

with the optimal constant $2$.

J. Kiefer{{cite journal |last1=Kiefer |first1=J. |title=Optimum Experimental Designs |journal=Journal of the Royal Statistical Society, Series B (Methodological) |date=July 1959 |volume=21 |issue=2 |pages=272–310 |doi=10.1111/j.2517-6161.1959.tb00338.x |url=https://doi.org/10.1111/j.2517-6161.1959.tb00338.x}}{{cite web |last1=Ruskai |first1=Mary Beth |title=Evolution of a Fundemental [sic] Theorem on Quantum Entropy |url=https://www.youtube.com/watch?v=P3-xI1u1Y2s |website=youtube.com |publisher=World Scientific |access-date=20 August 2020 |quote=Invited talk at the Conference in Honour of the 90th Birthday of Freeman Dyson, Institute of Advanced Studies, Nanyang Technological University, Singapore, 26–29 August 2013. The note on Kiefer (1959) is at the 26:40 mark.}} proved a peripherally related convexity result in 1959, which is a corollary of an operator Schwarz inequality proved by E.H.Lieb and M.B.Ruskai. However, these results are comparatively simple, and the proofs do not use the results of Lieb's 1973 paper on convex and concave trace functionals. It was this paper that provided the mathematical basis of the proof of SSA by Lieb and Ruskai. 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

.{{cite journal | last1 = Narnhofer | first1 = H. | year = 1985 | title = From Relative Entropy to Entropy | journal = Fizika | volume = 17 | pages = 258–262 }}

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

E. P. Wigner and M. M. Yanase {{cite journal | last1=Wigner | first1=E. P. | last2=Yanase | first2=M. M. | title=Information Content of Distributions | journal=Proceedings of the National Academy of Sciences | volume=49 | issue=6 | date=1 May 1963 | issn=0027-8424 | doi=10.1073/pnas.49.6.910 | pages=910–918| pmid=16591109 | pmc=300031 | bibcode=1963PNAS...49..910W |doi-access=free}} proposed a different definition of entropy, which was generalized by Freeman Dyson.

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

:$I\_p(\backslash rho,\; K)=\backslash frac\{1\}\{2\}\{\backslash rm\; Tr\}[\backslash rho^p,\; K^*][\backslash rho^\{1-p\},\; K],$

where $[A,B]=AB-BA$ is a commutator, $K^*$ is the

adjoint of $K$ and $0\backslash leq\; p\backslash leq\; 1$ is fixed.

It was conjectured by E. P. Wigner and M. M. Yanase in {{cite journal | last1=Wigner | first1=Eugene P. | last2=Yanase | first2=Mutsuo M. | title=On the Positive Semidefinite Nature of a Certain Matrix Expression | journal=Canadian Journal of Mathematics | publisher=Canadian Mathematical Society | volume=16 | year=1964 | issn=0008-414X | doi=10.4153/cjm-1964-041-x | pages=397–406|doi-access=}} that $p$- skew information is concave as a function of a density matrix $\backslash rho$ for a fixed $0\backslash leq\; p\backslash leq\; 1$.

Since the term $-\backslash tfrac\{1\}\{2\}\{\backslash rm\; Tr\}\backslash rho\; KK^*$ is concave (it is linear), the conjecture reduces to the problem of concavity of $Tr\backslash rho^p\; K^*\backslash rho^\{1-p\}K$. As noted in, this conjecture (for all $0\; \backslash leq\; p\; \backslash leq\; 1$) implies SSA, and was proved

for $p=\; \backslash tfrac\{1\}\{2\}$ in, and for all $0\backslash leq\; p\; \backslash leq\; 1$ in

in the following more general form: The function of

two matrix variables

{{NumBlk|:|$A,\; B\; \backslash mapsto\; \{\backslash rm\; Tr\}\; A^\{r\}K^*B^pK$ |{{EquationRef|1}}}}

is jointly concave in $A$ and $B,$

when $0\backslash leq\; r\backslash leq\; 1$ and $p+r\; \backslash leq\; 1$.

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 $p=\backslash tfrac\{1\}\{2\}$, which was disproved by Hansen{{cite journal | last=Hansen | first=Frank | title=The Wigner-Yanase Entropy is not Subadditive | journal=Journal of Statistical Physics | publisher=Springer Nature | volume=126 | issue=3 | date=18 January 2007 | issn=0022-4715 | doi=10.1007/s10955-006-9265-x | pages=643–648| arxiv=math-ph/0609019 | bibcode=2007JSP...126..643H | s2cid=119667187 }} by giving a counterexample.

It was pointed out in that the first statement below is equivalent to SSA and A. Ulhmann in A. Ulhmann, Endlich Dimensionale Dichtmatrizen, II, Wiss. Z. Karl-Marx-University Leipzig 22 Jg. H. 2., 139 (1973). showed the equivalence between the second statement below and SSA.

• $S(\backslash rho^1)+S(\backslash rho^3)-S(\backslash rho^\{12\})-S(\backslash rho^\{23\})\backslash leq\; 0.$ Note that the conditional entropies $S(\backslash rho^\{12\}|\backslash rho^1)$ and $S(\backslash rho^\{23\}|\backslash rho^3)$ do not have to be both non-negative.
• The map $\backslash rho^\{12\}\backslash mapsto\; S(\backslash rho^1)-S(\backslash rho^\{12\})$ is convex.

Both of these statements were proved directly in.

As noted by Lindblad{{cite journal|last1=Lindblad|first1=Göran|title=Expectations and entropy inequalities for finite quantum systems|journal=Communications in Mathematical Physics|volume=39|issue=2|year=1974|pages=111–119|issn=0010-3616|doi=10.1007/BF01608390|bibcode=1974CMaPh..39..111L|s2cid=120760667|url=http://projecteuclid.org/euclid.cmp/1103860161}} and Uhlmann,{{cite journal|last1=Uhlmann|first1=A.|title=Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory|journal=Communications in Mathematical Physics|volume=54|issue=1|year=1977|pages=21–32|issn=0010-3616|doi=10.1007/BF01609834|bibcode=1977CMaPh..54...21U|s2cid=15800519|url=http://projecteuclid.org/euclid.cmp/1103900757}} if, in equation ({{EquationNote|1}}), one takes $K=1$ and $r=1-p,\; A=\backslash rho$ and

$B=\backslash sigma$ and differentiates in $p$ at $p=0$, one

obtains the joint convexity of relative entropy:

i.e., if $\backslash rho=\backslash sum\_k\backslash lambda\_k\backslash rho\_k$, and $\backslash sigma=\backslash sum\_k\backslash lambda\_k\backslash sigma\_k$, then

{{NumBlk|:| $S\backslash Bigl(\backslash sum\_k\; \backslash lambda\_k\backslash rho\_k||\backslash sum\_k\backslash lambda\_k\; \backslash sigma\_k\; \backslash Bigr)\backslash leq\; \backslash sum\_k\backslash lambda\_k\; S(\backslash rho\_k||\backslash sigma\_k),$ |{{EquationRef|2}}}}

where $\backslash lambda\_k\backslash geq\; 0$ with $\backslash sum\_k\backslash lambda\_k=1$.

The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations $\backslash mathcal\{N\}$ on density matrices,

$S(\backslash mathcal\{N\}(\backslash rho)\backslash |\backslash mathcal\{N\}(\backslash sigma))\backslash leq\; S(\backslash rho\backslash |\backslash sigma)$.

This inequality is called Monotonicity of quantum relative entropy. Owing to the Stinespring factorization theorem, this inequality is a consequence of a particular choice of the CPTP map - a partial trace map described below.

The most important and basic class of CPTP maps is a partial trace operation $T:\backslash mathcal\{B\}(\backslash mathcal\{H\}^\{12\})$

\rightarrow \mathcal{B}(\mathcal{H}^{1}), given by $T=1\_\{\backslash mathcal\{H\}^1\}\backslash otimes\; \backslash mathrm\{Tr\}\_\{\backslash mathcal\{H\}^2\}$. Then

{{NumBlk|:| $S(T\backslash rho||T\backslash sigma)\backslash leq\; S(\backslash rho||\backslash sigma),$ |{{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

$T$ can be written in Uhlmann's representation as

:$T(\backslash rho^\{12\}\; )\; =\; N^\{-1\}\; \backslash sum\_\{j=1\}^N\; (1\_\{\backslash mathcal\{H\}^1\}\backslash otimes\; U\_j)\; \backslash rho^\{12\}(1\_\{\backslash mathcal\{H\}^1\}\backslash otimes\; U\_j^*),$

for some finite $N$ and some collection of unitary matrices on $\backslash mathcal\{H\}^2$ (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

and Uhlmann, whose proof is the one given here.

SSA is obtained from ({{EquationNote|3}})

with $\backslash mathcal\{H\}^1$ replaced by $\backslash mathcal\{H\}^\{12\}$ and

$\backslash mathcal\{H\}^2$ replaced $\backslash mathcal\{H\}^3$. Take $\backslash rho\; =\; \backslash rho^\{123\},$ $\backslash sigma\; =\; \backslash rho^1\backslash otimes\; \backslash rho^\{23\},$ $T=\; 1\_\{\backslash mathcal\{H\}^\{12\}\}\backslash otimes\; Tr\_\{\backslash mathcal\{H\}^3\}$.

Then ({{EquationNote|3}}) becomes

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

Therefore,

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

which is SSA. Thus,

the monotonicity of quantum relative entropy (which follows from ({{EquationNote|1}}) implies SSA.

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 $\backslash Rightarrow$ MPT: follows since the MPT is a particular case of MONO;
• MPT $\backslash Rightarrow$ MONO: was shown by Lindblad,{{cite journal | last=Lindblad | first=Göran | title=Completely positive maps and entropy inequalities | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=40 | issue=2 | year=1975 | issn=0010-3616 | doi=10.1007/bf01609396 | pages=147–151| bibcode=1975CMaPh..40..147L | s2cid=121650206 | url=http://projecteuclid.org/euclid.cmp/1103860462 }} using a representation of stochastic maps as a partial trace over an auxiliary system;
• MPT $\backslash Rightarrow$ SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";
• SSA $\backslash Rightarrow$ MPT: by choosing $\backslash rho\_\{123\}$ to be block diagonal, one can show that SSA implies that the map

$\backslash rho\_\{12\}\backslash mapsto\; S(\backslash rho\_1)-S(\backslash rho\_\{12\})$ is convex. In it was observed that this convexity yields MPT;

• MPT $\backslash Rightarrow$ JC: as it was mentioned above, by choosing $\backslash rho\_\{12\}$ (and similarly, $\backslash sigma\_\{12\}$) to be block diagonal matrix with blocks $\backslash lambda\_k\backslash rho\_k$ (and $\backslash lambda\_k\backslash sigma\_k$), the partial trace is a sum over blocks so that $\backslash rho:=\backslash rho\_2=\backslash sum\_k\backslash lambda\_k\backslash rho\_k$, so from MPT one can obtain JC;
• JC $\backslash Rightarrow$ SSA: using the 'purification process', Araki and Lieb,{{cite journal | last1 = Lieb | first1 = E. H. | year = 1975 | title = Some Convexity and Subadditivity Properties of Entropy | journal = Bull. Am. Math. Soc. | volume = 81 | pages = 1–13 | doi = 10.1090/s0002-9904-1975-13621-4 | doi-access = free }} observed that one could obtain new useful inequalities from the known ones. By purifying $\backslash rho\_\{123\}$ to $\backslash rho\_\{1234\}$ it can be shown that SSA is equivalent to

:$S(\backslash rho\_4)+S(\backslash rho\_2)\backslash leq\; S(\backslash rho\_\{12\})+S(\backslash rho\_\{14\}).$

Moreover, if $\backslash rho\_\{124\}$ is pure, then $S(\backslash rho\_2)=S(\backslash rho\_\{14\})$ and $S(\backslash rho\_4)=S(\backslash rho\_\{12\})$, 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,{{cite journal | last=Ruskai | first=Mary Beth | title=Inequalities for quantum entropy: A review with conditions for equality | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=43 | issue=9 | year=2002 | issn=0022-2488 | doi=10.1063/1.1497701 | pages=4358–4375| arxiv=quant-ph/0205064 | bibcode=2002JMP....43.4358R | s2cid=3051292 }} erratum 46, 019901 (2005) for a discussion.

In,{{cite journal | last=Petz | first=D. |author-link=Dénes Petz | title=Sufficient subalgebras and the relative entropy of states of a von Neumann algebra | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=105 | issue=1 | year=1986 | issn=0010-3616 | doi=10.1007/bf01212345 | pages=123–131| bibcode=1986CMaPh.105..123P | s2cid=18836173 | url=http://projecteuclid.org/euclid.cmp/1104115260 }}{{cite journal | last=Petz | first=D. |author-link=Dénes Petz | title=Sufficiency of Channels over von Neumann Algebras |journal=Quart. J. Math. Oxford |volume=35 |pages=475–483 |date=1986 |doi=10.1093/qmath/39.1.97}} Dénes Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel:

For all states $\backslash rho$ and $\backslash sigma$ on a Hilbert space $\backslash mathcal\{H\}$ and all quantum operators $T:\; \backslash mathcal\{B\}(\backslash mathcal\{H\})\backslash rightarrow\; \backslash mathcal\{B\}(\backslash mathcal\{K\})$,

:$S(T\backslash rho||T\backslash sigma)=\; S(\backslash rho||\backslash sigma),$

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

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

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

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

where $T^*$ is the adjoint map of $T$.

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

For all states $\backslash rho$ and $\backslash sigma$ on $\backslash mathcal\{H\}$ and all quantum operators $T:\; \backslash mathcal\{B\}(\backslash mathcal\{H\})\backslash rightarrow\; \backslash mathcal\{B\}(\backslash mathcal\{K\})$,

:$S(T\backslash rho||T\backslash sigma)=\; S(\backslash rho||\backslash sigma),$

if and only if the following equivalent conditions are satisfied:

• $T^*(T(\backslash rho)^\{it\}T(\backslash sigma)^\{it\})=\backslash rho^\{it\}\backslash sigma^\{-it\}$ for all real $t$.
• $\backslash log\backslash rho-\backslash log\backslash sigma=T^*\backslash Bigl(\backslash log\; T(\backslash rho)-\backslash log\; T(\backslash sigma)\; \backslash Bigr).$

where $T^*$ is the adjoint map of $T$.

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

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

:$\backslash mathcal\{H\}^B=\backslash bigoplus\_j\; \backslash mathcal\{H\}^\{B^L\_j\}\backslash otimes\; \backslash mathcal\{H\}^\{B^R\_j\}$

into a direct sum of tensor products, such that

:$\backslash rho^\{ABC\}=\backslash bigoplus\_j\; q\_j\backslash rho^\{AB^L\_j\}\backslash otimes\backslash rho^\{B^R\_jC\},$

with states $\backslash rho^\{AB^L\_j\}$ on $\backslash mathcal\{H\}^A\backslash otimes\backslash mathcal\{H\}^\{B^L\_j\}$ and $\backslash rho^\{B^R\_jC\}$ on $\backslash mathcal\{H\}^\{B^R\_j\}\backslash otimes\backslash mathcal\{H\}^C$, and a probability distribution $\backslash \{q\_j\backslash \}$.

E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality, namely,

$S(\backslash rho^\{12\})+S(\backslash rho^\{23\})-S(\backslash rho^\{123\})-S(\backslash rho^2)\; \backslash geq\; 2\backslash max\; \backslash \{0,\; S(\backslash rho^1)-S(\backslash rho^\{13\}),\; S(\backslash rho^3)-S(\backslash rho^\{13\})\backslash \}$

If $S(\backslash rho^1)-S(\backslash rho^\{13\})\backslash leq\; 0$ and $S(\backslash rho^3)-S(\backslash rho^\{13\})\backslash leq\; 0$, as is always the case for the classical Shannon entropy, this inequality has nothing to say. For the quantum entropy, on the other hand, it is quite possible that the conditional entropies satisfy $-S(\backslash rho^\{13\}|\backslash rho^1)=S(\backslash rho^1)-S(\backslash rho^\{13\})>0$ or $-S(\backslash rho^\{13\}|\backslash rho^3)=S(\backslash rho^3)-S(\backslash rho^\{13\})>0$ (but never both!). Then, in this "highly quantum" regime, this inequality provides additional information.

The constant 2 is optimal, in the sense that for any constant larger than 2, one can find a state for which the inequality is violated with that constant.

In his paper I. Kim, Operator Extension of Strong Subadditivity of Entropy, {{arXiv|1210.5190}} (2012). I. Kim studied an operator extension of strong subadditivity, proving the following inequality:

For a tri-partite state (density matrix) $\backslash rho^\{123\}$ on $\backslash mathcal\{H\}^1\backslash otimes\; \backslash mathcal\{H\}^2\backslash otimes\backslash mathcal\{H\}^3$,

:$Tr\_\{12\}\backslash Bigl(\backslash rho^\{123\}(-\backslash log(\backslash rho^\{12\})-\backslash log(\backslash rho^\{23\})+\backslash log(\backslash rho^2)+\backslash log(\backslash rho^\{123\}))\backslash Bigr)\; \backslash geq\; 0.$

The proof of this inequality is based on Effros's theorem,{{cite journal | last1 = Effros | first1 = E. G. | year = 2009 | title = A Matrix Convexity Approach to Some Celebrated Quantum Inequalities | journal = Proc. Natl. Acad. Sci. USA | volume = 106 | issue = 4| pages = 1006–1008 | doi = 10.1073/pnas.0807965106 | pmid = 19164582 | pmc = 2633548 | arxiv = 0802.1234 | bibcode = 2009PNAS..106.1006E | doi-access = free }} for which particular functions and operators are chosen to derive the inequality above. M. B. Ruskai describes this work in details in M. B. Ruskai, Remarks on Kim’s Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions, {{arXiv|1211.0049}} (2012). 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.

A significant strengthening of strong subadditivity was proved in 2014,{{cite journal |first1=O. |last1=Fawzi |first2=R. |last2=Renner |title=Quantum conditional mutual information and approximate Markov chains |journal=Communications in Mathematical Physics |volume=340 |issue=2 |pages=575–611 |doi=10.1007/s00220-015-2466-x |arxiv=1410.0664 |date=2015}} which was subsequently improved by Mark Wilde and coworkers.{{cite journal |first=M. M. |last=Wilde |title=Recoverability in quantum information theory |journal=Proceedings of the Royal Society A |volume=471 |issue=2182 |pages=338 |date=2015 |doi=10.1098/rspa.2015.0338 |arxiv=1505.04661}}{{cite journal |first1=Marius |last1=Junge |first2=Renato |last2=Renner |first3=David |last3=Sutter |first4=Mark M. |last4=Wilde |first5=Andreas |last5=Winter |title=Universal recovery maps and approximate sufficiency of quantum relative entropy |journal=Annales Henri Poincaré |volume=19 |issue=10 |pages=2955–2978 |date=2018 |doi=10.1007/s00023-018-0716-0

|arxiv=1509.07127}} In 2017,{{cite journal|last1=Carlen|first1=Eric A.|last2=Vershynina|first2=Anna|date=2020|title=Recovery map stability for the Data Processing Inequality |journal=J. Phys. A: Math. Theor. |volume=53 |pages=035204 |doi=10.1088/1751-8121/ab5ab7 |arxiv=1710.02409}} it was shown that the recovery channel can be taken to be the original Petz recovery map. These improvements of strong subadditivity have physical interpretations in terms of recoverability, meaning that if the conditional mutual information $I(A;B|E)=S(AE)\; +\; S(BE)\; -\; S(E)\; -\; S(ABE)$ of a tripartite quantum state $\backslash rho\_\{ABE\}$ is nearly equal to zero, then it is possible to perform a recovery channel $\backslash mathcal\{R\}\_\{E\backslash to\; AE\}$ (from system E to AE) such that $\backslash rho\_\{ABE\}\; \backslash approx\; \backslash mathcal\{R\}\_\{E\backslash to\; AE\}(\backslash rho\_\{BE\})$. These results thus generalize the exact equality conditions mentioned above.

• Von Neumann entropy
• Conditional quantum entropy
• Quantum mutual information
• Kullback–Leibler divergence

{{reflist}}

Category:Quantum mechanical entropy