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 reduced density matrices describing the subsystems.

Some useful references here are:

• "Quantum Computation and Quantum Information"[4]
• "Quantum Entropy and Its Use"[5]
• Trace Inequalities and Quantum Entropy: An Introductory Course[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 allows for the description of 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

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

Subadditivity of entropy

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 ${\displaystyle S(\rho ^{1})=S(\rho ^{2})>0}$. 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'.

Strong subadditivity (SSA)

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.

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

Statement

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

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

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

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]

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

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

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

where ${\displaystyle [A,B]=AB-BA}$ is a commutator, ${\displaystyle K^{*}}$ is the adjoint of ${\displaystyle K}$ and ${\displaystyle 0\leq p\leq 1}$ is fixed.

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

${\displaystyle A,B\mapsto {\rm {Tr}}A^{r}K^{*}B^{p}K}$

(1)

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.

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

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 (1), 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

${\displaystyle S{\Bigl (}\sum _{k}\lambda _{k}\rho _{k}||\sum _{k}\lambda _{k}\sigma _{k}{\Bigr )}\leq \sum _{k}\lambda _{k}S(\rho _{k}||\sigma _{k}),}$

(2)

where ${\displaystyle \lambda _{k}\geq 0}$ with ${\displaystyle \sum _{k}\lambda _{k}=1}$.

Monotonicity of quantum relative entropy

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

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

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 ${\displaystyle T:{\mathcal {B}}({\mathcal {H}}^{12})\rightarrow {\mathcal {B}}({\mathcal {H}}^{1})}$, given by ${\displaystyle T=1_{{\mathcal {H}}^{1}}\otimes \mathrm {Tr} _{{\mathcal {H}}^{2}}}$. Then

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

(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 ${\displaystyle T}$ 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 (3) now follows from (2). This theorem is due to Lindblad [16] and Uhlmann,[15] whose proof is the one given here.

SSA is obtained from (3) with ${\displaystyle {\mathcal {H}}^{1}}$ replaced by ${\displaystyle {\mathcal {H}}^{12}}$ and ${\displaystyle {\mathcal {H}}^{2}}$ replaced ${\displaystyle {\mathcal {H}}^{3}}$. Take ${\displaystyle \rho =\rho ^{123},}$ ${\displaystyle \sigma =\rho ^{1}\otimes \rho ^{23},}$ ${\displaystyle T=1_{{\mathcal {H}}^{12}}\otimes Tr_{{\mathcal {H}}^{3}}}$. Then (3) becomes

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

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 (1) implies SSA.

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 ${\displaystyle \Rightarrow }$ MPT: follows since the MPT is a particular case of MONO;
• MPT ${\displaystyle \Rightarrow }$ MONO: was shown by Lindblad,[18] using a representation of stochastic maps as a partial trace over an auxiliary system;
• MPT ${\displaystyle \Rightarrow }$ SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";
• SSA ${\displaystyle \Rightarrow }$ MPT: by choosing ${\displaystyle \rho _{123}}$ to be block diagonal, one can show that SSA implies that the map

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

• MPT ${\displaystyle \Rightarrow }$ JC: as it was mentioned above, by choosing ${\displaystyle \rho _{12}}$ (and similarly, ${\displaystyle \sigma _{12}}$) to be block diagonal matrix with blocks ${\displaystyle \lambda _{k}\rho _{k}}$ (and ${\displaystyle \lambda _{k}\sigma _{k}}$), the partial trace is a sum over blocks so that ${\displaystyle \rho$ :=\rho _{2}=\sum _{k}\lambda _{k}\rho _{k}} , so from MPT one can obtain JC;
• JC ${\displaystyle \Rightarrow }$ SSA: using the 'purification process', Araki and Lieb,[8][19] observed that one could obtain new useful inequalities from the known ones. By purifying ${\displaystyle \rho _{123}}$ to ${\displaystyle \rho _{1234}}$ it can be shown that SSA is equivalent to

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

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:

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

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

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

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

where ${\displaystyle T^{*}}$ is the adjoint map of ${\displaystyle T}$.

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

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

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

if and only if the following equivalent conditions are satisfied:

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

where ${\displaystyle T^{*}}$ is the adjoint map of ${\displaystyle T}$.

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

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

with states ${\displaystyle \rho ^{AB_{j}^{L}}}$ on ${\displaystyle {\mathcal {H}}^{A}\otimes {\mathcal {H}}^{B_{j}^{L}}}$ and ${\displaystyle \rho ^{B_{j}^{R}C}}$ on ${\displaystyle {\mathcal {H}}^{B_{j}^{R}}\otimes {\mathcal {H}}^{C}}$, and a probability distribution ${\displaystyle \{q_{j}\}}$.

Carlen-Lieb Extension

E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality,[9] namely,

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

If ${\displaystyle S(\rho ^{1})-S(\rho ^{13})\leq 0}$ and ${\displaystyle S(\rho ^{3})-S(\rho ^{13})\leq 0}$, 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 ${\displaystyle -S(\rho ^{13}|\rho ^{1})=S(\rho ^{1})-S(\rho ^{13})>0}$ or ${\displaystyle -S(\rho ^{13}|\rho ^{3})=S(\rho ^{3})-S(\rho ^{13})>0}$ (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.

Operator extension of strong subadditivity

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

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

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

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.

Extensions of strong subadditivity in terms of recoverability

A significant strengthening of strong subadditivity was proved in 2014,[27] which was subsequently improved in [28] and.[29] In 2017,[30] 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 ${\displaystyle I(A;B|E)=S(AE)+S(BE)-S(E)-S(ABE)}$ of a tripartite quantum state ${\displaystyle \rho _{ABE}}$ is nearly equal to zero, then it is possible to perform a recovery channel ${\displaystyle {\mathcal {R}}_{E\to AE}}$ (from system E to AE) such that ${\displaystyle \rho _{ABE}\approx {\mathcal {R}}_{E\to AE}(\rho _{BE})}$. These results thus generalize the exact equality conditions mentioned above.

See also

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

References

1. Robinson, Derek W.; Ruelle, David (1967). "Mean entropy of states in classical statistical mechanics". Communications in Mathematical Physics. Springer Science and Business Media LLC. 5 (4): 288–300. doi:10.1007/bf01646480. ISSN 0010-3616.
2. Lanford, Oscar E.; Robinson, Derek W. (1968). "Mean Entropy of States in Quantum‐Statistical Mechanics". Journal of Mathematical Physics. AIP Publishing. 9 (7): 1120–1125. doi:10.1063/1.1664685. ISSN 0022-2488.
3. Lieb, Elliott H.; Ruskai, Mary Beth (1973). "Proof of the strong subadditivity of quantum‐mechanical entropy" (PDF). Journal of Mathematical Physics. AIP Publishing. 14 (12): 1938–1941. doi:10.1063/1.1666274. ISSN 0022-2488.
4. M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambr. U. Press, (2000)
5. M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer (1993)
6. E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).
7. Umegaki, Hisaharu (1962). "Conditional expectation in an operator algebra. IV. Entropy and information". Kodai Mathematical Seminar Reports. Tokyo Institute of Technology, Department of Mathematics. 14 (2): 59–85. doi:10.2996/kmj/1138844604. ISSN 0023-2599.
8. Araki, Huzihiro; Lieb, Elliott H. (1970). "Entropy inequalities". Communications in Mathematical Physics. 18 (2): 160–170. doi:10.1007/BF01646092. ISSN 0010-3616.
9. Carlen, Eric A.; Lieb, Elliott H. (2012). "Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy". Letters in Mathematical Physics. 101: 1–11. arXiv:1203.4719. doi:10.1007/s11005-012-0565-6.
10. Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-X. ISSN 0001-8708.
11. Narnhofer, H. (1985). "From Relative Entropy to Entropy". Fizika. 17: 258–262.
12. Wigner, E. P.; Yanase, M. M. (1 May 1963). "Information Content of Distributions". Proceedings of the National Academy of Sciences. 49 (6): 910–918. doi:10.1073/pnas.49.6.910. ISSN 0027-8424.
13. Wigner, Eugene P.; Yanase, Mutsuo M. (1964). "On the Positive Semidefinite Nature of a Certain Matrix Expression". Canadian Journal of Mathematics. Canadian Mathematical Society. 16: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X.
14. Hansen, Frank (18 January 2007). "The Wigner-Yanase Entropy is not Subadditive". Journal of Statistical Physics. Springer Nature. 126 (3): 643–648. arXiv:math-ph/0609019. doi:10.1007/s10955-006-9265-x. ISSN 0022-4715.
15. A. Ulhmann, Endlich Dimensionale Dichtmatrizen, II, Wiss. Z. Karl-Marx-University Leipzig 22 Jg. H. 2., 139 (1973).
16. Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2): 111–119. doi:10.1007/BF01608390. ISSN 0010-3616.
17. Uhlmann, A. (1977). "Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory". Communications in Mathematical Physics. 54 (1): 21–32. doi:10.1007/BF01609834. ISSN 0010-3616.
18. Lindblad, Göran (1975). "Completely positive maps and entropy inequalities". Communications in Mathematical Physics. Springer Science and Business Media LLC. 40 (2): 147–151. doi:10.1007/bf01609396. ISSN 0010-3616.
19. Lieb, E. H. (1975). "Some Convexity and Subadditivity Properties of Entropy". Bull. AMS. 81: 1–13. doi:10.1090/s0002-9904-1975-13621-4.
20. Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. AIP Publishing. 43 (9): 4358–4375. arXiv:quant-ph/0205064. doi:10.1063/1.1497701. ISSN 0022-2488. erratum 46, 019901 (2005)
21. Petz, Dénes (1986). "Sufficient subalgebras and the relative entropy of states of a von Neumann algebra". Communications in Mathematical Physics. Springer Science and Business Media LLC. 105 (1): 123–131. doi:10.1007/bf01212345. ISSN 0010-3616.
22. D. Petz, Sufficiency of Channels over von Neumann Algebras, Quart. J. Math. Oxford 35, 475–483 (1986).
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. Effros, E. G. (2009). "A Matrix Convexity Approach to Some Celebrated Quantum Inequalities". Proc. Natl. Acad. Sci. USA. 106 (4): 1006–1008. doi:10.1073/pnas.0807965106.
26. M. B. Ruskai, Remarks on Kim’s Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions, arXiv:1211.0049 (2012).
27. O. Fawzi, R. Renner. Quantum conditional mutual information and approximate Markov chains. Communications in Mathematical Physics: 340, 2 (2015)
28. M. M. Wilde. Recoverability in quantum information theory. Proceedings of the Royal Society A, vol. 471, no. 2182, page 20150338 October 2015
29. Marius Junge, Renato Renner, David Sutter, Mark M. Wilde, Andreas Winter. Universal recovery maps and approximate sufficiency of quantum relative entropy. Annales Henri Poincare, vol. 19, no. 10, pages 2955--2978, October 2018 arXiv:1509.07127
30. Carlen, Eric A.; Vershynina, Anna (2017-10-06). "Recovery map stability for the Data Processing Inequality". arXiv:1710.02409 [math.OA].