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. E.A. Carlen and E.H. Lieb have contributed a strengthening of SSA in 2012. Renato Renner and Omar Fawzi proved a strengthening of strong subadditivity in 2014.
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.
- 1 Definitions
- 2 Subadditivity of entropy
- 3 Strong subadditivity (SSA)
- 4 Wigner–Yanase–Dyson conjecture
- 5 First two statements equivalent to SSA
- 6 Joint convexity of relative entropy
- 7 Monotonicity of quantum relative entropy
- 8 Relationship among inequalities
- 9 The case of equality
- 10 Carlen-Lieb Extension
- 11 Operator extension of strong subadditivity
- 12 See also
- 13 References
We will use the following notation throughout: A Hilbert space is denoted by , and denotes the bounded linear operators on . Tensor products are denoted by superscripts, e.g., . The trace is denoted by .
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., is a density matrix on .
The von Neumann quantum entropy of a density matrix is
A function of two variables is said to be jointly concave if for any the following holds
Subadditivity of entropy
Ordinary subadditivity  concerns only two spaces and a density matrix . It states that
This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies and are both non-negative. In the quantum case, however, both can be negative, e.g. can be zero while . Nevertheless, the subadditivity upper bound on continues to hold. The closest thing one has to is the Araki–Lieb triangle inequality 
which is derived in  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: . 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 on , we define a density matrix on as a partial trace: . Similarly, we can define density matrices: , , , , .
For any tri-partite state the following holds
where , for example.
Equivalently, the statement can be recast in terms of conditional entropies to show that for tripartite state ,
This can also be restated in terms of quantum mutual information,
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 
with the optimal constant .
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.
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
The Wigner–Yanase–Dyson -skew information of a density matrix . with respect to an operator is
where is a commutator, is the adjoint of and is fixed.
Concavity of p-skew information
It was conjectured by E. P. Wigner and M. M. Yanase in  that - skew information is concave as a function of a density matrix for a fixed .
Since the term is concave (it is linear), the conjecture reduces to the problem of concavity of . As noted in, this conjecture (for all ) implies SSA, and was proved for in, and for all in  in the following more general form: The function of two matrix variables
is jointly concave in and when and .
This theorem is an essential part of the proof of SSA in.
First two statements equivalent to SSA
- Note that the conditional entropies and do not have to be both non-negative.
- The map is convex.
Both of these statements were proved directly in.
Joint convexity of relative entropy
where with .
Monotonicity of quantum relative entropy
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 , given by . 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 can be written in Uhlmann's representation as
for some finite and some collection of unitary matrices on (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  and Uhlmann, whose proof is the one given here.
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 MPT: follows since the MPT is a particular case of MONO;
- MPT MONO: was shown by Lindblad, using a representation of stochastic maps as a partial trace over an auxiliary system;
- MPT SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";
- SSA MPT: by choosing to be block diagonal, one can show that SSA implies that the map
is convex. In  it was observed that this convexity yields MPT;
- MPT JC: as it was mentioned above, by choosing (and similarly, ) to be block diagonal matrix with blocks (and ), the partial trace is a sum over blocks so that , so from MPT one can obtain JC;
- JC SSA: using the 'purification process', Araki and Lieb, observed that one could obtain new useful inequalities from the known ones. By purifying to it can be shown that SSA is equivalent to
Moreover, if is pure, then and , 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;
The case of equality
Equality in monotonicity of quantum relative entropy inequality
For all states and on a Hilbert space and all quantum operators ,
if and only if there exists a quantum operator such that
Moreover, can be given explicitly by the formula
where is the adjoint map of .
D. Petz also gave another condition  when the equality holds in Monotonicity of quantum relative entropy: the first statement below. Differentiating it at we have the second condition. Moreover, M.B. Ruskai gave another proof of the second statement.
For all states and on and all quantum operators ,
if and only if the following equivalent conditions are satisfied:
- for all real .
where is the adjoint map of .
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 on a Hilbert space satisfies strong subadditivity with equality if and only if there is a decomposition of second system as
into a direct sum of tensor products, such that
with states on and on , and a probability distribution .
E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality, namely,
If and , 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 or (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  I. Kim studied an operator extension of strong subadditivity, proving the following inequality:
For a tri-partite state (density matrix) on ,
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.
- Von Neumann entropy
- Conditional quantum entropy
- Quantum mutual information
- Kullback–Leibler divergence
- D. W. Robinson and D. Ruelle, Mean Entropy of States in Classical Statistical Mechanis, Communications in Mathematical Physics 5, 288 (1967)
- O. Lanford III, D. W. Robinson, Jour. Mathematical Physics, 9, 1120 (1968)
- E. H. Lieb, M. B. Ruskai, Proof of the Strong Subadditivity of Quantum Mechanichal Entropy, J. Math. Phys. 14, 1938–1941 (1973).
- E.A. Carlen, E.H. Lieb. Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy, doi:10.1007/s11005-012-0565-6. arXiv:1203.4719 Lett. Math. Phys. 101, 1-11 (2012).
- O. Fawzi, R. Renner. Quantum conditional mutual information and approximate Markov chains. Communications in Mathematical Physics: 340, 2 (2015)
- M. Nielsen, I. Chuang Quantum Computation and Quantum Information, Cambr. U. Press, (2000)
- M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer (1993)
- E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).
- H. Umegaki, Conditional Expectation in an Operator Algebra. IV. Entropy and Information, Kodai Math. Sem. Rep. 14, 59–85, (1962)
- H. Araki, E. H. Lieb, Entropy Inequalities, Commun. Math. Phys. 18, 160–170 (1970).
- E. H. Lieb, Convex Trace Function and Proof of Wigner–Yanase–Dyson Conjecture, Adv. Math. 11, 267–288 (1973).
- H. Narnhofer, W.Thirring, From Relative Entropy to Entropy, Fizika 17, 258–262, (1985)
- E. P. Wigner, M. M. Yanase, Information Content of Distributions, Proc. Natl. Acad. Sci. USA 49, 910–918 (1963).
- E. P. Wigner, M. M. Yanase, On the Positive Semi-Definite Nature of a Certain Matrix Expression, Can. J. Math. 16, 397–406, (1964).
- F. Hansen, The Wigner-Yanase Entropy is Not Subadditive, J. Stat. Phys. 126, 643–648 (2007).
- A. Ulhmann, Endlich Dimensionale Dichtmatrizen, II, Wiss. Z. Karl-Marx-University Leipzig 22 Jg. H. 2., 139 (1973).
- G. Lindblad, Expectations and Entropy Inequalities for Finite Quantum Systems, Commun. Math. Phys. 39, 111–119 (1974).
- A. Ulhmann, Relative Entropy and the Wigner–Yanase–Dyson–Lieb Concavity in an Interpolation Theory, Comm. Math. Phys,54, 21–32, (1977).
- G. Lindblad, Completely Positive Maps and Entropy Inequalities, Commun. Math. Phys. 40, 147–151 (1975).
- E. H. Lieb, Some Convexity and Subadditivity Properties of Entropy, Bull. AMS 81, 1–13 (1975).
- M. B. Ruskai, Inequalities for Quantum Entropy: A Review with Conditions for Equality, J. Math. Phys. 43, 4358–4375 (2002); erratum 46, 019901 (2005)
- D. Petz, Sufficient Subalgebras and the Relative Entropy of States of a von Neumann Algebra, Commun. Math.Phys. 105, 123–131 (1986).
- D. Petz, Sufficiency of Channels over von Neumann Algebras, Quart. J. Math. Oxford 35, 475–483 (1986).
- 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).
- I. Kim, Operator Extension of Strong Subadditivity of Entropy, arXiv:1210.5190 (2012).
- E. G. Eﬀros. A Matrix Convexity Approach to Some Celebrated Quantum Inequalities. Proc. Natl. Acad. Sci. USA 106(4), 1006–1008 (2009).
- M. B. Ruskai, Remarks on Kim’s Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions, arXiv:1211.0049 (2012).