Strong subadditivity of quantum entropy

In quantum information theory, Strong Subadditivity of quantum entropy (SSA) concerns the relation among the von Neumann entropies of various quantum subsystems of a larger quantum system consisting of three subsystems (or of one quantum system with three degrees of freedom). It is a basic theorem in modern quantum information theory. It 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. In 2010 Ruskai found out that J. Kiefer had proved it back in 1959.
The classical version of SSA was long known and appreciated in classical probability theory and information theory. 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 quantum subsystems.
Some useful references here include:
Definitions.
We use the following notation throughout the following: A Hilbert space is denoted by formula_1, and formula_2 denotes the bounded linear operators on formula_1.
Tensor products are denoted by superscripts, e.g., formula_4. The trace 
is denoted by formula_5.
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., 
formula_6 is a density matrix on formula_7.
Entropy.
The von Neumann quantum entropy of a density matrix formula_8 is 
Relative entropy.
Umegaki's quantum relative entropy of two density matrices formula_8 and formula_11 is
Joint concavity.
A function formula_13 of two variables is said to be jointly concave if for any formula_14 the following holds
Subadditivity of entropy.
Ordinary subadditivity concerns only two spaces formula_7 and a density matrix formula_6. It states that
This inequality is true, of course, in classical probability theory, but the latter also contains the
theorem that the conditional entropies formula_19 and formula_20 are both non-negative. In the quantum case, however, both can be negative, 
e.g. formula_21
can be zero while formula_22. Nevertheless, the subadditivity upper bound on formula_21 continues to hold. The closest thing one has 
to formula_24 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: formula_26. 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 formula_27 on formula_1, we define a density matrix formula_6 on formula_30 as a partial trace: 
formula_31. Similarly, we can define density matrices: formula_32, formula_33, formula_34, formula_35, formula_36.
Statement.
For any tri-partite state formula_27 the following holds
where formula_39, for example.
Equivalently, the statement can be recast in terms of conditional entropies to show that for tripartite state formula_40,
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 formula_44.
As mentioned above, SSA was proved first by J. Kiefer in 1959 and independently by E.H.Lieb and M.B.Ruskai in 1973 using Lieb's theorem.
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.
Wigner–Yanase–Dyson conjecture.
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 formula_45-skew information of a density matrix formula_8. with respect to an operator formula_47 is
where formula_49 is a commutator, formula_50 is the
adjoint of formula_47 and formula_52 is fixed.
Concavity of "p"-skew information.
It was conjectured by E. P. Wigner and M. M. Yanase in that formula_45- skew information is concave as a function of a density matrix formula_8 for a fixed formula_52.
Since the term formula_56 is concave (it is linear), the conjecture reduces to the problem of concavity of formula_57. As noted in, this conjecture (for all formula_58) implies SSA, and was proved 
for formula_59 in, and for all formula_60 in 
in the following more general form: The function of 
two matrix variables 
is jointly concave in formula_61 and formula_62
when formula_63 and formula_64.
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 formula_45-skew information for formula_66, which was disproved by Hansen by giving a counterexample.
First two statements equivalent to SSA.
It was pointed out in that the first statement below is equivalent to SSA and A. Ulhmann in showed the equivalence between the second statement below and SSA.
Both of these statements were proved directly in.
Joint convexity of relative entropy.
As noted by Lindblad and Uhlmann
, if, in equation (), one takes formula_71 and formula_72 and 
formula_73 and differentiates in formula_74 at formula_75 one
obtains the Joint convexity of relative entropy :
i.e., if formula_76, and formula_77, then
where formula_78 with formula_79.
Monotonicity of quantum relative entropy.
The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations formula_80 on density matrices,
formula_81.
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 formula_82, given by formula_83. 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 
formula_84 can be written in Uhlmann's representation as
for some finite formula_86 and some collection of unitary matrices on formula_87 (alternatively, integrate over Haar measure). Since the trace (and hence the relative entropy) is unitarily invariant,
inequality () now follows from (). This theorem is due to Lindblad 
and Uhlmann, whose proof is the one given here.
SSA is obtained from () 
with formula_88 replaced by formula_89 and
formula_87 replaced formula_91. Take formula_92 formula_93 formula_94.
Then () becomes 
Therefore, 
which is SSA. Thus,
the monotonicity of quantum relative entropy (which follows from () 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:
The following implications show the equivalence between these inequalities.
formula_102 is convex. In it was observed that this convexity yields MPT;
Moreover, if formula_113 is pure, then formula_114 and formula_115, 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, for a discussion.
The case of equality.
Equality in monotonicity of quantum relative entropy inequality.
In, D. Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel:
For all states formula_8 and formula_11 on a Hilbert space formula_1 and all quantum operators formula_119,
if and only if there exists a quantum operator formula_121 such that 
Moreover, formula_121 can be given explicitly by the formula 
where formula_126 is the adjoint map of formula_127.
D. Petz also gave another condition when the equality holds in Monotonicity of quantum relative entropy: the first statement below. Differentiating it at formula_128 we have the second condition. Moreover, M.B. Ruskai gave another proof of the second statement.
For all states formula_8 and formula_11 on formula_1 and all quantum operators formula_119,
if and only if the following equivalent conditions are satisfied:
where formula_126 is the adjoint map of formula_127.
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 formula_40 on a Hilbert space formula_140 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 formula_143 on formula_144 and formula_145 on formula_146, and a probability distribution formula_147.
Carlen-Lieb Extension.
E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality, namely,
formula_148
If formula_149 and formula_150, 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 formula_151 or formula_152 (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) formula_27 on formula_154,
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.
Extensions of strong subadditivity in terms of recoverability.
A significant strengthening of strong subadditivity was proved in 2014, which was subsequently improved in and. In 2017, 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 formula_156 of a tripartite quantum state formula_157 is nearly equal to zero, then it is possible to perform a recovery channel formula_158 (from system E to AE) such that formula_159. These results thus generalize the exact equality conditions mentioned above.