1. Definitions
    Density matrix  Entropy  Relative entropy  Joint concavity 

2. Subadditivity of entropy

3. Strong subadditivity (SSA)
    Statement 

4. Wigner–Yanase–Dyson conjecture
    The Wigner–Yanase–Dyson p-skew information  Concavity of p-skew information 

5. First two statements equivalent to SSA

6. Joint convexity of relative entropy

7. {{anchor|Mono2016_10}}Monotonicity of quantum relative entropy

8. Relationship among inequalities

9. The case of equality
    Equality in monotonicity of quantum relative entropy inequality  Equality in strong subadditivity inequality 

10. Carlen-Lieb Extension

11. Operator extension of strong subadditivity

12. Extensions of strong subadditivity in terms of recoverability

13. See also

14. References

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:^[4][5][6]

Definitions

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 .

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.,

is a density matrix on .

Entropy

The von Neumann quantum entropy of a density matrix is

.

Relative entropy

Umegaki's^[7] quantum relative entropy of two density matrices and is

.

Joint concavity

A function of two variables is said to be jointly concave if for any the following holds

Subadditivity of entropy

Ordinary subadditivity ^[8] 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 ^[8]

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: . 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: , , , , .

Statement

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 ^[9]

,

with the optimal constant .

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 -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 ^[13] 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,^[10] this conjecture (for all ) implies SSA, and was proved

for in,^[13] and for all in ^[10]

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.^[3]

In their paper ^[13] E. P. Wigner and M. M. Yanase also conjectured the subadditivity of -skew information for , 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.

• 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.^[3]

Joint convexity of relative entropy

As noted by Lindblad ^[16] and Uhlmann

,^[17] if, in equation ({{EquationNote|1}}), one takes and and

and differentiates in at one

obtains the Joint convexity of relative entropy :

i.e., if , and , then

where with .

{{anchor|Mono2016_10}}Monotonicity of quantum relative entropy

The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations on density matrices,

.

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 ({{EquationNote|3}}) now follows from ({{EquationNote|2}}). This theorem is due to Lindblad ^[16]

and Uhlmann,^[15] whose proof is the one given here.

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

with replaced by and

replaced . Take .

Then ({{EquationNote|3}}) becomes

Therefore,

which is SSA. Thus,

the monotonicity of quantum relative entropy (which follows from ({{EquationNote|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,^[18] 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 ^[3] 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,^[8][19] 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;

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 and on a Hilbert space and all quantum operators ,

if and only if there exists a quantum operator such that

and

Moreover, can be given explicitly by the formula

where is the adjoint map of .

D. Petz also gave another condition ^[21] 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.^[23]

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 .

Carlen-Lieb Extension

E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality,^[9] 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 ^[24] 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,^[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] These improvements of strong subadditivity have physical interpretations in terms of recoverability, meaning that if the conditional mutual information of a tripartite quantum state is nearly equal to zero, then it is possible to perform a recovery channel (from system E to AE) such that . These results thus generalize the exact equality conditions mentioned above. In 2017,^[30] for the first time, it was shown that the recovery channel can be taken to be the original Petz recovery map.

See also

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

References

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