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.
Contents
- Definitions
- Density matrix
- Entropy
- Relative entropy
- Joint concavity
- Subadditivity of entropy
- Strong subadditivity SSA
- Statement
- WignerYanaseDyson conjecture
- The WignerYanaseDyson p skew information
- Concavity of p skew information
- First two statements equivalent to SSA
- Joint convexity of relative entropy
- Monotonicity of quantum relative entropy
- Relationship among inequalities
- Equality in monotonicity of quantum relative entropy inequality
- Equality in strong subadditivity inequality
- Carlen Lieb Extension
- Operator extension of strong subadditivity
- References
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.
Definitions
We will use the following notation throughout: A Hilbert space 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.,
Entropy
The von Neumann quantum entropy of a density matrix
Relative entropy
Umegaki's quantum relative entropy of two density matrices
Joint concavity
A function
Subadditivity of entropy
Ordinary subadditivity concerns only two spaces
This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies
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:
Given a density matrix
Statement
For any tri-partite state
where
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.
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
where
Concavity of p-skew information
It was conjectured by E. P. Wigner and M. M. Yanase in that
Since the term
is jointly concave in
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
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 (1), one takes
where
Monotonicity of quantum relative entropy
The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations
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
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
for some finite
SSA is obtained from (3) with
Therefore,
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:
The following implications show the equivalence between these inequalities.
Moreover, if
See, for a discussion.
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
if and only if there exists a quantum operator
Moreover,
where
D. Petz also gave another condition when the equality holds in Monotonicity of quantum relative entropy: the first statement below. Differentiating it at
For all states
if and only if the following equivalent conditions are satisfied:
where
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
into a direct sum of tensor products, such that
with states
Carlen-Lieb Extension
E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality, namely,
If
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)
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.