Strong subadditivity of quantum entropy
Strong subadditivity of entropy was long known and appreciated in classical probability theory and
information theory.
Its extension to quantum mechanical 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.
SSA concerns the relation between the entropies of various subsystems of a larger system consisting of three subsystems. 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"
- "Quantum Entropy and Its Use"
- Trace Inequalities and Quantum Entropy: An Introductory Course
Definitions
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 isRelative entropy
Umegaki's quantum relative entropy of two density matrices and isJoint concavity
A function of two variables is said to be jointly concave if for any the following holdsSubadditivity of entropy
Ordinary subadditivity concerns only two spaces and a density matrix. It states thatThis 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 canbe 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 holdswhere, 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.
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 -skew information of a density matrix. with respect to an operator iswhere 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, the conjecture reduces to the problem of concavity of. As noted in, this conjecture 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.
In their paper E. P. Wigner and M. M. Yanase also conjectured the subadditivity of -skew information for, 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.- Note that the conditional entropies and do not have to be both non-negative.
- The map is convex.
Joint convexity of relative entropy
As noted by Lindblad and Uhlmann, if, in equation, one takes and and
and differentiates in at one
obtains the Joint convexity of relative entropy :
i.e., if, and, then
where with.
Monotonicity of quantum relative entropy
The relative entropy decreases monotonically under completely positive trace preserving 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 . Since the trace 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 replaced by and
replaced. Take .
Then becomes
Therefore,
which is SSA. Thus,
the monotonicity of quantum relative entropy 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 ;
- Monotonicity of quantum relative entropy under partial trace ;
- Strong subadditivity ;
- Joint convexity of quantum relative entropy ;
- 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
- MPT JC: as it was mentioned above, by choosing to be block diagonal matrix with blocks , 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
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 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.
Equality in strong subadditivity inequality
, 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.
Carlen-Lieb Extension
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 . 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 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.