- Quantum mutual information
quantum information theory, quantum mutual information, or von Neumann mutual information, is a measure of correlation between subsystems of quantum state. It is the quantum mechanical analog of Shannon mutual information.
For simplicity, it will be assumed that all objects in the article are finite dimensional.
The definition of quantum mutual entropy is motivated by the classical case. For a probability distribution of two variables "p"("x", "y"), the two marginal distributions are
The classical mutual information "I"("X", "Y") is defined by
where "S"("q") denotes the
Shannon entropyof the probability distribution "q".
One can calculate directly
So the mutual information is
But this is precisely the
relative entropybetween "p"("x", "y") and "p"("x")"p"("y"). In other words, if we assume the two variables "x" and "y" to be uncorrelated, mutual information is the "discrepancy in uncertainty" resulting from this (possibly erroneous) assumption.
It follows from the property of relative entropy that "I"("X","Y") ≥ 0 and equality holds if and only if "p"("x", "y") = "p"("x")"p"("y").
The quantum mechanical counterpart of classical probability distributions are density matrices.
Consider a composite quantum system whose state space is the tensor product
Let "ρ""AB" be a density matrix acting on "H". The
von Neumann entropyof "ρ", which is the quantum mechanical analaog of the Shannon entropy, is given by
For a probability distribution "p"("x","y"), the marginal distributions are obtained by integrating away the variables "x" or "y". The corresponding operation for density matrices is the
partial trace. So one can assign to "ρ" a state on the subsystem "A" by
where Tr"B" is partial trace with respect to system "B". This is the reduced state of "ρAB" on system "A". The reduced von Neumann entropy' of "ρAB" with respect to system "A" is
"S"("ρB") is defined in the same way.
"Technical Note:" In mathematical language, passing from the classical to quantum setting can be described as follows. The "algebra of observables" of a physical system is a
C*-algebraand states are unital linear functionals on the algebra. Classical systems are described by commutative C*-algebras, therefore classical states are probability measures. Quantum mechanical systems have non-commutative observable algebras. In concrete considerations, quantum states are density operators. If the probability measure "μ" is a state on classical composite system consisting of two subsystem "A" and "B", we project "μ" onto the system "A" to obtain the reduced state. As stated above, the quantum analog of this is the partial trace operation, which can be viewed as projection onto a tensor component. "End of note"
We can see now the appropriate definition of quantum mutual information should be
Quantum mutual information can be interpretated the same way as in the classical case: it can be shown that
quantum relative entropy.
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