Definition
The quantum mechanical counterpart of classical probability distributions are density matrices.
Consider a composite quantum system whose state space is the tensor product
Let ρAB be a density matrix acting on H. The von Neumann entropy of ρ, which is the quantum mechanical analogy 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 TrB 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*-algebra and 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
It can now be seen that the appropriate definition of quantum mutual information should be
Quantum mutual information can be interpreted the same way as in the classical case: it can be shown that
where denotes quantum relative entropy.
Read more about this topic: Quantum Mutual Information
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