Editing Density matrix (section)

== Entropy ==
The [[von Neumann entropy]] <math>S</math> of a mixture can be expressed in terms of the eigenvalues of <math>\rho</math> or in terms of the [[Trace (linear algebra)|trace]] and [[Matrix logarithm|logarithm]] of the density operator <math>\rho</math>. Since <math>\rho</math> is a positive semi-definite operator, it has a [[spectral theorem|spectral decomposition]] such that <math>\rho = \textstyle\sum_i \lambda_i |\varphi_i\rangle \langle\varphi_i|</math>, where <math>|\varphi_i\rangle</math> are orthonormal vectors, <math>\lambda_i \ge 0</math>, and <math>\textstyle \sum \lambda_i = 1</math>. Then the entropy of a quantum system with density matrix <math>\rho</math> is
: <math>S = -\sum_i \lambda_i \ln\lambda_i = -\operatorname{tr}(\rho \ln\rho).</math>

This definition implies that the von Neumann entropy of any pure state is zero.<ref name=Rieffel>{{Cite book|title-link= Quantum Computing: A Gentle Introduction |title=Quantum Computing: A Gentle Introduction|last1=Rieffel|first1=Eleanor G.|last2=Polak|first2=Wolfgang H.|date=2011-03-04|publisher=MIT Press|isbn=978-0-262-01506-6|language=en|author-link=Eleanor Rieffel}}</ref>{{Rp|217}} If <math>\rho_i</math> are states that have support on orthogonal subspaces, then the von Neumann entropy of a convex combination of these states,
: <math>\rho = \sum_i p_i \rho_i,</math>
is given by the von Neumann entropies of the states <math>\rho_i</math> and the [[Shannon entropy]] of the probability distribution <math>p_i</math>:
: <math>S(\rho) = H(p_i) + \sum_i p_i S(\rho_i).</math>
When the states <math>\rho_i</math> do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combination <math>\rho</math>.<ref name="mikeandike" />{{rp|518}}

Given a density operator <math>\rho</math> and a projective measurement as in the previous section, the state <math>\rho'</math> defined by the convex combination
: <math>\rho' = \sum_i P_i \rho P_i,</math>
which can be interpreted as the state produced by performing the measurement but not recording which outcome occurred,<ref name=":2">{{Cite book|last=Wilde|first=Mark M.|title=Quantum Information Theory|publisher=Cambridge University Press|year=2017|isbn=978-1-107-17616-4|edition=2nd|doi=10.1017/9781316809976.001|arxiv=1106.1445|s2cid=2515538 |oclc=973404322}}</ref>{{Rp|159}} has a von Neumann entropy larger than that of <math>\rho</math>, except if <math>\rho = \rho'</math>. It is however possible for the <math>\rho'</math> produced by a ''generalized'' measurement, or [[POVM]], to have a lower von Neumann entropy than <math>\rho</math>.<ref name="mikeandike">{{Citation | last1=Nielsen | first1=Michael | last2=Chuang | first2=Isaac | title=Quantum Computation and Quantum Information | title-link=Quantum Computation and Quantum Information | publisher=[[Cambridge University Press]] | isbn=978-0-521-63503-5 | year=2000}}.</ref>{{rp|514}}