Editing Quantum entanglement (section)

=== Reduced density matrices ===
The idea of a reduced density matrix was introduced by [[Paul Dirac]] in 1930.<ref name="Dirac1930">
{{cite journal
 | last = Dirac | first = Paul Adrien Maurice | author-link = Paul Dirac
 | title = Note on exchange phenomena in the Thomas atom
 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]
 | volume = 26
 | number = 3
 | pages = 376–385
 | doi = 10.1017/S0305004100016108
 | bibcode=1930PCPS...26..376D
 | year = 1930
 | url = https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6C5FF7297CD96F49A8B8E9E3EA50E412/S0305004100016108a.pdf/note-on-exchange-phenomena-in-the-thomas-atom.pdf
 | doi-access=free
}}</ref> Consider as above systems {{mvar|A}} and {{mvar|B}} each with a Hilbert space {{mvar|H<sub>A</sub>, H<sub>B</sub>}}. Let the state of the composite system be
: <math> |\Psi \rangle \in H_A \otimes H_B. </math>

As indicated above, in general there is no way to associate a pure state to the component system {{mvar|A}}. However, it still is possible to associate a density matrix. Let
: <math>\rho_T = |\Psi\rangle \; \langle\Psi|</math>.
which is the [[projection operator]] onto this state. The state of {{mvar|A}} is the [[partial trace]] of {{mvar|ρ<sub>T</sub>}} over the basis of system {{mvar|B}}:
: <math>\rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j^{N_B} \left( I_A \otimes \langle j|_B \right) \left( |\Psi\rangle \langle\Psi| \right)\left( I_A \otimes |j\rangle_B \right) = \hbox{Tr}_B \; \rho_T.</math>

The sum occurs over <math>N_B := \dim(H_B)</math> and <math>I_A</math> the identity operator in <math>H_A</math>. {{mvar|ρ<sub>A</sub>}} is sometimes called the reduced density matrix of {{mvar|ρ}} on subsystem {{mvar|A}}. Colloquially, we "trace out" or "trace over" system {{mvar|B}} to obtain the reduced density matrix on {{mvar|A}}.<ref name="Rieffel2011"/>{{rp|207–212}}<ref name="Rau2021"/>{{rp|133}}<ref name="Zwiebach2022"/>{{rp|§22.4}}

For example, the reduced density matrix of {{mvar|A}} for the entangled state
: <math>\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right),</math>
discussed above is<ref name="Zwiebach2022"/>{{rp|§22.4}}
: <math>\rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right ).</math>

This demonstrates that the reduced density matrix for an entangled pure ensemble is a mixed ensemble. In contrast, the density matrix of {{mvar|A}} for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math> discussed above is<ref name="Nielsen-2010"/>{{rp|106}}
: <math>\rho_A = |\psi\rangle_A \langle\psi|_A,</math>
the projection operator onto <math>|\psi\rangle_A</math>.

In general, a bipartite pure state ''ρ'' is entangled if and only if its reduced states are mixed rather than pure.<ref name="Rau2021"/>{{rp|131}}