Editing Quantum entanglement (section)

=== Ensembles ===
As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a [[density matrix]], which is a [[positive-semidefinite matrix]], or a [[trace class]] when the state space is infinite-dimensional, and which has trace 1. By the [[spectral theorem]], such a matrix takes the general form:
: <math>\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,</math>
where the ''w''<sub>''i''</sub> are positive-valued probabilities (they sum up to 1), the vectors {{math|''α''<sub>''i''</sub>}} are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret {{mvar|ρ}} as representing an ensemble where <math> w_i </math> is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need [[#Reduced density matrices|density matrices]] to represent the state.<ref name="Peres1993"/>{{rp|73–74}}<ref name="Holevo2001"/>{{rp|13–15}}<ref name="Zwiebach2022"/>{{rp|§22.2}}

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits [[electron]]s towards an observer. The electrons' Hilbert spaces are [[identical particles|identical]]. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with spins aligned in the positive {{math|'''z'''}} direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative {{math|'''y'''}} direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

Following the definition above, for a bipartite composite system, mixed states are just density matrices on {{math|''H<sub>A</sub>'' ⊗ ''H<sub>B</sub>''}}. That is, it has the general form
: <math>\rho =\sum_{i} w_i\left[\sum_{j} \bar{c}_{ij} (|\alpha_{ij}\rangle\otimes|\beta_{ij}\rangle)\right]\left[\sum_k c_{ik} (\langle\alpha_{ik}|\otimes\langle\beta_{ik}|)\right]
</math>
where the ''w''<sub>''i''</sub> are positively valued probabilities, <math display="inline">\sum_j |c_{ij}|^2=1</math>, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as<ref name=Laloe>{{cite journal|last=Laloe|first=Franck|year=2001|title=Do We Really Understand Quantum Mechanics|journal=American Journal of Physics |volume=69 |issue=6|pages=655–701 |arxiv=quant-ph/0209123 |bibcode=2001AmJPh..69..655L |doi=10.1119/1.1356698|s2cid=123349369 }}</ref>{{rp|131–132}}
: <math>\rho = \sum_i w_i \rho_i^A \otimes \rho_i^B, </math>
where the {{math|''w''<sub>''i''</sub>}} are positively valued probabilities and the <math>\rho_i^A</math>s and <math>\rho_i^B</math>s are themselves mixed states (density operators) on the subsystems {{mvar|A}} and {{mvar|B}} respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are themselves pure ensembles. A state is then said to be entangled if it is not separable.

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be [[NP-hard]].<ref>{{cite book |last=Gurvits |first=L. |title=Proceedings of the thirty-fifth annual ACM symposium on Theory of computing |year=2003 |isbn=978-1-58113-674-6 |page=10 |language=en |chapter=Classical deterministic complexity of Edmonds' Problem and quantum entanglement |doi=10.1145/780542.780545 |arxiv=quant-ph/0303055 |s2cid=5745067}}</ref> For the {{math|2 × 2}} and {{math|2 × 3}} cases, a necessary and sufficient criterion for separability is given by the famous [[Peres-Horodecki criterion|Positive Partial Transpose (PPT)]] condition.<ref>{{cite journal |vauthors=Horodecki M, Horodecki P, Horodecki R |title=Separability of mixed states: necessary and sufficient conditions |journal=Physics Letters A |volume=223 |issue=1 |page=210 |year=1996 |doi=10.1016/S0375-9601(96)00706-2 |bibcode=1996PhLA..223....1H|arxiv = quant-ph/9605038 |citeseerx=10.1.1.252.496 |s2cid=10580997 }}</ref>