This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(April 2022) (Learn how and when to remove this message)

Inquantum information theory, anentanglement witness is afunctional which distinguishes a specificentangled state from separable ones. Entanglement witnesses can be linear or nonlinear functionals of thedensity matrix. If linear, then they can also be viewed asobservables for which the expectation value of the entangled state is strictly outside the range of possible expectation values of anyseparable state.

Let a composite quantum system have state space${\displaystyle H_A\otimes H_B}$. Amixed stateρ is then atrace-class positive operator on the state space which has trace 1. We can view the family of states as a subset of the realBanach space generated by the Hermitian trace-class operators, with the trace norm. A mixed stateρ isseparable if it can be approximated, in the trace norm, by states of the form

${\displaystyle \xi =\sum _{i=1}^k{p}_i{\rho }_i^A\otimes {\rho }_i^B,}$

where${\displaystyle {\rho }_i^A}$ and${\displaystyle {\rho }_i^B}$ are pure states on the subsystemsA andB respectively. So the family of separable states is the closedconvex hull of pure product states. We will make use of the following variant ofHahn–Banach theorem:

Theorem Let${\displaystyle S_1}$ and${\displaystyle S_2}$ be disjoint convex closed sets in a real Banach space and one of them iscompact, then there exists a boundedfunctionalf separating the two sets.

This is a generalization of the fact that, in real Euclidean space, given a convex set and a point outside, there always exists an affine subspace separating the two. The affine subspace manifests itself as the functionalf. In the present context, the family of separable states is a convex set in the space of trace class operators. Ifρ is an entangled state (thus lying outside the convex set), then by theorem above, there is a functionalf separatingρ from the separable states. It is this functionalf, or its identification as an operator, that we call anentanglement witness. There is more than one hyperplane separating a closed convex set from a point lying outside of it, so for an entangled state there is more than one entanglement witness. Recall the fact that the dual space of the Banach space of trace-class operators is isomorphic to the set ofbounded operators. Therefore, we can identifyf with a Hermitian operatorA. Therefore, modulo a few details, we have shown the existence of an entanglement witness given an entangled state:

Theorem For every entangled stateρ, there exists a Hermitian operator A such that, and${\displaystyle \mathrm{Tr}(A\sigma )\ge 0}$ for all separable statesσ.

When both${\displaystyle H_A}$ and${\displaystyle H_B}$ have finite dimension, there is no difference between trace-class andHilbert–Schmidt operators. So in that caseA can be given byRiesz representation theorem. As an immediate corollary, we have:

Theorem A mixed stateσ is separable if and only if

${\displaystyle \mathrm{Tr}(A\sigma )\ge 0}$

for any bounded operator A satisfying${\displaystyle \mathrm{Tr}(A\cdot P\otimes Q)\ge 0}$, for all product pure state${\displaystyle P\otimes Q}$.

If a state is separable, clearly the desired implication from the theorem must hold. On the other hand, given an entangled state, one of its entanglement witnesses will violate the given condition.

Thus if a bounded functionalf of the trace-class Banach space andf is positive on the product pure states, thenf, or its identification as a Hermitian operator, is an entanglement witness. Such af indicates the entanglement of some state.

Using the isomorphism between entanglement witnesses and non-completely positive maps, it was shown (by the Horodeckis) that

Theorem Assume that${\displaystyle H_A,H_B}$ are finite-dimensional. A mixed state${\displaystyle \sigma \in L(H_A)\otimes L(H_B)}$ is separable if for every positive map Λ from bounded operators on${\displaystyle H_B}$ to bounded operators on${\displaystyle H_A}$, the operator${\displaystyle (I_A\otimes \mathrm{\Lambda })(\sigma )}$ is positive, where${\displaystyle I_A}$ is the identity map on${\displaystyle L(H_A)}$, the bounded operators on${\displaystyle H_A}$.

• Terhal, Barbara M. (2000). "Bell inequalities and the separability criterion".Physics Letters A.271 (5–6):319–326.arXiv:quant-ph/9911057.Bibcode:2000PhLA..271..319T.doi:10.1016/S0375-9601(00)00401-1.ISSN 0375-9601. Also available atquant-ph/9911057
• R.B. Holmes.Geometric Functional Analysis and Its Applications, Springer-Verlag, 1975.
• M. Horodecki, P. Horodecki, R. Horodecki,Separability of Mixed States: Necessary and Sufficient Conditions, Physics Letters A 223, 1 (1996) andarXiv:quant-ph/9605038
• Z. Ficek, "Quantum Entanglement Processing with Atoms", Appl. Math. Inf. Sci. 3, 375–393 (2009).
• Barry C. Sanders and Jeong San Kim, "Monogamy and polygamy of entanglement in multipartite quantum systems", Appl. Math. Inf. Sci. 4, 281–288 (2010).
• Gühne, O.; Tóth, G. (2009). "Entanglement detection".Phys. Rep.474 (1–6):1–75.arXiv:0811.2803.Bibcode:2009PhR...474....1G.doi:10.1016/j.physrep.2009.02.004.

• Quantum information theory

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from April 2022
• All articles lacking in-text citations
• Use American English from February 2019
• All Wikipedia articles written in American English
• Use mdy dates from February 2019