Yao's principle

In quantum information theory, an entanglement witness is a functional which distinguishes a specific entangled state from separable ones. Entanglement witnesses can be linear or nonlinear functionals of the density matrix. If linear, then they can also be viewed as observables for which the expectation value of the entangled state is strictly outside the range of possible expectation values of any separable state.

Details

Let a composite quantum system have state space $H_{A}\otimes H_{B}$. A mixed state ρ is then a trace-class positive operator on the state space which has trace 1. We can view the family of states as a subset of the real Banach space generated by the Hermitian trace-class operators, with the trace norm. A mixed state ρ is separable if it can be approximated, in the trace norm, by states of the form

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

where $\rho _{i}^{A}$'s and $\rho _{i}^{B}$'s are pure states on the subsystems A and B respectively. So the family of separable states is the closed convex hull of pure product states. We will make use of the following variant of Hahn–Banach theorem:

Theorem Let $S_{1}$ and $S_{2}$ be disjoint convex closed sets in a real Banach space and one of them is compact, then there exists a bounded functional f 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 functional f. In the present context, the family of separable states is a convex set in the space of trace class operators. If ρ is an entangled state (thus lying outside the convex set), then by theorem above, there is a functional f separating ρ from the separable states. It is this functional f, or its identification as an operator, that we call an entanglement witness. There are more than one hyperplane separating a closed convex set and a point lying outside of it. So for an entangled state there are more than one entanglement witnesses. Recall the fact that the dual space of the Banach space of trace-class operators is isomorphic to the set of bounded operators. Therefore we can identify f with a Hermitian operator A. Therefore, modulo a few details, we have shown the existence of an entanglement witness given an entangled state:

Theorem For every entangled state ρ, there exists a Hermitian operator A such that $\operatorname {Tr} (A\,\rho )<0$, and $\operatorname {Tr} (A\,\sigma )\geq 0$ for all separable states σ.

When both $H_{A}$ and $H_{B}$ have finite dimension, there is no difference between trace-class and Hilbert–Schmidt operators. So in that case A can be given by Riesz representation theorem. As an immediate corollary, we have:

Theorem A mixed state σ is separable if and only if

$\operatorname {Tr} (A\,\sigma )\geq 0$

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

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 functional f of the trace-class Banach space and f is positive on the product pure states, then f, or its identification as a Hermitian operator, is an entanglement witness. Such a f indicates the entanglement of some state.

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

Theorem A mixed state $\sigma \in L(H_{A})\otimes L(H_{B})$ is separable if for every positive map Λ from bounded operators on $H_{B}$ to bounded operators on $H_{A}$, the operator $(I_{A}\otimes \Lambda )(\sigma )$ is positive, where $I_{A}$ is the identity map on $\;L(H_{A})$, the bounded operators on $H_{A}$.

References

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• Barry C. Sanders and Jeong San Kim, "Monogamy and polygamy of entanglement in multipartite quantum systems", Appl. Math. Inf. Sci. 4, 281–288 (2010).
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
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  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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