ProbCons

Gleason's theorem is a mathematical result which is of particular importance for the field of quantum logic. It proves that the Born rule for the probability of obtaining specific results to a given measurement, follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space. The essence of the theorem is that:

For a Hilbert space of dimension 3 or greater, the only possible measure of the probability of the state associated with a particular Linear subspace a of the Hilbert Space will have the form Tr(P(a) W), where Tr is a trace class operator of the matrix product of the projection operator P(a) and the density matrix for the system W.

Context

Quantum logic treats quantum events (or measurement outcomes) as logical propositions, and studies the relationships and structures formed by these events, with specific emphasis on quantum measurement. More formally, a quantum logic is a set of events that is closed under a countable disjunction of countably many mutually exclusive events. The representation theorem in quantum logic shows that these logics form a latticeTemplate:Dn which is isomorphic to the lattice of subspaces of a vector space with a scalar product.

It remains an open problem in quantum logic to prove that the field K over which the vector space is defined, is either the real numbers, complex numbers, or the quaternions. This has negative implications for the possibility of a P-adic quantum mechanics. This is a necessary result for Gleason's theorem to be applicable, since in all these cases we know that the definition of the inner product of a non-zero vector with itself will satisfy the requirements to make the vector space in question a Hilbert space. Soler's Result, the restriction of the field to just these three fields [1], has negative implications for the possibility of a P-adic quantum mechanics.

Application

The representation theorem allows us to treat quantum events as a lattice L = L(H) of subspaces of a real or complex Hilbert space. Gleason's theorem allows us to "attach" these events to probabilities. This section draws extensively from the analysis presented in Pitowsky (2005).

We let A represent an observable with finitely many potential outcomes: the eigenvalues of the Hermitian operator A, i.e. ${\alpha }_1,{\alpha }_2,{\alpha }_3,...,{\alpha }_n$. An "event", then, is a proposition $x_i$, which in natural language can be rendered "the outcome of measuring A on the system is ${\alpha }_i$". The events $x_i$ generate a sublattice of the Hilbert space which is a finite Boolean algebra, and if n is the dimension of the Hilbert space, then each events is an atom.

A state, or probability function, is a real function P on the atoms in L, with the following properties:

1. $P(0)=0$, and $P(y)\ge 0$ for all $y\in L$
2. $\sum _{j=1}^{n}P(x_j)=1$, if $x_1,x_2,x_3,...,x_n$ are orthogonal atoms

This means for every lattice element y, the probability of obtaining y as a measurement outcome is fixed, since it may be expressed as the union of a set of orthogonal atoms: $P(y)=\sum _{j=1}^{r}P(x_j)=1$

Here, we introduce Gleason's theorem itself:

Given a state P on a space of dimension $\ge 3$, there is an Hermitian, non-negative operator W on H, whose trace is unity, such that $P(x)=⟨\mathbf{x},W\mathbf{x}⟩$ for all atoms $x\in L$, where < , > is the inner product, and $\mathbf{x}$ is a unit vector along $x$. In particular, if some $x_0$ satisfies $P(x_0)=1$, then $P(x)={|⟨{\mathbf{x}}_{\mathbf{0}},\mathbf{x}⟩|}^2$ for all $x\in L$.

This is, of course, the Born rule for probability in quantum mechanics. The theorem presumes that the underlying set of numbers that the functions are defined over are real numbers or complex numbers. A constructive proof exists.

Implications

Gleason's theorem highlights a number of fundamental issues in quantum measurement theory. The fact that the logical structure of quantum events dictates the probability measure of the formalism is taken by some to demonstrate an inherent stochasticity in the very fabric of the world. To some researchers, such as Pitowski, the result is convincing enough to conclude that quantum mechanics represents a new theory of probability. Alternatively, such approaches as relational quantum mechanics make use of Gleason's theorem as an essential step in deriving the quantum formalism from information-theoretic postulates.

The theorem is often taken to rule out the possibility of hidden variables in quantum mechanics. This is because the theorem implies that there can be no bivalent probability measures, i.e. probability measures having only the values 1 and 0. Because the mapping $u\to ⟨Wu,u⟩$ is continuous on the unit sphere of the Hilbert space for any density operator W. Since this unit sphere is connected, no continuous function on it can take only the value of 0 and 1. (Wilce (2006), pg. 3) But, a hidden variables theory which is deterministic implies that the probability of a given outcome is always either 0 or 1: either the electron's spin is up, or it isn't (which accords with classical intuitions). Gleason's theorem therefore seems to hint that quantum theory represents a deep and fundamental departure from the classical way of looking at the world, and that this departure is logical, not interpretational, in nature.

See also

• Quantum logic
• Born's rule
• Measurement in quantum mechanics
• Hilbert space

References

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  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
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  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
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  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
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  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
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Wilce, A. (2006). "Quantum Logic and Probability Theory". In The Stanford Encyclopedia of Philosophy (Spring 2006 Edition), Edward N. Zalta (ed.).
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534