Vasil Penchev. Negative or Complex Probability in Quantum Information

8/14/2019 Vasil Penchev. Negative or Complex Probability in Quantum Information

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Negative and ComplexProbability in

Quantum InformationVasil Penchev,

Institute of PhilosophicalResearch Bulgarian Academy

of Science


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Negative probability inNegative probability inpracticepractice

tica l m e a n in g is co n n e cte d to th e a p p ltica l m e a n in g is co n n e cte d to th e a p p ltu m in fo rm a tio nu m in fo rm a tio n ,n d m o re e xa ctly o,n d m o re e xa ctly o f: .in q u a n tu m co m m u n ica tio n V e ry sm.in q u a n tu m co m m u n ica tio n V e ry sma ce re g io n s tu rn o u t to b e th e rm o d yna ce re g io n s tu rn o u t to b e th e rm o d yn.ica l to th o se o f su p e rco n d u cto r M a cro.ica l to th o se o f su p e rco n d u cto r M a crosig n a ls m ig h t e x ist in co h e re n t o r e n -sig n a ls m ig h t e x ist in co h e re n t o r e n.ta te S u ch p h y sica l o b je cts h a vin g sh.ta te S u ch p h y sica l o b je cts h a vin g shn a ry p ro p e rtie s co u ld b e in th e b a se on a ry p ro p e rtie s co u ld b e in th e b a se oo m m u n ica tive ch a n n e ls o r e v e n m ao m m u n ica tive ch an n e ls o r e v e n m a


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Negative probability

I.Why does it appear inquantum mechanics?

II.It appears in phase-spaceformulated quantummechanics

III.Next, in quantumcorrelations

IV. and for wave-


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Mathematically: A ratio oftwo measures (of sets),

which are not collinear

Physically: f ratio of themeasuring of two physical

quantities, which are not


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THE MAPPING

PHASE HILBERT

SPACE

Since both are :

phase space is a of cells:

and the Hilbert of qubits:

the mapping is reduced to:


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The problem:(2) (3)p ro b le mp ro b le m :


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ncexan paresimultaneously

measurable, then it can

P :IG N E R F U N C T IO N IG N E R F U N C T IO N( )9 3 2 )9 3 2 INN


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The original Weyltransformation (1927)

[ ] =f ?] =f ?


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The original Weyltransformation (1927)

[ ]f tu rn s o u t to b e p a rtlyn a lo g ica l to D ira c s -fu n ctio n s ( ch w a rtz


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Groenewoldsstatistical ideas(1946)

Our problems are about:the correspondencea A

between physical quantities aand quantum operators A(quantization) and

the possibility ofunderstanding the statisticalcharacter of quantum mechanicsby averaging over uniquely

determined processes as in


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Groenewolds statisticalideas (1946)

the correspondencea A(quantization), in fact,

generates two kinds ofproblems abut the physicalquantities a:

a is not continuous function (itis either continuous, orgeneralized one, distribution);

there exist uantities a whose


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Groenewolds statisticalideas (1946)

The difficulties in ( heuantization of physical)uantities eflect at theame rate in ( tatisticaldescription):

egative probability ofome states appears , ut theyre easily interpreted


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(1949)

Classical statistical mechanicsis, however, only a special case inthe general theory of dynamicalstatistical (stochastic) processes.In the general cam, there is thepossibility of 'diffusion' of theprobability ' fluid', so that thetransformation with time of theprobability distribution need notbe deterministic in the classicalsense. In this a er, we shall


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approach(1949)

A single

system

Statistical

description

Description by

-functionDeter-minism

Quantumobjects

Probabilstic(P ) distribution

Indeter-

minism

Probabilstic(P ,


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approach(1949)

phase-space distributionsare not unique for a givenstate, but depend on the

variables one is going tomeasure. In Heisenberg'swords (5), 'the statistical

predictions of quantum theoryare thus significant only whencombined with experiments

which arc actually capable of


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approach(1949)sin g lesin g le u a n tu m syste m u a n tu m syste m s as a

-o n sta n d a rd B o ltzm a n no n sta n d a rd B o ltzm a n ne n se m b len se m b le

?P+ P P_


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StatisticalStatistical

descriptiondescription

DescriptionDescription

byby--functionsfunctionsNon-standard Boltz-Non-standard Boltz-

mann ensemblemann ensembleGibbsGibbs

enesembleenesemble

u va ences e ween :

artsarts ((elementselements))

f the systemf the systemPossible statesPossible states

((worldsworlds)) of the systemof the system

ssential partsssential partselements)elements) of the systemof the system

OrthogonalOrthogonal possible statespossible states((separatedseparated worldsworlds))

xternalxternal partsparts

elementselements)) of the systemof the systemNon-orthogonal possibleNon-orthogonal possible

statesstates ((interacting worldsinteracting worlds))

Again to Moyals statistical


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Again to Moyals statisticalapproach

symmetry(orantisymmetry) conditionsintroduce a probabilitydependence between any twoparticles in B. E . (or F .D.)

assemblies even in theabsence of any energyinteraction. It is this

dependence which gives rise



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The parameter accepts thefollowing values: in a Maxwell

Boltzmann ensem-ble (theclassical case) =0; in a Bose Einstein ensemble: =1 ; in aFermi Dirac: =-1 . ni, nk areaverage frequencies of thenumber of articles a

i, resp. a

k,


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?

hase space

ilbertspace


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III. Negative probability

in quantum correlations

h e b a ttle fo r o r a g a in sth e b a ttle fo r o r a g a in st id d e n p a ra m e te rsid d e n p a ra m e te rs n nu a n tu m m e ch a n icsu a n tu m m e ch a n icsocal hiddenparameterssality) againstnonloces (quantumcorrelation


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(1935)

Einstein Podolsky Rosen!

Link
http://www.phys.uu.nl/~stiefelh/epr_latex.pdfhttp://www.phys.uu.nl/~stiefelh/epr_latex.pdfhttp://www.phys.uu.nl/~stiefelh/epr_latex.pdf


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Zustnde of Scrdinger

and the poor cat (1935)Link
http://www.tu-harburg.de/rzt/rzt/it/QM/cat.htmlhttp://www.tu-harburg.de/rzt/rzt/it/QM/cat.htmlhttp://www.tu-harburg.de/rzt/rzt/it/QM/cat.html


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The sets of quantum quantities are

considered, and the conclusion is:There are no ensembles whichare free from dispersion. There are

ho-mogeneous ensemblesConsequently, there are nohomogeneous ensembles, i.e. for

exam-ple those of a single


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A: ( ) Erw 0 : ( . + . +) =Erw a b . ( )+ . ( )+ , ,Erw b Erw where a b : s a dispersion free

uantity (rw R1)=1[ ( ,rw )=(R , )] : s a homogenous one{ , b


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I. { } { ( ) ( )}f f R . { , , } { + +I R S + +}SThere corresponds to each physical

quantity of a quantum mechanical

system, a unique hyper-maximalHermitian operator, as we know andit is convenient to assume that this

correspon-dence is one-to-one -- that


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By introducing negativeprobability, then expectation is notadditive in general, then the

premises of the theorem are notfulfilled and the deduction is notvalid

artsarts ((elementselements))

f the systemf the systemPossible statesPossible states


ssential partsssential parts

elementselements)) of the systemof the system OrthogonalOrthogonal

possible statespossible states

((separated worldsseparated worlds))

xternalxternal partsparts

elementselements)) of the systemof the systemNon-orthogonal possibleNon-orthogonal possible

statesstates ((interacting worldsinteracting worlds))

eorem anegative probabilityegative probability


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A few equivalent expressions ofthem:

1. Non-negative probability

2. Orthogonal possible states

3. Separated worlds

4. An isolated quantum

system


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The demonstrations of vonNeumann and others, that

quantum mechanics does notpermit a hidden variable

interpretation, are reconsidered. I

is shown that their essentialaxioms are unreasonable. It is

urged that in further examination

of this roblem an interestin

)


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His essential assumption" is:Anyreal linear

combination of any two Hermitianoperators represents an

observable, and the same linearcombination of expectation values

is the expec-tation value of thecombination. This is true for

uantum mechanical states it is

(55))


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inequalities

by negative probabilitySince von Neumanns theorem isvalid only about nonnegative

probability (expectationadditivity), and quantum

mechanics permits negativeprobability, the idea is the domain

of the theorem validity to bedescribed by an inequality of theexpectation of two quantities (the

-


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-local hidden parameter:

clearing1. Von Neumannns theoremas well as the theories of

hidden parameters interpretthem as local ones implicitly.

2. Bells inequalities discussthe distinction between loaland nonlocal parameter

because quantum mechaics StatisticalStatistical Description byDescription by


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StatisticalStatistical

descriptiondescriptionDescription byDescription by--functionfunction

A non-standardnon-standardBoltzmann ensembleBoltzmann ensemble--

A GibbsA Gibbsensembleensemble

artsarts ((elements)elements)

f the systemf the system

PossiblePossible statesstates


ssential parts (elements)ssential parts (elements)

OrthogonalOrthogonal possible statespossible states((separated worldsseparated worlds))

xternalxternal partspartselementselements)) of the systemof the system

Non-othogonal statesNon-othogonal states((interacting worlds)interacting worlds)

-


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clearingThe notion ofnon-local hiddenparameter, the notion of the

externality of a system:1.Not any externalneighborhood, but on-ly a

small one(of the order ofa few), at that correspondingly, onlin phase space

-


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clearing

X

p near -eighbor hood of

x

pxpberg s uncertainty isberg s uncertainty is Grooroo

-


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clearing

xpthe lighthe lightconeone

he lighthe lightconeone

smallneighborhoodMx22

-


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Lorentzinvarianceis valid/meanut of a small neighborho

ncertainty relation ismeaningful / valid ( )ithin a small neigborhood


clearing

xpthe lighthe lightconeone

he lighthe lightconeone

smallneigborhoodMx22

An absolutely immovableAn absolutely immovable


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An absolutely immovableAn absolutely immovablebodybody

Heisenbergs uncertaintyHeisenbergs uncertaintyexcludes any absolutelyexcludes any absolutelyimmovable body as well as anyimmovable body as well as any

exactly constant phase volumeexactly constant phase volume..Any body is outlinedAny body is outlined rather byrather byan undetermined auraan undetermined aura ororhalo, than by a sharp outlinehalo, than by a sharp outline..The aura is within phase spaceThe aura is within phase spaceand its magnitude isand its magnitude iscomparable with the Planckcomparable with the Planck

constantconstant.. It consists of theIt consists of the

TheThe halohalo of negativeof negative


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TheThe halohalo of negativeof negativeprobability in phase spaceprobability in phase space

h e h a lo o f n e g a tiv e p r h e h a lo o f n e g a tiv e p r

S= .x.xPp

-he probability of non standard Boltzman-he probability of non standard Boltzman

ga n a ou e compar son o a


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ga n a ou e compar son o aGibbs and of a non-standardGibbs and of a non-standard

Boltzmann ensembleBoltzmann ensembleGibbs ensembleGibbs ensemble A non-standardA non-standard

Boltzmann oneBoltzmann onen ensemble of the states ofn ensemble of the states of

he system as a wholehe system as a wholeAn ensemble ofAn ensemble of

parts (P>0parts (P>0,, P


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probability violate Bells

inequalities?1 + [b(),c()] |[a(),b()]

[a(),c()]|P

P< bc

hase space :he halo ofhe halo ofegativeegative robability statesrobability statesaround any component

of the system ushes awayushes away hehethers in itsthers in its( - )roper non common( - )roper non commonartart of phasespace

he notion ofS

((i di t b bili tii di t b bili ti


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((immediate probabilisticimmediate probabilisticinteraction)interaction))

A corollary of von Neumanns theorem afterits genera-lizingfrom an isolated to two ormore interacting sys-tems is the immediateinteraction of probabilities ::

The necessary and sufficient condition ofThe necessary and sufficient condition ofimmediate probabilistic interactionimmediate probabilistic interactionis tois to

shareshare ommon possible states ofommon possible states of

Isolated systemsIsolated systems InteractingInteractingsystemssystems

P P

a bc

a b=

probabilityprobability


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probabilityprobability((immediate probabilisticimmediate probabilistic

interaction)interaction)11 ++ [b([b(),c(),c()] )] |[a([a(),b(),b()] )] [a([a(),c(),c()])]|P

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Vasil Penchev. Negative or Complex Probability in Quantum Information