quantum versus classical probabilitymuller/ctms/lectures/rau_090818.pdf · classical probability...

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Quantum versus classical probability

Jochen Rau

Goethe University, Frankfurt

Duke University, August 18, 2009

arXiv:0710.2119v2 [quant-ph]

http://www.fh-darmstadt.de/../all/kontakt.htm

http://www.fh-darmstadt.de/all/kontakt.htm

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Motivation from quantum foundations

Reconstructing quantum theory:

In search of a physical principle

Duke University, August 2009 1J. Rau | Quantum vs classical probability

Special relativity Quantum mechanics

Mathematical

apparatus

Physical

principle

Lorentz transformations • States

• Unitary evolution

• Measurement

• space and time not absolute

but observer-dependent

• causality (light cone)

preserved in all frames?



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Relevance for quantum gravity

The issue is pertinent in the quest for a theory of

quantum gravity


General relativity Quantum mechanics

Mathematical

apparatus

Physical

principle

• Pseudo-Riemannian manifold

• Einstein equation

• States


• Measurement

• local flatness

• equivalence principle

• ? ?



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Examples for reconstruction

There have been many reconstructive approaches –

none of them entirely satisfactory


Name Protagonists Principal idea Shortcomings

Quantum logic• Birkhoff, v. Neumann

• Jauch et al

(„Geneva School“)

• Generalise classical logic: lattice theory

• Propositions subspaces of Hilbert sp.

• Skew field remains unspecified

(R, C, H)

• Proof only for d≥3

Algebraic

• Jordan, v. Neumann,

Wigner

• Gelfand, Neumark, Segal

• Haag, Kastler

• Algebra of operators (C* algebra)

• Hermitian elements observables

• Physical understanding?

Operational,

convex cone

• Mackey

• Ludwig et al

(„Marburg School“)

• Varadarajan, Gudder

• Focus on primitive laboratory operations

• Convex geometry of state space

• Convexity does not suffice

• Additional mathematical

assumptions needed

Test spaces

• Foulis, Randall et al

(„Amherst School“)

• Generalise probability theory • How to single out quantum theory

from the multitude of conceivable

generalisations of classical

probability?



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Motivation from quantum computing

Interest in quantum foundations has been revived

by the advent of quantum computation


classical probability

theory

sum rule, Bayes rule

classical information

Shannon

classical computing

Turing, network model

quantum probability

interference, entan-

glement, Bell,

Kochen-Specker

quantum

information

Holevo, Schumacher

other non-classical

probability theories

generalised

information

quantum computing

no-cloning,

teleportation

other non-classical

forms of informa-

tion processing

Physics / computer

science / logic

?

?

Today‘s talk



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Major differences

Quantum theory and classical probability are often seen

as two very different theories...


classical

probability

quantum

theory

• amplitudes, interference

• non-commutativity

• uncertainty relations

• entanglement

• violation of Bell inequalities

• Kochen-Specker theorem

• ...



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Overlap

...yet they share a substantial common core


class-

ical

quan-

tum

common

core



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Logical structure

Both classical and quantum theory deal with propositions

and logical relations


orthomodular poset / orthoalgebra / test space

weaker than classical: , defined iff jointly decidable



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Reasoning in the face of uncertainty

Probabilities satisfy sum and Bayes rules –

central to ensure consistency


Bayesian probability

embodies agent‘s

state of knowledge

degree of belief rather

than limit of relative

frequency

can be legitimately

assigned not just to

ensembles but also to

individual systems

Consistency

Different ways of using the

same information must lead

to the same conclusions,

irrespective of the particular

path chosen+

Cox 1946, Jaynes 2003

Quantitative rules

Sum rule

Bayes rule

quantum:

only if probabilities

pertain to propositions

that are jointly decidable



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Agent intervention

Agents can intervene by selection or interrogation


machine

Generic setup

000

11..

input

register

r

100

01..

output

register

p

1-p

Interrogation

finite or infinite sequence of reprod‘le

measurements, possibly stochastic

register = record of results

measurement setups may be

controlled by intermediate results

(feed forward, as in 1-way q. comp.)

no selection, no waste

I

Two examples

Selection

sequence of reprod‘le measurements

register = record of results

selection rule, possibly stochastic

keep or discard the system

S

1111

10..

algorithm



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S(ki) = dim M({ki}) + S(ki)

Most general selection

Arbitrary stochastic selection rules are allowed –

selection can prepare arbitrary states


classical 0 ki

quantum Σijkikj ki2

prob(x|r) = in...i1 prob(x|in...i1,mn...m1,)prob(in...i1|mn...m1,)prob(s|in...i1,S)

record of results

(string of integers)

sequence of n

measurement setups

record remains

internal to machine

{{xi}|xi=I,d(xi)=ki} parameters of a non-normalised state

normalised toprob(s|S,mn...m1,)-1 selected



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Most general interrogation

Arbitrary interrogation algorithms are allowed


record of results

(string of integers)

prob(x|r) = I,r prob(x|r,I,)prob(r|I,)prob(I|r,I)

sequence of

meas‘t setups

normalised

to one

Convexity

arbitrary mixtures constitute valid posteriors

states form a convex set

feed forwardstochastic

record remains

internal to machine



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Key question

Which additional assumptions are needed to arrive

at the classical and quantum case, respectively?


class-

ical

quan-

tum

common

core

? ?

Not ħ: We consider probability

theory, not dynamics!

variable

value 1 value 2ideal:



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Response to interrogation

Only in the quantum case it is possible to prepare arbitrary

states by mere interrogation


Preceding interrogation can make

any measurement emulate any other

measurement of the same resolution:

{xi},{yi}M({ki}) I:

prob(yi|I,) = prob(xi|) i,

based on

quantum Zeno effect

Interrogation can steer a system from

any pure state to any other pure state:

e,f I: prob(x|I,e) = prob(x|f) x

v. Neumann 1932

Quantum malleability

Interrogation has no effect:

prob(x|I,) = prob(x|) I, x,

Classical invariability

based on

joint decidability

Deterministic, static interrogation



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Agent-dependency

Responsiveness to interrogation adds another layer

to agent-dependency


Classical

Quantum

elicitmay

inform

answers

(data)

prior

expectation + conclusion

questions

data conclusionOrthodox:

Bayes: dataprior

expectation + conclusion

: agent-dependent



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Wheeler‘s game of 20 questions (1/2)

In quantum theory conclusions depend on questions asked


Wheeler 1983 („Law without law“)



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The word does not already exist „out

there“ – rather, information about the

word is brought into being through the

questions raised

1

The answers given are internally

consistent; convergence to some

word is possible

2

Wheeler‘s game of 20 questions (2/2)

In quantum theory conclusions depend on questions asked


Wheeler 1983 („Law without law“)

The questioner has influence on the

outcome: a different sequence of

questions will lead to a different word

3

...like in quantum theory:

•There is no preexisting reality that is

merely revealed, rather than

influenced, by the act of measurement

•The image of reality that emerges

through acts of measurement reflects

as much the history of intervention as it

reflects the external world



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World view

This new form of agent-dependency leads to a radically

different world view


Classical Quantum

reality with

independent

existence

manipulate in

experiment: grab,

dissect into parts,

select, transform

observe without

disturbance

?

!

(partial)

record of

answers:

0111001

1101100

....ongoing

interrogation

image of reality



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Conjecture

Can agent-dependency serve as a foundational principle

for quantum theory?


Special relativity Quantum mechanics

Mathematical

apparatus

Physical

principle

Lorentz transformations • States


• Measurement

• space and time not absolute

but observer-dependent

• causality (light cone)

preserved in all frames

• image of reality not absolute

but agent-dependent

• consistency of reasoning

preserved for all agents

?



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Response to interrogation

Only in the quantum case it is possible to prepare arbitrary

states by mere interrogation


Preceding interrogation can make

any measurement emulate any other

measurement of the same resolution:

{xi},{yi}M({ki}) I:

prob(yi|I,) = prob(xi|) i,

based on

quantum Zeno effect

Interrogation can steer a system from

any pure state to any other pure state:

e,f I: prob(x|I,e) = prob(x|f) x

v. Neumann 1932

Quantum malleability

Interrogation has no effect:

prob(x|I,) = prob(x|) I, x,

Classical invariability

based on

joint decidability

Hidden, deterministic, static interrogation



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Smoothness

Maximum agent-dependency rests on the Zeno effect,

which in turn presupposes smoothness


For finite granularity d:

1 Set of pure states X(d) is a continuous, compact, simply connected manifold

X(d) Robustness under small preparation

inaccuracies

>0 >0: prob(x|e)>1- eB(e0;)

Continuity condition:

ee0

B(e0;)

prob(x|e)

(x e0)

2 Probabilities change in a continuous fashion:



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Analysis of symmetry group

Continuity in combination with common core leads to

quantum theory in Hilbert space over R, C or H


Constraint Implication for group

probabilities change in a

continuous fashion

dim G(d) =

dim X(2)/2d(d-1) + dim G(1)d

X(d) continuous Lie group

dimensions match dim X(d) =

dim G(d) - dim G(d-1) - dim G(1)

Definition

Automorphisms of a

proposition system

preserve

logical relations

, , \

granularity d

Irreducible building

block

Group acts transitively

on collections M({ki}):

M({ki}) ~ G(ki)/iG(ki)

X(d) compact and simply

connected (hull of convex set)

compact and simple

up to Abelian factor

X(d) ~ M({1,d-1}) transitive on X(d)

G(d) {SO(nd), U(nd), Sp(nd) | nN}



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Conclusion

Classical and quantum case derive from common core

via additional assumptions on agent-dependency


class-

ical

quan-

tum

common

core

agent-

dependencymin

prob(x|I,)=prob(x|)

max

e,f I: prob(x|I,e)=prob(x|f)

R, C or H?1

?

intermediate

cases2



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Summary


Quantum theory shares with classical probability theory

a substantial core of common properties

Quantum theory and classical probability theory mark opposite

extremes in their response to interrogation – quantum theory

maximises agent-dependency

No reference to unitary evolution – primitive operations are

measurements only1

1 see also: T. Rudolph; measurement-based quantum computing



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Further details:

arXiv:0710.2119v2 [quant-ph]



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