What Quantum Communication Complexity has to say on Non … · 2017-03-15 · Source: (Winter,...

What Quantum

Communication Complexity

has to say on Non-Locality

and on Macroscopic Realism

Rethinking Foundations of Physics Workshop 2017

History of Quantum mechanics in a nutshell


First Quantum Revolution – „old school‟, 1900–mid 1920sSemi–classical (phenomenological) corrections to explain new

physical fenomena, which were impossible to include in the current

paradigm (anomalies).

Planck (1900) – Black body radiation

Einstein (1905) – Photoelectric effect

Bohr (1913) – Atomic model

De Broglie (1923) – Matter waves

Second Quantum Revolution – „Knabenphysik’,1925–1935Awareness of a completely new physics. Formal formulation of the

new postulates, theorems, quantities (i.e. Spin) and foundamental

interpretations.

Heisenberg (1925, 1927) – Matrix mechanics, Uncertainty principle

Schrödinger (1926) – Wave Mechanics (Schrödinger Equation)

Pauli (1925) – Exclusion principle

Born(1926) – Probabilistic interpretation

Bohr (1927) – Principle of Complementarity

Dirac (1928) – relativistic QM (Dirac equation)

1935: a year of scepticisms


1) Einstein-Podolsky-Rosen (EPR) paradox, May 1935Gedankenexperiment aimed at confuting Copenhagen interpretaion of QM

Under the assumption of:

• maximally entangled states (perfect correlations)

• locality

• realism

• completeness

And showing the ‗absurdity‘ of the conclusion, should have proven the

incompleteness of the theory

2) “Schrödinger‟s cat” (EPR), November 1935Gedankenexperiment aimed at confuting Copenhagen interpretation of QM

Under the assumption of:

• quantum superpositions of macroscopic objects (a dead and an alive cat)

And showing the ‗absurdity‘ of the

existence itself of such a system,

should have confuted the idea of

the wave function collapse

After 1935: a missing revolution?


EPR and “Schrödinger‟s cat” thought experiments, could have

immediately triggered a new revolution in foundations of

(quantum) physics

BUT: Nürnberger Gesetze (1935) and War World II (1939):

All the Jewish/socialist

scientists left the Mitteleuropa • The debate about the Foundations of

QM was ―frozen‖

Physicists became tools for

develop war weapons and

devices

Any philosophical speculation was

rejected. Only immediately practical

applications were supported

• Nearly a hundred of them went to the USA

The quantum scientific revolution was stopped by the social –

economical – political environment.

USA, postwar (Cold War in force)

A new ―paradigm‖ was imposed: Shut up and calculate!

Struggling to have foundations back into physics


Overlooked early (although pivotal) contributions:

David Bohm, 1952 – Hidden variables theories

John Bell, 1964 – Bell‘s inequalities

1970s – few outsiders questioned the foundations of QM (O. Freire: ―Quantum Dissidents‖, 2014):

USA: „Fundamental Physik Group’ (D. Kaiser: ―How the

Hippies Saved Physics‖, 2011)

Italy: F. Selleri, M. Cini, A. Baracca, S. Bergia, G. C.

Ghirardi (Baracca, Bergia & Del Santo, 2017)

Paris (France): J. P. Vigier, L. de Broglie (and his

‗Fondation’)

London (UK): D. Bohm, K. Popper (Del Santo, 2017)

Source

BertaAlice

A typical “Bell Scenario” (Brunner et al., 2014)


Two systems, which may have previously interacted (e.g. they

have been created by a common source), are then spatially

separated at large distance and measured by two observers,

Alice and Berta.

Source

BertaAlice

They both freely choose one setting

of their local measurement apparatusx

y

Alice chooses x

out of X possible

settings

Berta chooses y

out of Y possible

settings

Outcome of the

measurement: a

Outcome of the

measurement: b

A typical “Bell Scenario”


Source

BertaAlice


From one run to the other of the experiment, different values of the

outcomes a and b can be obtained, although the chosen settings x and y

stay the same (strict determinism is not assumed).

The outcomes a and b are in general governed by a probability distribution:

Repeating the experiment very

many times, one can get a fair

extimator of the distribution


Source

BertaAlice


From one run to the other of the experiment, different values of the

outcomes a and b can be obtained, although the chosen settings x and y

stay the same (strict determinism is not assumed).

The outcomes a and b are in general governed by a probability distribution:

Experimental result:




Explanation: there is in principle nothing strange: correlations can

be an emergent effect due to our ignorance (cf. statistical mechanics).

In fact one can consider a theory with the following

assumptions:

1. [Free will]: Alice and Berta are completely free to chose

independently their settings

2. Locality: there is not any direct influence between distant

(space-like separated) systems

3. Realism: ―external reality is assumed to exist and have definite

properties, whether or not they are observed by someone‖ (Clauser

and Shimony, 1978)




Explanation: there is in principle nothing strange: correlations can

be an emergent effect due to our ignorance (cf. statistical mechanics).

In fact one can consider a theory with the following

assumptions:

1. [Free will]: Alice and Berta are completely free to chose

independently their settings

2. Locality: there is not any direct influence between distant

(space-like separated) systems

3. Realism: ―external reality is assumed to exist and have definite

properties, whether or not they are observed by someone‖ (Clauser

and Shimony, 1978)

There exists a set Λ of past factors, described by some (hidden)

variables λ, which fully account for the correlations between a and b:



But the hidden variables λ can be not fully controllable, therefore

they also may change from one run to another of the experiment:

λ are governed by a probability distribution q(λ)

Condition of Local Realism (LR)

Fundamental Remarks:

• LR follows directly from conditions 1. 2. and 3.

• No assumption of determinism has been considered to derive LR

• No assumption of ―classical behavior‖ has been considered to

derive LR



Philosophy put to the test: Bell‟s Inequalities

Let a ,b {-1,1} and x, y {0,1}.

Consider the expectation values of the product ab for every setting

choice (x,y):

Defining the quantity S, one finds:

CHSH inequality (a Bell‟s inequality)

Local Realism is experimentally testable!

LR:

LR

Quantum Mechanics

For a maximally entangled states, e.g.

And certain settings of the apparatus (x,y), one finds the maximum

value of S to be:

Nature (i.e. empirical evidences):

Since 1970s, empirical tests do violate Bell‟s inequalities,

agreeing with the predictions of QM!

• Freedman and Clauser, 1972

• Aspect et al., 1982

• Weihs et al., 1998 (locality loophole - free)

• Rowe et al., 2001 (detection loophole - free)

• Giustina et al., 2015 (loophole - free)


QM formalism

violates Bell‟s

inequalities!

Philosophy put to the test: Bell‟s Inequalities


Bell‟s Inequalities: philosophy or physics?

―The assumptions behind Bell inequalities are often criticized as being

philosophical.‖ (Brukner and Zukowsky, 2012)

BUT:

(Relativistic) classical physics is realistic and local

THEREFORE:

The validity of LR is an actual problem of nature

―Philosophical propositions could be defined as those which are not

observationally or experimentally falsifiable at the given moment of the

development of human knowledge, or in pure mathematical theory are not

logically derivable. Therefore, the conjunction of all assumptions of Bell

inequalities is not a philosophical statement, as it is testable both

experimentally and logically‖ (ibid.)

―Violations of Bell‘s inequalities imply that the underlying conjunction of

assumptions of realism, locality and ‘free will‘ is not valid, and nothing more.

It is often said that the violations indicate ‗(quantum) non-locality‘. However if

one wants non-locality to be the implication, one has to assume ‗free will‘

and realism. But this is only at this moment a philosophical choice‖ (ibid.)

All correlations

Without any restrictions, all the possible theoretical correlations can

violate any physical constrains (including Einstein‘s relativity)

Possibly unphysical


A general overview

All correlations

(unphysical?)

No-signaling correlations:

All correlations

Physical interpretation: local probabilities of Alice are

independent of Berta‟s measuring settings y

Alice and Berta cannot signal to each

other by the choice of the inputs


All correlations

(unphysical?)

No-signaling correlations

Quantum correlations

Where are measurement

operators (POVM elements):


All correlations

(unphysical?)



Local realist correlations

λ are the hidden-variablestaking values in the space Λ


All correlations

(unphysical?)



Bell‟s inequalities

Local realist correlations



Bell‟s Inequalities as non-local games

Source: (Winter, 2010)

(Berta)(Alice)

0. Players can communicate and

agree on a strategy

1. A referee assigns an input to

each player (x to Alice, y to

Berta)

2. The referee challenges the

players to guess the right value

of a function of the inputs:

f(x,y)

3. The players answer the request

with a and b, respectively,

without any communication

QUANTUM ADVANTAGE

Can the parties increase the

success in solving the problem if

they share prior entanglement?


The CHSH game

BertaAlicex y

ba

Game:

Players win if they

guess the value of

f(x,y):=xy=a b

where

...

S (CHSH inequality)

SQuant = 2√2

SClass = 2


New games based on quantum superpositions

COMMUNICATION COMPLEXITY

What is the highest possible probability for the parties to arrive at the

correct value for the function if only a restricted amount of

communication is allowed?

Question: Are there advantages in exploiting genuine quantum effects?

Answer: YES!

Entanglement

Very well known

problem: hundreds of

games/tasks which

reduce communication

complexity (with

respect to the best

classical strategy)

Why?

Violation of LR

Quantum superpositions

Rather new problem: mere superpositions can be a

resource for communication.

e.g. "Exponential communication complexity

advantage from quantum superposition of the

direction of communication‖ (Guérin et al. 2015);

Spatial superposition of single particles (Del Santo &

Dakic, in preparation)

Why?

Violation of Macroscopic realism?


Macroscopic realism (Kofler & Brukner, 2013)

Consider a system described by macroscopically distinct variables,

At the macrovariable is measured


Assuming:

Macrorealism per se: “A macroscopic object, which has available to it two or

moremacroscopically distinct states, is at any given time in a definite one of those

states.”

Noninvasive measurability: “It is possible in principle to determine which of

these states the system is in without any effect on the state itself, or on the

subsequent system dynamics.”

Condition of

Macroscopic Realism






Assuming:



states.”




Condition of

Macroscopic Realism

Recall… Condition of

Local Realism (LR)






Assuming:



states.”




Simplest case: ,

defining the temporal correlations:

Condition of

Macroscopic Realism

Leggett-Garg inequality

(in the form of CHSH)

• Schrödinger, Erwin. "Die gegenwärtige Situation in der Quantenmechanik

„.Naturwissenschaften 23.48 (1935): 807-812.

• Einstein, Albert, Boris Podolsky, and Nathan Rosen. "Can quantum-mechanical

description of physical reality be considered complete?." Physical review 47.10 (1935):

777.

• Junior, Olival Freire. The Quantum Dissidents: Rebuilding the Foundations of

Quantum Mechanics (1950-1990). Springer, 2014.

• Kaiser, David. How the hippies saved physics: Science, counterculture, and the

quantum revival. WW Norton & Company, 2011.

• Baracca, Angelo, Silvio Bergia, and Flavio Del Santo. "The origins of the research on

the foundations of quantum mechanics (and other critical activities) in Italy during the

1970s.― Studies in History and Philosophy of Modern Physics 57 (2017): 66-79.

• Del Santo, Flavio. "Genesis of Karl Popper's EPR-Like Experiment and its Resonance

amongst the Physics Community in the 1980s." arXiv preprint arXiv:1701.09178 (2017).

• Brukner, Časlav, and Marek Żukowski. "Bell's Inequalities—Foundations and Quantum

Communication." Handbook of Natural Computing. Springer Berlin Heidelberg, (2012).

1413-1450.

• Brunner, Nicolas, et al. "Bell nonlocality." Reviews of Modern Physics 86.2 (2014):

419.

• Clauser, John F., and Abner Shimony. "Bell's theorem. Experimental tests and

implications." Reports on Progress in Physics 41.12 (1978): 1881.

References (1/2)


•Freedman, Stuart J., and John F. Clauser. "Experimental test of local hidden-variable

theories." Physical Review Letters 28.14 (1972): 938.

• Aspect, Alain, Philippe Grangier, and Gérard Roger. "Experimental realization of

Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's

inequalities." Physical review letters 49.2 (1982): 91.

• Weihs, Gregor, et al. "Violation of Bell's inequality under strict Einstein locality

conditions." Physical Review Letters 81.23 (1998): 5039.

• Rowe, Mary A., et al. "Experimental violation of a Bell's inequality with efficient

detection." Nature 409.6822 (2001): 791-794.

• Giustina, Marissa, et al. "Significant-loophole-free test of Bell‘s theorem with entangled

photons." Physical review letters 115.25 (2015): 250401.

• Winter, Andreas. "Quantum mechanics: The usefulness of uselessness.‖ .

Nature 466.7310 (2010): 1053-1054.

• Guérin, Philippe Allard, et al. "Exponential communication complexity advantage from

quantum superposition of the direction of communication." Physical Review

Letters 117.10 (2016): 100502.

• Kofler, Johannes, and Časlav Brukner. "Condition for macroscopic realism beyond the

Leggett-Garg inequalities." Physical Review A 87.5 (2013): 052115.

References (2/2)


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