what can and cannot be said about randomness using quantum physics acín

ICREA Colloquium, Barcelona, 27 September 2016

Antonio Acín ICREA Professor at ICFO-‐InsDtut de Ciencies Fotoniques, Barcelona

What can and cannot be said about randomness in quantum physics

Our goal: to “prove” the existence of randomness in nature.

DefiniDon of randomness

bi

Observer


bi

Observer Eve

A process is (perfectly) random if it is unpredictable, not only to the observer, but to any observer, called Eve in what follows and possibly correlated to the process.


bi

Observer Eve

A process is (perfectly) random if it is unpredictable, not only to the observer, but to any observer, called Eve in what follows and possibly correlated to the process.

This definiDon is saDsfactory both from a fundamental and applied perspecDve. •  From a fundamental perspecDve it is difficult to argue that a process is random

if there could exist an observer able to predict its outcomes. •  PracDcally, by demanding that the results should look random to any observer,

the generated randomness is guaranteed to be private.

No randomness from scratch

The generaDon of randomness from scratch is impossible!

No randomness from scratch

The generaDon of randomness from scratch is impossible!

This follows from the non-‐falsifiable hypothesis of the existence of a super-‐determinisDc model in which everything, including all the history of our universe, was pre-‐determined in advance and known by the external observer. Any protocol for randomness genera5on must be based on assump5ons.

CerDfiable physical randomness

Our working assumpDon is that processes are physical and therefore obey the laws of physics.

The random numbers should be unpredictable to any physical observer, that is, any observer whose acDons are constrained by the laws of physics.

¿Does randomness exist in classical physics?

Randomness in classical physics In the macroscopic world, there is no such thing as true randomness. Any random process is simply a consequence of:

1) ImperfecDons in the preparaDon of the system and/or

2) ParDal knowledge

Randomness in classical physics In the macroscopic world, there is no such thing as true randomness. Any random process is simply a consequence of:

1) ImperfecDons in the preparaDon of the system and/or

2) ParDal knowledge

Example:

One can never exclude the existence of an observer with perfect knowledge of the iniDal posiDon and speed of the ball and the size and shape of the roule^e, who can predict the result with certainty.

Randomness is, thus, a simple consequence of our limitaDons, for instance in our observaDon and computaDonal capabiliDes, informaDon storage and the preparaDon of the systems.

Randomness in classical physics


However, the theory does not incorporate any form of intrinsic randomness. Given a perfect knowledge of the iniDal condiDons in a system, it is in principle possible to predict its future (and past) behaviour.



However, the theory does not incorporate any form of intrinsic randomness. Given a perfect knowledge of the iniDal condiDons in a system, it is in principle possible to predict its future (and past) behaviour.

LAPLACE We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in moDon, and all posiDons of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the Dniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.


¿What happens when we move to the quantum world?

Quantum randomness Textbook: the outputs of a quantum measurement are random.


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The outputs of this experiment are random because: 1.  Devices are quantum…


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The output randomnesss does not rely only on the “quantumness” of the process.

The outputs of this experiment are random because: 1.  Devices are quantum… but also 2.  It is a pure single-‐photon state; 3.  The transmission coefficient of

the mirror is exactly ½; 4.  The detectors do not have

memory effects; 5.  …


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The output randomnesss does not rely only on the “quantumness” of the process.

The outputs of this experiment are random because: 1.  Devices are quantum… but also 2.  It is a pure single-‐photon state; 3.  The transmission coefficient of

the mirror is exactly ½; 4.  The detectors do not have

memory effects; 5.  …

It is unsaDsfactory that the random character of the process relies on our knowledge of it. How can we know if our descripDon is correct? If not correct, is the observed randomness again an arDfact of ignorance?

Can the presence of randomness be guaranteed by any physical mechanism?

Known soluDons •  Classical Random Number Generators (CRNG). All of them are of

determinisDc Nature. •  Quantum Random Number Generators (QRNG). There exist different

soluDons, but the main idea is encapsulated by the following example:

•  In any case, all these soluDons have three problems, which are important both from a fundamental and pracDcal point of view.

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Single photons are prepared and sent into a mirror with transmilvity equal to ½. The random numbers are provided by the clicks in the detectors.

Problem 1: cerDficaDon

•  Good randomness is usually verified by a series of staDsDcal tests.

•  There exist chaoDc systems, of determinisDc nature, that pass all exisDng randomness tests.

•  Do these tests really cerDfy the presence of randomness? It is well known that no finite set of tests can do it.

•  Do these tests cerDfy any form of quantum randomness? Classical systems pass them!

RANDU RANDU is an infamous linear congruenDal pseudorandom number generator of the Park–Miller type, which has been used since the 1960s.

Three-‐dimensional plot of 100,000 values generated with RANDU. Each point represents 3 subsequent pseudorandom values. It is clearly seen that the points fall in 15 two-‐dimensional planes.

Problem 2: privacy

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1r2r

nr

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Classical Memory

The provider has access to a proper RNG. The provider uses it to generate a long sequence of good random numbers, stores them into a memory sDck and sells it as a proper RNG to the user.

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1r2r

nr

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Classical Memory 1r 2r nr…

The provider has access to a proper RNG. The provider uses it to generate a long sequence of good random numbers, stores them into a memory sDck and sells it as a proper RNG to the user. The numbers generated by the user look random. However, they can be perfectly predicted by the adversary. How can one be sure that the observed random numbers are also random to any other observer, possibly adversarial?

Problem 2: privacy

Problem 3: device dependence All the soluDons rely on the details of the devices used in the generaDon.

How can imperfecDons in the devices affect the quality of the generated numbers? Can these imperfecDons be exploited by an adversary?

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Single photons are prepared and sent into a mirror with transmilvity equal to ½. The random numbers are provided by the clicks in the detectors.

No randomness for single systems

It is impossible to cerDfy that the outcomes produced by a single system are random without making assumpDons about its internal working.



This follows form the simple fact that any observed probability distribuDon can be wri^en in terms of determinisDc assignments:

P b =1,b = 2,…,b = r( ) = p b = i( )i=1

r

∑ δb.i

The observed randomness is just a consequence of the ignorance of: p b = i( )



This follows form the simple fact that any observed probability distribuDon can be wri^en in terms of determinisDc assignments:

P b =1,b = 2,…,b = r( ) = p b = i( )i=1

r

∑ δb.i

The observed randomness is just a consequence of the ignorance of: p b = i( )

Any staDsDcs obtained by measuring a quantum systems can be simulated classically.

Let’s then move to more than one system…

CerDfied randomness y

a b

x P(a,b x, y)

e=a?

z

Eve

Observer

The observer can now observe the correlaDons between the two systems. From the point of view of correlaDons, classical and quantum physics differ!

A crash course on Bell inequaliDes

Example: CHSH Bell inequality

CHSH = A1B1 + A1B2 + A2B1 − A2B2

+1 -1 +1 -1

1 2 1 2

Source

Example: CHSH Bell inequality

CHSH = A1B1 + A1B2 + A2B1 − A2B2

+1 -1 +1 -1

1 2 1 2

In classical physics, observables have well-‐defined values, now +1 or -‐1. Under this assumpDon: Example: So, the expectaDon value of this quanDty also saDsfies

Source

CHSH ≤ 2

A1 = A2 = B1 = B2 = +1⇒CHSH = +2

CHSH ≤ 2

Quantum Bell inequality violaDon

A2

A1

B1

B2

Φ =1200 + 11( )

Classical values are now replaced by operators.

-1 +1 -1

1 2 1 2

Source

+1

Quantum Bell inequality violaDon

CHSH = A1B1 Φ+ A1B2 Φ

+ A2B1 Φ− A2B2 Φ

= 2 2 > 2

A2

A1

B1

B2

Φ =1200 + 11( )

Classical values are now replaced by operators.

-1 +1 -1

1 2 1 2

Source

+1

Quantum non-‐locality

•  Bell inequaliDes are condiDons saDsfied by classical models in which measurement outputs are pre-‐determined.

•  CorrelaDons observed when measuring entangled states may lead to a violaDon of Bell inequality and, therefore, do not have a classical counterpart. These correlaDons are usually called non-‐local.

•  If some observed correlaDons violate a Bell inequality, the outcomes could not have pre-‐determined in advance è They are random.

•  If some observed correlaDons violate a Bell inequality, they cannot be reproduced classically è The devices are quantum.


a b

x P(a,b x, y)

e=a?

z

Eve

Observer

Ask the provider not one but two devices. If a Bell inequality violaDon is observed, the outputs contain some randomness.


a b

x P(a,b x, y)

e=a?

z

Eve

Observer

Ask the provider not one but two devices. If a Bell inequality violaDon is observed, the outputs contain some randomness. The cerDficaDon is device-‐independent, in the sense that it does not rely on any assumpDon on the internal working of the device.

CerDfied randomness The randomness in the outputs can be esDmated from the amount of observed Bell violaDon. At no violaDon, there is no randomness.

CerDfied randomness The randomness in the outputs can be esDmated from the amount of observed Bell violaDon. At no violaDon, there is no randomness. This randomness is not a consequence of ignorance! This region is impossible within quantum physics.

What did we use?

•  We assume the validity of the whole quantum formalism.

•  We needed two different systems. What does it mean? Do two devices define two systems? Not if they could be jointly prepared in advance.

a b

What did we use?

•  We assume the validity of the whole quantum formalism.

•  We needed two different systems. What does it mean? Do two devices define two systems? Not if they could be jointly prepared in advance.

•  We need the inputs, processes that happen in one locaDon and is not known at the other locaDon. Then, we can idenDfy two separate systems.

a b

x y

What are two systems?

•  All this is related to the noDon of causality and space-‐Dme. We think of regions in space-‐Dme that are staDsDcally meaningful and independent of the rest. We need these noDons for making scienDfic predicDons!

•  It may however argued that assuming that something happens in a region in space-‐Dme means that cannot be predicted by the rest of observers in the remaining space-‐Dme and, therefore, it is random.

•  It adds a form of circularity in the argument: randomness is needed to run the Bell test, which is in turn used to cerDfy the presence of randomness.

•  From a pracDcal perspecDve, or even from a reasonable fundamental point of view, we can assume that there are independent events (again, the whole noDon of causality is based on it, otherwise everything would be connected).

•  Yet, it is a logical possibility and a limitaDon in the proofs of randomness.

Randomness expansion

a b

x y

Source

Perfect random bits are available to choose the inputs. One can prove that one can generate using the outputs more random bits than are used for the inputs. There even exist protocols for unbounded randomness expansion. Colbeck, Kent, Pironio, Massar and others…

Randomness amplificaDon

k

Santha-‐Vazirani source: a device that produces bits with the promise ε

12−ε ≤ P k = 0 rest( ) ≤ 12 +ε


k


12−ε ≤ P k = 0 rest( ) ≤ 12 +ε

0 1/2

εiε f

Randomness amplificaDon: improve the randomness of the source.


k


12−ε ≤ P k = 0 rest( ) ≤ 12 +ε

0 1/2

εiε f

Randomness amplificaDon: improve the randomness of the source.

Randomness amplificaDon is impossible classically.


0 1/2

εiε f

Randomness amplificaDon is possible using quantum non-‐local correlaDons. Colbeck and Renner Idea: use the imperfect source to choose the inputs in a Bell test define the final source from the outputs of the experiment. Full randomness amplificaDon is possible: arbitrarily weak random bits can be mapped into arbitrarily good random bits. Gallego et al.

Experimental realizaDon

•  The two-‐box scenario is performed by two atomic parDcles located in two distant traps.

•  Using our theoreDcal techniques, we can cerDfy that 42 new random bits are generated in the experiment.

• It is the first Dme that randomness generaDon is cerDfied without making any detailed assumpDon about the internal working of the devices.

•  Similar Bell experiments with photons have recently been performed.

NIST Randomness Beacon NIST is implemenDng a source of public randomness. The service (at h^ps://beacon.nist.gov/home) uses two independent commercially available sources of randomness, each with an independent hardware entropy source and SP 800-‐90-‐approved components.

Commercially available physical sources of randomness are adequate as entropy sources for currently envisioned applicaDons of the Beacon. However, demonstrably unpredictable values are not possible to obtain in any classical physical context. Given this fact, our team established a collaboraDon with NIST physicists from the Physical Measurement Laboratory (PML). The aim is to use quantum effects to generate a sequence of truly random values, guaranteed to be unpredictable, even if an a^acker has access to the random source. In August 2012, this project was awarded a mulD-‐year grant from NIST's InnovaDons in Measurement Science (IMS) Program.

NIST Randomness Beacon

Bell cer5fied randomess!

Conclusions •  Randomness can be cerDfied using the non-‐local correlaDons observed when

measuring quantum states.

•  The cerDficaDon is device-‐independent: it does not rely on any assumpDon on the internal working of the devices.

•  The argument requires two different devices.

•  Independent (random?) inputs are needed to define the two different devices. This requirement may introduce some circularity in the argument.

•  Despite this circularity, using non-‐local quantum correlaDons randomness can be arbitrarily expanded or amplified.

•  The device-‐independent approach can be used to design novel devices producing cer5fied quantum randomness.