EP4508523A1

EP4508523A1 - System for generation of non-classically correlated strings of numbers and a method for confirming non-classical features of the generated strings of numbers

Info

Publication number: EP4508523A1
Application number: EP22723834.2A
Authority: EP
Inventors: Adan CABELLO QUINTERO; Nikolai MIKLIN; Marcin Pawlowski; Mohamed Bourennane
Original assignee: Universidad de Sevilla; University of Gdansk
Current assignee: Universidad de Sevilla; University of Gdansk
Priority date: 2022-04-12
Filing date: 2022-04-12
Publication date: 2025-02-19
Also published as: WO2023200344A1

Abstract

The invention deals with system for generation of non-classically correlated strings of numbers in three options and a method for confirming non-classical features of the generated strings of numbers. The present invention is in the field of quantum technologies that produce strings of correlated random numbers and method for testing or, in other words, confirming the non-classicality of generated strings of numbers. One of variant is based on system for generating correlated strings of numbers with a test of non-classicality without communication, second system for generating correlated strings of numbers with a test of non-classicality with communication and a source of non-classically correlated particles, the third system for generating correlated strings of numbers with a test of non-classicality with communication and no source of non-classically correlated particles

Description

System for generation of non-classically correlated strings of numbers and a method for confirming non-classical features of the generated strings of numbers

The present invention is in the field of quantum technologies that produce strings of correlated random numbers and method for testing or, in other words, confirming the non-classicality of generated strings of numbers. These strings of numbers can be later used for certified random number generation or quantum key distribution.

The presented invention is based on a physical processes involving production and detection of quantum states with high Hilbert space dimension. The invention can be implemented in generation of random numbers in lotteries and gaming as well as in cybersecurity.

Symbols and abbreviations used in the description of state of the art and description of the invention thereinafter:

A,B - for systems based on test without communication denote detection stations, which are parts of a system in which quantum states are measured and outcomes obtained; for systemss based on test with communication A is sender which prepares a quantum state and sends it to receiver B.

POVM - Positive Operator Valued Measure x - input of A chosen by Processing Unit y - input of B chosen by Processing Unit a - measurement outcome for A b- measurement outcome for B ρAB - a quantum state measured by A and B POVM element representing measurement done by A after receiving input x and producing outcome a a POVM element representing measurement done by B after receiving input y and producing outcome b - detection efficiency, which is the probability with which A and B produce any outcome in a round of nonlocality test α(x) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was x β(y) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was y - conditional probability distribution for outcomes of A and B conditioned on their inputs “ parameters specifying the test of nonclassicality

C - classical bound of a test of non-classicality. It is a number such that C implies nonclassical behaviour of the system.

Pcrit - critical detection efficiency, which is the lowest detection efficiency allowing for conclusive outcome of a test of non-classicality

CHSH - Clauser-Horne-Shimony-Holt Bell inequality.

Non-classically correlated string of numbers refers to a string of numbers for which there exists a test of non-classicality yielding positive result.

Test of non-classicality refers to an inequality involving a probability distribution, such that for every classical probability distribution it is satisfied and there exist quantum probability distributions that violate the inequality. In the latter case the test is said to yield a positive result. Classical probability distribution refers to a probability distribution of measurement results on a system, which can be explained by a theory other than quantum mechanics.

Quantum probability distribution refers to a probability distribution of measurement results on a system, which can not be explained by a theory other than quantum mechanics.

Method of testing non-classicality refers to a procedure in which a probability distribution of measurements on a system is estimated and a test of non-classicality performed on the distribution.

Tests of nonclassical correlations are a basis of many quantum technologies. Bell inequalities, arguably the most important of these tests enable: quantum key distribution [A. Ekert, "Quantum cryptography based on Bell’s theorem", Phys. Rev. Lett., 67, 661 (1991)], secure randomness generation [S. Pironio, et al. "Random numbers certified by Bell’s theorem", Nature 464, 1021, (2010)], secret sharing [M. Hillery, V. Buzek, A. Berthiaume, Phys. Rev. A, 59, 1829 (1999)] and communication complexity reduction [H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010)]. Other tests of nonclassicality also found use in the same range of applications. Entanglement witnesses are known to be useful for quantum key distribution [C. H. Bennett, G. Brassard, and N. D. Mermin, Phys. Rev. Lett. 68, 557 (1992)] and discord tests for communication complexity reduction [T. K. Chuan, T. Paterek, New J. Phys. 16, 093063 (2014)].

The biggest challenge with realizing these technologies in practice lies in efficient detection of particles carrying quantum states. In scientific literature this problem is called detection efficiency loophole and was first brought up by Pearle already in 1970 [P. M. Pearle, “Hidden-variable example based upon data rejection," Phys. Rev. D, 2, 1418-1425 (1970)]. In the case of the most commonly used Bell inequality, for the test to be loophole-free, overall detection efficiency of 83% is required, when using maximally entangled states. This is extremely difficult to achieve with current technology. Eberhard showed that we can lower this requirement to 67% by using non-maximally entangled states [P. H. Eberhard, “Background level and counter efficiencies required for a loophole-free Einstein- Podolsky-Rosen experiment," Phys. Rev. A, 47(2), 747-750, (1993)] but it comes at the price of higher requirements on the quality of prepared quantum states and worse overall performance.

For two-partite Bell inequalities involving quantum states with local Hilbert space dimension d=2 it is known that the results mentioned above, i.e.: 83% for maximally entangled states and 67% in general, cannot be improved. While difficulty of preparing states which can violate Bell inequalities involving more parties grows very quickly with their number, increasing local Hilbert space dimension, though far from trivial, is still considerably easier. Moreover, if is known [T. Vertesi, S. Pironio, and N. Brunner, “Closing the detection loophole in bell experiments using qudits," Phys. Rev. Lett., 104, 060401, (2010)] that for d>2 it is possible to find Bell inequalities with lower critical detection efficiency. However the lowest known bound is 62% which is still very challenging for current hardware to achieve. Moreover, the quantum states that need to be prepared and measurements to be performed are quite difficult to realize in practice.

The main reason why better results have not been found is that that the amount of possible Bell inequalities increases exponentially with d and so does the time required to analyse each of them. Because of that even the case of d=3 has not been yet fully analysed, which is in stark contrast with current technological capabilities of controlling more than a thousand dimensions [E.A. Aguilar, et al., Phys. Rev. Lett. 120, 230503 (2018)]. Since some hardware can produce quantum states of different dimensions it is also desirable for tests of non-classicality to be performed and analysed with variable d.

Other tests of non-classicality including dimension witnesses, entanglement witnesses, Leggett-Garg inequalities, steering inequalities, etc..., suffer from the same problems.

The known systems for generation of non-classically correlated strings of numbers necessarily have at least two active components called the parties. The systems fall into two main categories depending on the information flow within the system. In the first category there is no direct communication between the parties. These systems use Bell inequalities, entanglement witnesses or steering inequalities as the non- classicality tests. In the second category classical or quantum communication between the parties is involved. The system of the second category use dimension witnesses or Leggett-Garg inequalities as non-classicality tests. The second category can be further divided into two variants: one that involves a common source of correlations for the parties, and one that does not. Therefore there are three possible variants of systems for generation of non-classically correlated strings of numbers.

Variant 1

The known system for generating correlated strings of numbers for a test of non- classicality without communication between the parties is presented in fig. 1 and comprised of parts as disclosed below.

Source of non-classically correlated particles: It encodes a quantum state p into several carriers and sends one carrier to each of measurement stations. The number of stations is arbitrary. Here, for simplicity, we assume that there are two of them. The stations are labelled A and B and the quantum state they receive ρAB.

Measurement stations: The measurement stations are parts of the system which perform measurements of the quantum state encoded by the source of non-classically correlated particles. Both of measurement stations receive inputs from Processing Unit and perform one measurement based on it. We denote measurement choice of A by x and choice of B by y. The outcomes of the measurements are a for A and b for B. The POVM elements specifying the measurements performed by A and B are The measurement outcomes a and b are sent to Processing Unit.

Processing Unit: Processing unit is an electronic controller for the other components. It estimates conditional probability distribution P(a,b|x,y) by repeatedly sending inputs x and y to measurement stations and receiving the outcomes a and b. This is repeated a large number of times (typically more than 10.000). If the measurement procedure failed to produce an outcome (which happens with probability 1- η ) the processing unit assigns an outcome using functions α(x) and β(y) for A and B, respectively. The processing unit performs a test of non-classicality. It is done by taking linear combination of P(a,b|x,y) and comparing to a value C, which is the classical bound, i.e:

If the above inequality is violated the we say that the probability distribution P(a,b|x,y) cannot have a classical explanation and therefore its non-classicality is confirmed.

Variant 2:

The known system for generating correlated strings of numbers with the test of non- classicality with communication between the parties and a source of non-classically correlated particles is presented in fig. 2 and comprised of

A source of non-classically correlated particles: The source prepares a quantum state p and sends the whole or part of it to the sender and to the receiver. It encodes a quantum state p into several carriers and sends one carrier to each of measurement stations. The number of stations is arbitrary. Here, for simplicity, we assume that there are two of them. The stations are labelled A and B .

Sender (A): A quantum state p is prepared by the sender. It is done by a modification of the state the sender received from the source. The sender can keep a part of the quantum state and sends the rest to the receiver. The quantum state A and B have after communication from the source and between them is ρAB. The sender receives inputs from Processing Unit and the operation it performs is based on it. We denote its choice by x. If the preparation involves measurement then the outcome of the measurement is denoted by a. The POVM elements specifying the measurements performed by the sender are A^x _a. In some tests the sender does not perform any measurement. This case is modelled by the measurement outcome assigned a constant. The measurement outcome a is sent to the Processing Unit.

Receiver (B): The receiver receives quantum states from the sender and the source of non-classically correlated particles. The receiver also receives input y from Processing Unit and performs measurement based on it. We denote the measurements outcome of the receiver by b. The POVM elements specifying the measurements are . The measurement outcome b is sent to the Processing Unit.

Processing Unit: Processing unit is an electronic controller for the other components. It estimates conditional probability distribution P(a,b|x,y) by repeatedly sending inputs x and y to sender A and receiver B and receiving the outcomes. This is repeated a large number of times (typically more than 10.000). If the measurement procedure failed to produce an outcome (which happens with probability 1- η) the processing unit assigns an outcome using functions α(x) and β(y) for A and B, respectively. The processing unit performs a test of non-classicality. It is done by taking linear combination of P(a,b|x,y) and comparing to a value C, which is the classical bound, i.e:

Variant 3

The known system for generating correlated strings of numbers with the test of non- classicality with communication between the parties and no source of non-classically correlated particles is presented in fig. 3 and comprised of Sender (A): A quantum state p is prepared by the sender. The sender can keep a part of the quantum state and sends the rest to the receiver. The quantum state A and B have after communication from the source and between them is ρAB. The sender receives inputs from Processing Unit and the operation it performs is based on it. We denote its choice by x. If the preparation involves measurement then the outcome of the measurement is denoted by a. The POVM elements specifying the measurements performed by the sender are In some tests the sender does not perform any measurement. This case is modelled by the measurement outcome assigned a constant. The measurement outcome a is sent to the Processing Unit.

Receiver (B): The receiver receives quantum states from the sender. The receiver also receives input y from Processing Unit and performs measurement based on it. We denote the measurements outcome of the receiver by b. The POVM elements specifying the measurements are . The measurement outcome b is sent to the Processing Unit.

Processing Unit: Processing unit is an electronic controller for the other components. It estimates conditional probability distribution P(a,b|x,y) by repeatedly sending inputs x and y to sender A and receiver B and receiving the outcomes. This is repeated a large number of times (typically more than 10.000). If the measurement procedure failed to produce an outcome (which happens with probability 1- rj) the processing unit assigns an outcome using functions α(x) and β(y) for A and B, respectively. The processing unit performs a test of non-classicality. It is done by taking linear combination of P(a,b|x,y) and comparing to a value C, which is the classical bound, i.e:

If the above inequality is violated the we say that the probability distribution P(a,b|x,y) cannot have a classical explanation and therefore its non-classicality is confirmed. Known systems and tests which can reliably perform a test of nonclassicality are specified by a set of parameters which are explained below:

The test itself is specified by the numbers The bound C is then calculated from these numbers. The method of calculation and its result depend on the definition of nonclassicality. For example, for a dimension witness it will be different than for entanglement witness. For the most popular test of Bell inequalities C =

The behaviour of the detection stations is specified by the state ρAB and measurements defined by POVM elements such that the probability distribution P passes the test of nonclassicality, i.e.

The behaviour of the processing unit must allow for imperfect detectors. If the detectors are imperfect then, sometimes measurement station yields no result since the carrier of the quantum state avoided detection. In such a case the test has to have a well-defined strategy to assign an outcome for it. We denote by α(x) the outcome assigned by A if its choice of measurement is x. Analogously for B it is β(y). If measurement stations produce outcome with probability q, then the left hand side of the nonclassicality test will be The desired property of the test is for to exceed C for as low value of as possible. This lowest value of is called critical detection efficiency and denoted

Therefore to specify the system it suffices to provide parameters: and β(y). From them C is computed by

In WO2019125733 method of generating a random bit string, the method comprising: providing a weak source of randomness; providing a quantum device configured to entangle a plurality of quantum particles in a quantum state; measuring each of the plurality of particles in two different bases; determining a level of violation of a Bell inequality; determining whether to accept or abort based at least partly on the determined level of violation; and extracting the random bit string via a two-source extractor.is described, which postprocesses results from a test of non- classicality i.e. Bell inequality to obtain randomness. The device specifies a couple of tests of non-classicality in exemplary embodiments but will operate with any Bell inequality and the component used for Bell inequality test can be considered as a black box.

In US7428562 it is described method for generating a random series, comprising: generating an entangled quantum state of a first system and a second system; measuring the first system and the second system; determining a random value from a result of measurement of one of the first system and the second system; and evaluating results from measurements of both of the first system and the second system to determine whether the results are consistent with the first and second systems being in the entangled state.. This is a particular choice of test which fails immediately if the detectors are imperfect. To avoid this so-called “fair sampling assumption” is typically invoked, which assumes that the events registered by the detectors are a fair sample of all the events. However it is known that using this assumption significantly reduces the security of the test which can then give false- positive results. In CN108984153 it is described - a device based on a specific Bell inequality test with critical detection efficiency of 78%.

In CN108365955 it is described use of hyperentanglement (local Hilbert space dimension is 4) but only uses a single qubit per party for non-classicality test which is the standard CHSH inequality with very high critical detection efficiency.

In CN107786280 involves hyperentangled Bell state but it is used as a resource in a trusted protocol but no non-classicality test is involved.

Using non-classicality tests for generating correlated strings of numbers is known but there are still drawbacks to the known methods. The two most important ones are:

(a) high requirements for detection efficiency;

(b) low robustness to experimental imperfections.

Therefore there is a need for new tests of non-classicality involving quantum states of high Hilbert space dimension without described disadvantage and which:

(a) require low critical detection efficiency;

(b) outcomes can be analysed for a wide range of dimensions;

(c) involve quantum states easy to prepare and measurements easy to perform in practice;

(d) produce strings of numbers with properties desirable from quantum information protocols point of view;

(e) are robust to small imperfections in the hardware used.

This sets goals of the invention.

The object of the invention was to provide system generating correlated strings of numbers with the characteristics a-e listed below. The invention has all the properties (a)-(e) but its most important characteristic is the critical detection efficiency significantly lower than the initial tests that they are based on. The invention yields tests with significantly lower (which is better) critical detection efficiencies than in state of the art - eg. CN108984153.

The invention is a system that generates a string of numbers and performs a test of non-classicality on them, which:

(a) requires low critical detection efficiency;

(b) outcomes can be analysed for a wide range of dimensions;

(c) involves quantum states easy to prepare and measurements easy to perform in practice;

(d) produces strings of numbers with properties desirable from quantum information protocols point of view;

(e) is robust to small imperfections in the hardware used.

Symbols and abbreviations used in the description of the invention thereinafter:

A,B - for systems based on test without communication denote detection stations, which are parts of a system in which quantum states are measured and outcomes obtained; for systems based on test with communication A is sender which prepares a quantum state and sends it to receiver B.

POVM - Positive Operator Valued Measure x - input of A chosen by Processing Unit y - input of B chosen by Processing Unit a - measurement outcome for A b- measurement outcome for B ρAB - a quantum state measured by A and B a POVM element representing measurement done by A after receiving input x and producing outcome a a POVM element representing measurement done by B after receiving input y and producing outcome b - detection efficiency, which is the probability with which A and B produce any outcome in a round of nonlocality test α(x) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was x β(y) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was y - conditional probability distribution for outcomes of A and B conditioned on their inputs “ parameters specifying the test of nonclassicality

C - classical bound of a test of non-classicality. It is a number such that implies nonclassical behaviour of the system. - critical detection efficiency, which is the lowest detection efficiency allowing for conclusive outcome of a test of non-classicality

CHSH - Clauser-Horne-Shimony-Holt Bell inequality

To design a system which generates strings of non-classically correlated numbers a test is needed first. For this test concrete parameters of the system can be established in such a way that the numbers generated by the system pass the test.

Let T be a known test of non-classicality. It is specified by a type (eg. Bell inequality, dimension witness, entanglement witness, Leggett-Garg inequality, steering inequality, etc. . .,) and parameters A system D which can generate a correlated string of numbers, which passes test T is schematically shown in fig.1,2 or 3 depending on the test type. It is parametrized by The invention is based on a modification of any existing test of non-classicality T and quantum sates and measurements that pass it. The layout of the system is exactly like in fig.1,2 or 3 depending on the type of T. What is changed according to the invention is the method used by the processing unit, the quantum states prepared by the source or the sender and the measurements performed by the detector stations or the receiver.

The core concept of our invention is that from every test T and system D which generates strings of numbers, which passes T we obtain a novel test T_N and a new system D_N with much better desired properties.

Below the details of obtaining new test T_N and system D_N are presented.

Description how to obtain from the parameters of test T and system D the parameters of T_N and D_N:

The initial test T is described by parameters and B and have the critical detection efficiency - Our invention includes a method which creates a sequence of new tests T_N. All tests T_N follow exactly the same steps 1-6 as T but use different parameters and can be viewed as a parallel repetition of T repeated N times. Therefore Ti is identical to T. For N>1 the quantum state of T_N is Iⁿ TN the measurements for A and B are defined by where and and are defined analogously. For all i the new functions α and β are The classical bound of T_N is C^N. The parameters are given by formula

Apart from these modifications a non-linear correction — K(X + K) is added to the left hand side of the inequality. Therefore the formula for T_N is: Where and m and n represent the number of possible values the variables a and b can take, respectively. The sum over is taken such that and match on i-th element, but are not the same. The same holds for the sum over Coefficient \ where M is the number of possible values x and y can take and is the algebraic bound of initial test.

For T_N the critical detection efficiency is

Where decays exponentially with N.

A system which performs the test T_N is any device which can reliably prepare states ^e measurements and process the measurement results. The layout of the system is exactly like in the initial system D, either like in the fig 1,2 or 3 depending on the test type. Therefore the invention has three variants.

Variant 1 :

Variant 1 is schematically presented in fig. l. The system comprises of parts disclosed below.

(a) Source: Source produces non-classically correlated particles which are the carriers. It encodes a quantum state into several carriers and sends one carrier to each of measurement stations. The number of stations is arbitrary. Here, for simplicity, we assume that there are two of them. The stations are labelled A and B and the quantum state they receive

(b) Measurement stations: The measurement stations are parts of the system which perform measurements of the quantum state encoded by the source of non-classically correlated particles. Both of measurement stations receive inputs from Processing Unit and perform one measurement based on it. We denote measurement choice of A by and choice of B by The outcomes of the measurements are for A and for B. The POVM elements specifying the measurements performed by A and B are . The POVM elements specifying the measurements performed by A and B are and The measurement outcomes a and b are sent to the

Processing Unit.

(c) Processing Unit: Processing unit is an electronic controller for the other components. It estimates conditional probability distribution by repeatedly sending inputs x and to measurement stations and receiving the outcomes. This is repeated a large number of times (typically more than 10.000). If the measurement procedure failed to produce an outcome (which happens with probability 1- η) the processing unit assigns an outcome using functions α(x) and β(y) for A and B, respectively. The processing unit performs a test of non-classicality. It is done by taking linear combination of and comparing to a value C^N, which is the classical bound, i.e:

If the above inequality is violated the we say that the probability distribution cannot have a classical explanation and therefore its nonclassicality is confirmed.

Variant 2:

Variant 2 is schematically presented in fig.2. The system comprises of

(a) A source: Source produces non-classically correlated particles which are the carriers. The source prepares a quantum state and sends the whole or part of it to the sender and (optionally) to the receiver.

(b) Sender (A): A quantum state p ^w is prepared by the sender. It is done by a modification of the state the sender received from the source. The sender can keep a part of the quantum state and sends the rest to the receiver. The quantum state A and B have after communication from the source and between them is The sender also receives inputs from Processing Unit and performs measurement based on it. We denote its measurement choice by The outcome of the measurement is denoted by The POVM elements specifying the measurements performed by the sender are A In some tests the sender does not perform any measurement. This case is modelled by the measurement outcome assigned a constant. The measurement outcome a is sent to the Processing Unit.

(c) Receiver (B): The receiver receives quantum states from the sender and the source of non-classically correlated particles. The receiver also receives input from Processing Unit and performs measurement based on it. We denote the measurements outcome of the receiver by The POVM elements specifying the measurements are The measurement outcome is sent to the Processing Unit.

(d) Processing Unit: Processing unit is an electronic controller for the other components. It estimates conditional probability distribution by repeatedly sending inputs and to sender A and receiver B and receiving the outcomes. This is repeated a large number of times (typically more than 10.000). If the measurement procedure failed to produce an outcome (which happens with probability 1- η) the processing unit assigns an outcome using functions α(x) and β(y) for A and B, respectively. The processing unit performs a test of non-classicality. It is done by taking linear combination of and comparing to a value C^N, which is the classical bound, i.e:

Variant 3:

Variant 3 is schematically presented in fig.3. The system comprises of

(a) Sender (A): A quantum state is prepared by the sender. The sender can keep a part of the quantum state and sends the rest to the receiver. The quantum state A and B have after communication from the source and between them is p The sender also receives inputs from Processing Unit and performs measurement based on it. We denote its measurement choice by The outcome of the measurement is denoted by a. The POVM elements specifying the measurements performed by the sender are In some tests the sender does not perform any measurement. This case is modelled by the measurement outcome assigned a constant. The measurement outcome is sent to the Processing Unit.

(b) Receiver (B): The receiver receives quantum states from the sender. The receiver also receives input from Processing Unit and performs measurement based on it. We denote the measurements outcome of the receiver by The POVM elements specifying the measurements are The measurement outcome is sent to the Processing Unit.

(c) Processing Unit: Processing unit is an electronic controller for the other components. It estimates conditional probability distribution by repeatedly sending inputs and to sender A and receiver B and receiving the outcomes. This is repeated a large number of times (typically more than 10.000). If the measurement procedure failed to produce an outcome (which happens with probability 1- η) the processing unit assigns an outcome using functions α(x) and β(y) for A and B, respectively. The processing unit performs a test of non-classicality. It is done by taking linear combination of ) and comparing to a value C^N, which is the classical bound, i.e:

In all varaints the method for estimating by the Processing Unit is the following. The processing unit stores in its memory values of and for the last M repetitions. M is a free parameter chosen by the user. In each repetition next set of and is added. After M rounds the probability is estimated using where is Kronecker’s function equal to 1 if α = b and 0 otherwise.

The inventors of the patent confirm that part of the research work on the invention was carried out at the International Center for Theory of Quantum Technologies of the University of Gdansk, based on funding obtained from the Foundation for Polish Science as part of the "International Centre for Theory of Quantum Technologies" project (agreement number MAB/ 2018/5). The mentioned project is carried out within the International Research Agendas Programme co- financed by the European Union from the funds of the Smart Growth Operational Programme, axis IV: Increasing the research potential (Measure 4.3).

The invention can be applied whenever nonclassical correlations (with a test certifying their non-classicality) are needed. One of such cases is the generation of certified random numbers. It has application in lotteries, gaming and basic science but the most important one lies in cybersecurity. Random numbers of high quality are necessary for secure operation of computers, choices of passwords and data encryption.

Another case requiring non-classical correlations is quantum key distribution - a technology, which uses tests of non-classicality to prove the security of communication.

Two properties of the method are of particular importance: (a) there exists an efficient method of checking if the correlations in the strings of numbers produced by the system of devices have desired properties; (b) the requirements on the hardware used to detect quantum states are lower than in the current state-of-the-art solutions. The latter one is practically appealing as it not only makes it much easier (and cheaper) to construct devices capable of running the tests of non-classicality but also increases their efficiency. In the case of quantum random number generation and key distribution mentioned above it translates to faster and more secure production of the random numbers or cryptographic keys.

Inventions are further disclosed in example and drawing where fig 1-3 shown the known system in three different variants, figure 1 : system for generating correlated strings of numbers with a test of non-classicality without communication; figure 2: system for generating correlated strings of numbers with a test of non- classicality with communication and a source of non-classically correlated particles, figure 3: system for generating correlated strings of numbers with a test of non- classicality with communication and no source of non-classically correlated particles, figure 4: source of non-classically correlated particles: a pulsed laser pumps a nonlinear crystal to generate time bin entanglement, figure 5 : detector station for A. The one for B is the same. It consists of set of phase shifters in unbalanced interferometric optical setup. Then the photon detection signal and measurement choice are recorded on a local FPGA chip with a time tagging unit.

Example - preferred embodiment - practical use of the invention in the embodiment with a test of a Bell inequality violation is presented below.

Symbols and abbreviations used in the description of the preferred embodiment of the invention thereinafter:

POVM - Positive Operator Valued Measure x - input of A chosen by Processing Unit y - input of B chosen by Processing Unit a - measurement outcome for A b- measurement outcome for B ρAB - a quantum state measured by A and B a POVM element representing measurement done by A after receiving input x and producing outcome a - a POVM element representing measurement done by B after receiving input y and producing outcome b - detection efficiency, which is the probability with which A and B produce any outcome in a round of nonlocality test α(x) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was x β(y) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was y

P(a, b|x,y) - conditional probability distribution for outcomes of A and B conditioned on their inputs “ parameters specifying the test of nonclassicality

CHSH - Clauser-Horne-Shimony-Holt Bell inequality

Test.

Establishing parameters for T_N: In the preferred embodiment the initial test T is based on CHSH inequality and N=10. We take the initial ρAB to be maximally entangled state of two qubits, operators correspond to projections on eigenstates of Pauli matrix to and - The coefficients for CHSH inequality are defined by

They imply the classical bound and the algebraic bound which yields K = 256. The functions a and P are constant and equal to 0. Such parameters lead to

According to our invention the test T ₁₀ uses the state , the measurements The classical bound is The functions α and β are constant and equal to 0. The coefficients are given by and A=B=0. This gives n

System

The device is build according to the schematic in fig.1.

The source is composed of pump pulsed laser at wavelength 775 nm. The laser is pumped d=1024 times per preset time window. Its beam is directed at SPDC nonlinear crystal producing photon pairs with their spectrum centered at 1550nm. The emitted photons are coupled into two single-mode optical fibers that direct one photon each to the measurement stations. Since there is in principle no information available in which of the pulses the photon pair has been created, the resulting state can be written as a maximally entangled state of 10 qubit pairs:

Where

The source is shown in fig. 4

The measurement stations A and B are depicted in fig.5. They use a structured interferometer which translates information encoded into the amplitudes of the pulses into time and polarization registered by the tagging unit. Both of measurement stations receive inputs from Processing Unit and perform one measurement based on it. We denote measurement choice of A by and choice of B by y The outcomes of the measurements are for A and for B. The POVM elements specifying the measurements performed by A and B are where operators correspond to projections on eigenstates of Pauli matrix to The choice of the measurement operators and are chosen by suitable modifications of the phase modulation properties of the mirrors in the interferometer. The measurement outcomes and are sent to the Processing Unit.

The processing unit takes form of an Field Programable Gate Array (FPGA) chip, which performs the test TN. It estimates conditional probability distribution by repeatedly sending inputs and to sender A and receiver B and receiving the outcomes. If the measurement procedure failed to produce an outcome (which happens with probability 1- η) the processing unit assigns an outcome using functions for A and B, respectively. This is repeated more than 10.000 times. The processing unit stores in its memory values of and for the last 10. 000 repetitions. In each repetition next set of a and is added. After M rounds the probability ) is estimated using where is Kronecker’s function equal to 1 if α = b and 0 otherwise.

Then the processing unit performs a test of non-classicality. It is done by taking linear combination of and comparing to a value C^N, which is the classical bound, i.e: where

Claims

Claims System for generation of correlated strings of non-classically correlated numbers, comprising: sender (A) receiver (B) a processing unit (PU), characterized that the device comprises a processing unit (PU), which is connected to the sender (A) and the receiver (B) by electrical wires while the processing unit (PU) is connecting in such a way that enables sending commands to at least two detector stations the sender (A) and the receiver (B) after which the sender (A) prepares a quantum state specified by the command from the Processing Unit (PU) and sends the state to the receiver and the receiver B makes measurements specified by processing unit (PU), while the choice of measurement for B is and the sender (A): produces states of the form N is a parameter chosen by the user, receiver B is configured to send the measurement results to the processing unit (PU), and the result is denoted by processing unit assigns , as a result of the sender (A). System for generation of correlated strings of non-classically correlated numbers, comprising: a source of entangled states (S), at least two detector stations (A and B), a processing unit (PU), characterized that the system comprises a processing unit (PU), which is connected to the source of entangled states (S) and at least two detector stations (A and B) by electrical wires while the processing unit (PU) is connecting in such a way that enables sending commands to the source of entangled states (S) and at least two detector stations (A and B) after which the source of entangled states (S) produces a state sending to at least two detector stations ((A and B) and the detector stations (A and B) make measurements specified by processing unit (PU), while the choice of measurement for A is and for B is and the source of entangled states (S): produces states of the form where N is a parameter chosen by the user, encodes the states onto particles in such a way that the number of particles is equal to the number of detector stations, estimates probability distribution ) and returns it to the user, and detector stations A and B are configured to send the measurement results to the processing unit (PU), the result from first station is denoted by and from second station System for generation of correlated strings of non-classically correlated numbers, comprising: a source of entangled states or classical correlations (S), sender (A) receiver (B) a processing unit (PU), characterized that the system comprises a processing unit (PU), which is connected to the source of entangled states or classical correlations (S) and the sender (A) and the receiver (B) by electrical wires while the processing unit (PU) is connecting in such a way that enables sending commands to the source of entangled states or classical correlations (S) and at least two detector stations the sender (A) and the receiver (B) after which the source of entangled states or classical correlations (S) produces a signal to the sender (A) and the receiver (B), the sender (A) prepares a message specified by the command from the Processing Unit (PU) and the signal from the source of entangled states or classical correlations (S) and sends the message to the receiver and the sender and receiver (A and B) make measurements specified by processing unit (PU), while the choice of measurement for A is and for B is and he source of entangled states (S): produces states of the form p where N is a parameter chosen by the user, encodes the states onto particles in such a way that the number of particles is equal to the number of detector stations, estimates probability distribution and returns it to the user, and sender and receiver A and B are configured to send the measurement results to the processing unit (PU), the result from first station is denoted by and from second station The system according to claim 2 and 3 wherein entangled states produced by the entangled states (S) is of the form A method for testing non-classicality of generated string of numbers comprising steps of a) taking a known test of non-classicality specified by parameters and B where the non-classicality is defined by violating , while are functions which return the outcomes a and b respectively when the detector stations fail to produce an outcome and are state produced by an entanglement source S and measurements performed by stations A and B respectively, such that Creating a new test of non-classicality specified by parameters where and and are defined analogously and where the non-classicality is defined by violating, where and m and n represent the number of possible values the variables a and b can respectively take, wherein the sum over is taken such that and match on i-th element, but are not the same while the same holds for the sum over and where M is the number of possible values x and y can take and

- testing non-classicality of P by checking if is violated, where x - input of A chosen by Processing Unit for a known test of non-classicality T y - input of B chosen by Processing Unit for a known test of non-classicality T ρAB - a quantum state measured by A and B for a known test of non-classicality T a - measurement outcome for A for a known test of non-classicality T b- measurement outcome for B for a known test of non-classicality T - a POVM element representing measurement done by A after receiving input x and producing outcome a for a known test of non-classicality T - a POVM element representing measurement done by B after receiving input y and producing outcome b for a known test of non-classicality T - input of A chosen by Processing Unit for the new test of non-classicality - input of B chosen by Processing Unit for the new test of non-classicality - measurement outcome for A for the new test of non-classicality - measurement outcome for B for the new test of non-classicality a POVM element representing measurement done by A after receiving input x and producing outcome a for the new test of non-classicality a POVM element representing measurement done by B after receiving input y and producing outcome b for the new test of non-classicality α(x) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was x β(y) - a function specifying which outcome Processing Unit will assign to a measurement performed by A if it did not produce any outcome and its input was y - conditional probability distribution for outcomes of A and B conditioned on their inputs “ parameters specifying the test of nonclassicality

C - classical bound of a test of non-classicality, wherein C implies nonclassical behaviour of the system.