WO2011033543A1

WO2011033543A1 - Description of the patent application entitled: quantum signals receiver with noise compensation, quantum cryptography communications system and method

Info

Publication number: WO2011033543A1
Application number: PCT/IT2010/000133
Authority: WO
Inventors: Marco Lucamarini; Giovanni Di Giuseppe; Kumar Rupesh Parapatil Subramanian; Paolo Tombesi; Sajeev Damodarakurup; Andrea Lazzari; Gabriele Gradoni; David Vitali
Original assignee: Criptocam S.R.L.
Priority date: 2009-09-18
Filing date: 2010-03-29
Publication date: 2011-03-24
Also published as: IT1396183B1; ITMI20091594A1

Abstract

There is described a receiver (300) comprising an input connectable to a transmission channel (OFl) for receiving signal quantum states having an asymmetric distribution and associated to a cryptographic key to be determined. The receiver also comprises: a decoding and managing block (SSD, 60, 70) responsive to the quantum states distribution symmetry for measuring the signal quantum states and returning a quantity representing a channel introduced error, an actuator (90) adapted to vary physical parameters of signal quantum states for reducing the channel introduced error.

Description

Description of the patent application entitled: "Quantum signals receiver with noise compensation, quantum cryptography communications system and method"

Technical field of the invention

The present invention relates to the quantum cryptography sector and, in particular and without limitation, refers to quantum key distribution techniques .

Background art

As it is known, quantum cryptography allows the distribution of a secret key between two remote entities, the emitter and the receiver, usually called "Alice" and "Bob", with a security which is, in principle, absolute.

Typically, the key is encoded on elementary quantum systems, such as photons, that are exchanged on a quantum channel, such as an optical fiber. The security of this method derives from the well-known fact that the measurement of an unknown quantum state modifies this state.

Some of the fundamental principles of quantum cryptography can be found in the following prior art documents :

B. Huttner et al., "Quantum cryptography with co- herent states", Physical Review A (Atomic, Molecular, and Optical Ph-ysics), Mar. 1995, USA, vol. 51, No. 3, pp. 1863-1869.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum Cryptography", Rev. of Mod. Phys . 74, (2002) .

An important problem in this field is related to remote alignment of two devices dedicated to quantum key distillation in a QKD session. Alignment of two devices at the ends of a quantum channel provides for reduction of the noise level present on the channel that disturbs the delicate quantum states during passage thereof.

For example, in the case of phase encoding, if the receiver and the transmitter are not correctly aligned, it is impossible for the receiver to understand the information sent by the transmitter, as there is no precise reference phase for establishing the decoding procedure.

Various procedures have been adopted to date for alignment of a quantum channel.

The document A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden and N. Gisin, Appl . Phys. Lett. 70, 793 (1997) describes a "Plug-and-Play" technique. Plug-and-Play devices are bidirectional devices in which the noise is passively compensated: the quantum signal present- on the return path automatically compensates the noise encountered by the classical signal along the forward path.

The Applicant observes that the problem with this technique is that the intensity of the forward pulse is very high, and becomes a quantum signal only after it has been strongly attenuated at one of the ends of the channel. This causes a part of the photons to be reflected back as a result of "Rayleigh back- scattering" and increases transmission noise. Moreover, due to the fact that it utilizes classical pulses, this technique cannot be used for exclusively quantum signals.

The article by G. B. Xavier, G. Vilela de Faria, G. P. Temporao, and J. P. von der Weid, Opt. Expr. 16, 1867 (2008) refers to a multiplexing frequency technique. According to this technique, the reference pulses with frequency slightly different to that of the quantum signal are sent in the channel together with the quantum signal to extrapolate the noise level of the quantum signal.

The Applicant has noticed that the problem with this technique is that, as the reference pulses are very intense, they can generate, through non-linear phenomena (Raman scattering) , photons at the same frequencies as the quantum signals, thereby causing noise in Bob's detectors.

A time-division multiplexing technique with classical reference signals is described in Z . L. Yuan, A. R. Dixon, J. F. Dynes, A. W. Sharpe and A. J. Shields, Appl. Phys . Lett. 92, 201104 (2008) and a time-division multiplexing technique with quantum reference signals is described in J. Chen, G. Wu, L. Xu, X. Gu, E. Wu and H. Zeng, New J. Phys. 11, 065004 (2009) .

According to the techniques of these documents, reference pulses at the same frequency as the quantum signal are input in the channel at different times to those corresponding to the quantum signals. The noise encountered by the quantum signal is the same as that of the reference pulses, if these latter are sufficiently fast.

The Applicant noticed that this technique shows a problem due to the fact that the light packets linked to the classical reference pulses can expand and superpose the quantum packets. To prevent this, the reference pulses must be greatly distanced from the quantum pulses, thereby decreasing the transmission rate. Moreover, the space, the time or the frequen- cies occupied by the reference signals could be more advantageously used for QKD.

The document V. Makarov, A. Brylevski, and D. R. Hjelme, Appl . Opt. 43, 4385 (2004) describes a technique based on interruption of communication. In this technique Alice and Bob initially align their devices and start the QKD session. Initial alignment is performed, at quantum level, by sending predetermined information so that the receiver can align itself by synchronizing its receiving device. After alignment has taken place, the QKD session starts. Then, at a certain point, the users interrupt the communication to re-align their devices, as occurred initially.

The Applicant observed that this technique has two drawbacks: on the one hand the quantum flow must be interrupted to align the devices, thus decreasing the transmission rate; on the other, interruption is entirely random. Therefore, the communication could be interrupted when the noise level is still very low, losing time that could have been used for QKD.

Some documents considered useful for full understanding of the description below are:

• C. H. Bennett, Phys . Rev. Lett. 68, 3121 (1992) .

• K. Tamaki, M. Koashi, and N. Imoto, Phys. Rev. A 67, 032310 (2003) .

• D. S-ubacius, A. Zavriyev, and A. Trifonov, Appl. Phys. Lett. 86, 011103 (2005).

• C. H. Bennett and G. Brassard, in Proceedings IEEE Int. Conf. on Computers, Systems and Signal Processing, Bangalore, India (IEEE, New York, 1984), pp. 175-179.

• GB-A-2439771;

• Italian patent application MI-A-2009-000849 Brief summary of the invention

The Applicant has addressed the problem correlated to alignment between receiver and transmitter in a quantum cryptography communications system.

The invention relates to a quantum signal receiver defined by the appended claim 1 and to preferred embodiments thereof described in the dependent claims 2 to 7. According to another aspect, the invention also relates to a quantum cryptography communications system as defined by claim 8 and to particular embodiments thereof, as defined by the independent claims 9-11.

Moreover, the invention also relates to a quantum cryptography method as described in claim 12.

Brief description of the drawings

Further characteristics and advantages of the in- vention will be apparent from the description below of a preferred embodiment and of variants thereof, provided by way of example with reference to the accompanying drawings, wherein:

FIG. 1 shows, by means of functional blocks, an example of a quantum cryptography communications system;

FIG. 2 represents a schematization of the system of FIG. 1 showing a feedback loop included in a receiver;

FIG. 3 shows another schematization of the system of FIG. 1 in relation to a decoding operation;

FIG. 4 shows curves that place in relation results of decoding operations with the noise present on the transmission channel;

FIG. 5 schematically shows a control algorithm implementable by said feedback loop;

FIG. 6 shows graphs relative to a condition with no noise and in two different conditions with noise;

FIG. 7 shows curves relative to the behavior of the noise and of the compensation in accordance with an example of the invention;

FIG. 8 shows a Poincare sphere for defining angles that identify a quantum state.

Detailed description of the invention Structural description

FIG. 1 shows an example of a quantum cryptography communications system 100. The system 100 comprises a transmitter (also called Alice) and a receiver 300 (also called Bob) connected by a first quantum transmission channel OF1 and by a second quantum transmission channel 0F2.

In the present example, reference will be made to signals at typical frequencies of optical fiber transmissions, but electromagnetic radiation at any frequency adapted for guided propagation or propagation in free space for the purposes of communication between the transmitter 200 and the receiver 300 can also be used alternatively.

The transmitter 200 comprises, according to an example, a management and control system 5A, an electronic device for synchronization and conversion 10A, in turn comprising a synchronization module 20 (for example, with TTL, Transistor Transistor Logic, technology) , a first analog-digital and viceversa conversion module 30 and a second digital-analog conversion module 50.

According to a particular example, shown in FIG. 1, the quantum cryptography communications system 100 is such as to perform transmission of cryptographic keys according to a protocol that provides for encoding of logic values in quantum states.

According to a first embodiment, the communications system 100 is based on encoding of logic values in the optical radiation phase. This encoding can be implemented, in particular, according to the B92 protocol, which provides both for polarization encoding and relative phase encoding.

In accordance with a second embodiment, the communications system operates with encoding of logic values in distinct polarization values of the optical radiation. For example, encoding is performed in accordance with the polarization encoding established by the B92 protocol.

In greater detail, the transmitter 200 is provided with a source 80, for example of laser type, capable of generating electromagnetic radiation (i.e. photons) PI for the purpose of quantum communication. This source 80 is connected to the second digital- analog conversion module 50 to receive analog electrical control signals S64.

As photon source 80 it is possible to use, for example, SPDC (Spontaneous Parametric Down Conversion) sources that generate correlated or "entangled" photons, pulsed or continuous wave laser sources to generate coherent states, LED (Light Emitting Diode) sources to generate coherent or thermal states, or other single or non-single photon pulse sources.

Moreover, the transmitter 200 is advantageously provided with an optical attenuator 81 having a relative input connected to an output of the laser source 80. The optical attenuator 81 is connected to the first conversion module 30 to receive a control signal S40, for example, in voltage that allows setting of the desired attenuation of the laser pulses PI obtaining attenuated pulses P2.

Moreover, an output of the optical attenuator 81 is connected to an input port of a modulator of quantum state transmission 82 capable of modulating the quantum states with non-symmetrical (or equivalently, asymmetric) distribution.

In accordance with the first embodiment, the modulator of quantum state transmission 82 is a relative phase modulator of the electromagnetic radiation comprising, for example, one or more Mach-Zehnder interferometers with unbalanced arms. This type of modulation introduces encoding also known with the term "relative phase" encoding of the optical radiation, where "relative" is intended as the phase accumulated by the radiation along one of the paths of the interferometer relative to that accumulated along the other path.

The modulator of quantum state transmission 82 can be a polarization modulator, in the case of the second embodiment. This polarization modulator can, for example, be produced by means of one or more appropriate polarization controllers.

The transmission modulator 82 has an output port optically coupled to the first transmission channel OF1, for example an optical fiber, and is connected to the second conversion module 50 to receive an appropriate modulating signal S56 of the quantum states. The transmission modulator 82 is therefore capable of sending optical signals P3 on the first optical fiber OF1. The signal quantum states have encoded bits representative of the cryptographic key.

According to a particular example, the transmitter 200 is also provided with a further radiation source 83 (for example, a laser or a photodiode) , adapted to receive respective analog control signals S65 generated by the second conversion module 50 and to generate optical synchronization signals P5 to send on the second transmission channel OF2, for example a second optical fiber.

The management and control unit 5A, which can be, for example, a microprocessor provided with a relative CPU, contains hardware and/or software modules capable of completely managing operation of the transmitter 200. In particular, the management and control unit 5A is such as to send appropriate command signals SI and S2 and data signals, on buses Dl and D2, towards the converters 30 and 50 so as to apply the desired encoding and transmission methods of the quantum states.

Reference is now made to the receiver 300 (Bob) which comprises a further management and control unit 5B and a further electronic synchronization and conversion device 10B, for example, analogous to the electronic synchronization and conversion device 10A included in the transmitter 200. The further management and control unit 5B can be, for example, a microprocessor provided with a relative CPU, and comprises a calculation module 60 and a control module 70.

According to the example shown in FIG. 1, the further electronic synchronization and conversion device 10B is provided with a further synchronization module 20' , a third analog-digital-analog conversion module 30' and a fourth digital-analog conversion module 50' . Moreover, according to the example described, the receiver 300 -includes a photodetector 84 (i.e. at least one photodiode PIN) connected to an output of the second optical fiber OF2, which is capable of sending electrical synchronization signals S21' , corresponding to the optical signals P5, to the further synchronization module 20' .

The further synchronization module 20' allows electrical synchronization signals S21' to be processed and sent in the form of TTL signals S3 to the further management and control unit 5B.

An output of the first optical fiber OF1 is connected to an optical input of an actuator device 90 which is controlled by the control module 70 and adapted to act directly on optical signals.

An optical output of the actuator device 90 is optically coupled to a receiving modulator 85, in turn connected to the fourth digital-analog conversion module 50' to receive relative control signals S56' .

According to the first embodiment, the receiving modulator 85 is a phase modulator of the electromagnetic radiation. For example, this phase modulator 85 can be a Mach-Zehnder interferometer, of known type and provided with two arms that introduce appropri- ately unbalanced paths for the optical radiation. This Mach-Zehnder interferometer 85, on the basis of the value of the control signal S56' , determines the ratios between its outputs, selecting in this way the measurement base of the receiver 300. This method of selecting the measurement base can be considered as a filtering method controlled by the digital-analog conversion module 50' .

Alternatively, in accordance with the second embodiment, the receiving modulator 85 comprises one or more polarization controllers and relative polarization filters that select the radiation with predetermined polarization. The polarization controller 85 can be controlled to vary (that is, rotate) the polarization transmitted by the same controller at the measurement bases selected, for example in accordance with the B92 protocol.

With reference to the actuator device 90, in the case of the first embodiment, it can be, for example, a phase modulator placed, in particular, along one of the arms of the Mach-Zehnder interferometer included in the receiving modulator 85.

If the communications system 100 is fiber optic integrated in the third telecom window, the model X5 produced by the company JDSU (ww . j dsu . com) , which operates up to 10 Gb/s by means of a remote control, can, for example, be used as phase modulator 90.

In accordance with the second embodiment, the actuator device 90 can be produced by means of a polarization controller. If the communications system 100 is fiber optic integrated, the model PCD-M02-3X- NC-7 by Laser 2000 (www.laser2000.com), which operates at a repetition rate in the order of 10 mb/s, can, for example, be used as polarization controller.

The receiving modulator 85, which is such as to receive transmitted optical signals P3 and to return output optical signals P4, has an output connected to a detector module 86 (for example at least one suitable photodiode) adapted to convert the output optical signals P4 into detected electrical signals SD.

The fourth digital-analog conversion module 50' is such as to send to the further management and control unit 5B a receiving electrical signal S3 that activates this unit for processing of the detected electrical signals SD.

The further management and control unit 5B contains hardware and/or software modules capable of managing all operations of the receiver 300. In particular, this further management and control unit 5B is such as to sent appropriate command signals S4 and data on a bus D3 to the converter 50' , so as. to apply the desired measurement and decoding methods of the quantum states.

The set of components including the actuator 90, the receiving modulator 85, the detector module 86 and the control module 70 form a feedback loop.

It should be noted that in the case of phase encoding, the optional presence of a plurality of phase modulators inside the interferometer used in the communications system 100 advantageously also allows implementation of a modulation of the global phase of the radiation that propagates in the interferometer and not only of the relative phase.

Modulation of the global phase, which can be implemented in the embodiment that provides for phase encoding of the electromagnetic radiation, can be obtained synchronizing the transmission and/or receiving phase modulators of the system so that they can operate with particular velocity acting, for example, substantially on all the pulses that pass through them and not only on a part of these. As is apparent to those skilled in the art, global phase encoding has advantages in terms of security against attacks by eavesdroppers with respect to relative phase encoding alone . Description of examples of the cryptography method

An example of a quantum cryptography method will be described with reference to FIG. 2, which illustrates the communications system of FIG. 1 in simplified form.

The quantum cryptography method sued by the communications system 100 is of the QKD (Quantum Key Distribution) type and provides for transmission of a cryptography key by the transmitter 200 to the receiver 300. The cryptography key is used to encrypt and decrypt messages transmitted, for example, by means of conventional (i.e. not quantum) cryptography techniques .

This key can be expressed with a string of bits which will be encoded in quantum states and then associated with quantum parameters of the electromagnetic radiation or, equivalently, of the photons.

The transmitter 300 (Alice) generates the optical signals P3 with which initial "signal" quantum states are associated, i.e. quantum states that convey encoded information relative to the key to be transmitted .

According to the first embodiment, the signal, quantum states are obtained by means of phase modulation. In particular, this encoding can be obtained by means of the transmission modulator 82 such as, for example, a Maeh-Zehnder interferometer with unbalanced arms and therefore adapted to vary the phase of the optical signal P2 generated by the laser source 80 and attenuated by the attenuator 81.

In accordance with the second embodiment, the initial quantum states are obtained by means of polarization encoding of the electromagnetic radiation. In particular, this encoding can be obtained by means of the transmission modulator 82 produced as polarization controller and therefore adapted to vary the polarization of the optical signal P2 generated by the laser source 80 and then attenuated by the attenuator 81.

Moreover, it is observed that for both embodiments (phase and polarization encoding) the transmitter 200 prepares a set of signal quantum states with asymmetric distribution (DNS) .

Quantum states having asymmetric distribution are intended as quantum states that are not distributed symmetrically with respect to the origin of a circle obtained by intersection of a plane passing through the origin of the Poincare sphere with the sphere itself.

FIG. 8 shows a Poincare sphere, often used in op- tics to describe polarization states and the axes X, Y and Z are shown. FIG. 2 shows these quantum states |φ₀) and |φ,) represented on a circle A lying on a plane that intersects the Poincare sphere passing from its centre and provides an example of DNS in dimension d= 2. In fact, on the circle there are two states, indicated with |φ₀) and |φ,) , which are drawn in the right part of the circle, while the left part of the circle contains no states, and this means that the two states are not distributed symmetrically with respect to the origin of the circle.

If a weight is hypothetically associated with the two states |φ₀) and |φ,) , then the right side of the circle will "weigh" more than the left side, and the circle will tend to rotate clockwise to bring its centre of gravity (and therefore the two states) to the lowest possible position.

On the contrary, if the two states |φ₀) and |φ,) were in symmetric positions with respect to the centre of the circle, so as to form a diameter of the circle, their distribution would be symmetric, and the centre of gravity of the circle would coincide with its geometric centre. In this case, the circle would not tend to rotate, and would remain in its original position. We note here that the two states introduced above, |φ₀) and |φ,) , are those typically encoded in the QKD protocol known as "B92", as described in the previously cited articles:

C. H. Bennett, Phys. Rev. Lett. 68, 3121 (1992);

K. Tamaki, M. Koashi, and N. Imoto, Phys. Rev. A 67, 032310 (2003) .

Therefore, in the case of DNS the circle tends to bring its centre of gravity as low as possible, reaching a position of stable equilibrium. This means that, if the circle is hypothetically moved from its position of stable equilibrium, it will tend to return there, restoring the previous balance.

A mathematical explanation of asymmetric distribution DNS is provided below. The initial pure quan^¬ tum states prepared by the transmitter 200 can be defined as follows:

with k = {0, 1, 2, 3, ...d-1} . In the example indi^¬ cated above d was equal to 2. Initial distribution will be given by:

It is noted that in many QKD protocols, the dis^¬ tribution of the Eq. (2) is symmetric. This means that it is equal to the identity matrix in the Hil- bert space of the states |p_k): p = I. ( 3 )

However, in accordance with the example described, the transmitter 200 can use the protocol known as "B92" and prepare with equal probability one of the two quantum states |φ₀) and |φ₍) , as shown in FIG . 2.

Assuming that y^' = {0, l] , the two states prepared by the transmitter 200 can be expressed as follows:

|_9j) = P| 0_v) + (-iya| l_t) (4) wherein

O<a = sin(0 / 2) < 1 /V2, ≡Vl-a² (5) and {| 0 ) 'I } ^are *-he quantum states, also called "qubit", of the base X, which will be called "calculation base" for future references. The angle Θ, also indicated in FIG. 2 on the circle A, considering a particular example wherein 9=60^°, is the angle between the state \ θ_χ) and the state |φ₀) . The angle Θ can be selected, more in general, so that

0<θ<π/ 2 (6) The state |φ,) has an angle equal to -Θ with respect to the state | 0_r) . In accordance with the B92 protocol, choosing the quantum states |φ₀) and |φ,) so that their scalar product differs from zero, there is no measurement that allows perfect discrimination of the two quantum states generated by the transmitter. Therefore, in this case the density matrix is represented by- the following asymmetric distribution:

p = (7)

where I₂ is the identity in a two-dimensional space (d=2). The transmitter 200 will associate the logic information to be transmitted to the receiver 300 (BOB) with the two prepared states, in the following way :

after which it will send a series of quantum states on the first optical fiber OFl, so that the receiver 300 (Bob) can detect them and decode the information transmitted.

The optical signals P3 having associated the quantum states transmitted by Alice (by the transmitter 200) propagate along the first optical fiber OFl and reach the receiver 300 (that is, Bob) .

In FIG. 2 the receiver 300 has been represented, for greater clarity of description, only by means of the set of components that form the feedback loop: the actuator 90, the receiving modulator 85, the detector module 86 and the control module 70.

The set of the receiving modulator 85 and of the detector module 86 (and of other optional equipment, known to those skilled in the art) is structured so that it can perform decoding of the quantum signals received that is sensitive to symmetry, and is therefore a Symmetry-Sensitive-Decoder (SSD) .

According to the example represented in FIG. 2, the decoder SSD is configured for a dimension d=2. The circle B illustrated in FIG. 2 shows two crossed segments containing the states sent by the transmitter. Said segments represent the two axes (i.e. the bases) along which the decoder SSD performs its measurement, chosen randomly, with equal probability.

The results of measurement of the receiver 300 are, in this simple case, four, and can be indicated, with reference to the segments of FIG. 2, with the following two pairs:

In the following part of the description two cases are considered by way of example (again with reference to the case of θ= 60°), namely: a first case that refers to the absence of noise and a second case that refers to the presence of noise. It is noted that noise can occur as modulation error in the transmission modulator 82 or in the receiving modulator 85, or can be present in various forms on the first optical fiber OF1, on the second optical fiber OF2 or in the- optical fiber that joins OF1 and OF2. Therefore, the expression "error present" or "error introduced" on the channel can also include errors due to devices included in the transmitter (typically the transmission modulator) or in the receiver (typically the receiving modulator) .

First case: absence of noise

Reference is made to FIG. 3, which shows in a simplified manner part of the communications system 100 in which the decoder SSD is, for simplicity, illustrated by means of a first filter F₀ connected to a first conversion and decoding block DO and a second filter Fi connected to a second conversion and decoding block Dl . The first filter F₀ and the second filter Fi correspond to two different measurement bases performed at the receiver 300. According to a first embodiment, the first and the second filter, F₀ and Fi, represent phase filters, while in accordance with the second embodiment these filters are polarization filters .

It is assumed that the transmitter 200 sends on the first optical fiber OF1 the state |φ₀) . In this case if the decoder SSD performs the measurement along the base (see circle B) , associated in

FIG. 3 with the first filter F₀, the result will always be the state |φ₀) (shown at the output of a measurement block DOinc ) · A question mark has been placed at the measurement block D0i_nc in FIG. 3, the meaning of which will be explained below.

If instead the transmitter 200 sends on the first optical fiber OFl the state | p, ) and the decoder SS D again measures along the base || Ψο) »| Ψο } (first filter F₀) , it will find the state |φ₀) with probability |(ψι|ψο)| ^anc* the state φ₀^ with probability ^φ, φ₀^ (shown in FIG. 3 at the output of a measurement block D0_con) .

Therefore, as the decoder SS D can obtain the result |φ₀) both in the case of effective preparation by the transmitter 200 of the state |φ₀) , and in the case of preparation of the state |φ,) , it is clear that ob- tainment of |φ₀) by the receiver 300 (Bob) does not indicate in a conclusive and certain manner which quantum state the transmitter 200 initially prepared. This explains the question mark shown in FIG. 3 associated with the inconclusive result |φ₀) .

On the contrary, a result equal to φ₀^ measured by the decoder S SD , indicates in a conclusive and certain manner that the transmitter^' 200 prepared the state |φ,) , and in fact the logic bit "1" is associ- ated therewith (see the output of the symbolic measurement block D0_con) .

The situation described can be repeated analogously for the measurement base j|<p,),|(p J (filter Fi and measurement blocks Di_con and Dli_nc) . In other terms, the results φ₀^ and φ,^ are "conclusive" as they indicate the initial state prepared by the transmitter 200 to the receiver 300 with certainty, while the results |φ₀) and |φ,) are "inconclusive" as they do not provide the receiver 300 with this certainty.

As a consequence of the fact that the two initial quantum states were prepared following a DNS, the conclusive and inconclusive results of the receiver 300 do not have the same probability of occurrence. In fact, the probability of a conclusive result P_con and that of an inconclusive result Pj._nc are respectively:

As expressed by the formulae (9), the two probabilities P_con and Pine depend on the angle Θ of the initial states |φ₀) , |φ,} ; these are equal to each other only if but this value lies outside the interval established by the Eq. (6) . Therefore, in the absence of noise on the channel, the two probabilities are always different. FIG. 3 also shows a histogram indi- eating the count probability of the decoder associated with a conclusive result (D_con) and that associated with an inconclusive result (Dj__nc) . It can be noted that the histogram relative to the inconclusive result D_inc is higher than that relative to a conclusive result. In fact, when θ=60^° and the channel is without noise, then:

in accordance with the relations (9), from which it is easily obtained that P_con=3/8 and P_inc=5/8.

It is convenient to introduce for the continuation the following quantities expressed as ratios between the conclusive and inconclusive count probabilities associated with the two measurement bases taken into consideration:

R^-^ R, (11) 0 DO con Dl con

Therefore, the equation (10) can be rewritten (ignoring the subscripts, not essential in this context) as: R= D_inc/D_con = 1.667 (12)

Second Case : presence of noise

The case is considered in which there is sufficient noise to cause misalignment between the transmitter 200, in particular the transmission modulator 82, and the receiver 300, in particular the decoder SSD, even if the transmission and receiving modulators (for example, comprising a respective Mach- Zehnder interferometer) were initially aligned.

In fact, for example, the two Mach-Zehnder inter^¬ ferometers are located in two different places, remotely from each other, and are therefore subject to different environmental conditions such as temperature, vibrations of the supporting surface, stresses of the optical components of which they are composed.

Moreover, it is possible that a hacker could attempt to intercept information from the communications channel; if the channel in question is a quan^¬ tum channel, this attack by the hacker introduces further noise on the channel, causing misalignment of the two Mach-Zehnder interferometers. It should be noted that the users of the transmitter 200 and of the receiver 300, Alice and Bob, are unable to dis^¬ tinguish between "natural" noise on the channel and noise introduced by a hacker during an espionage op^¬ eration .

Any misalignment during transmission is particu^¬ larly detrimental for the transmission, as often the transmission must be interrupted to allow suitable re-alignment of the communication devices.

Another example of noise that causes misalignment between transmitter 200 and receiver 300 occurs when polarization e-ncoding of information is performed and an optical fiber is used as transmission medium. In this case the birefringence of the optical fiber will tend to cause spontaneous and unpredictable rotation of the polarization state prepared by the transmitter 200, disturbing Bob's receiving, which was aligned along Alice's initial states.

In the presence of noise on the transmission channel, the probabilities set down in the relations (9) will no longer be equal to those expressed by the histogram of FIG. 2 and in the equations (10) and (12) .

In this connection, the Applicant has found a link, which can be expressed mathematically, between a variable representative of the noise ε (which can be expressed as measurement of an angle) and the conclusive D_con and inconclusive Di_nc count probabilities, and the angle Θ.

In particular, it is possible to express this relation by means of the ratios R₀ and Ri defined above by the relations (11), that is:

FIG. 4 gives the trends of the curves relative to the ratios R₀ and Ri upon variation of the noise variable ε, maintaining 6=60^°, and considering for the receiver 300 a receiving modulator 85 produced by means of unbalanced Mach-Zehnder interferometers.

Both from the expressions (13) and (14) and from FIG. 4 it can be deduced that the pair of values of the ratios R₀ , Ri unequivocally identifies all the values of ε from n a +n. It can be observed that the equation (13) alone or the equation (14) alone is not in one-to-one correspondence and therefore does not allow a single value of the noise variable to be obtained, as can instead be obtained using both the relations (13) and (14) .

The receiver 300, by means of the decoder SSD, performs the measurements of the quantum states received and determines the number of conclusive and inconclusive counts for each logic level: D0_con, D0i_nc Dl_con, Dlin_c - Through the calculation module 60, to which the decoder SSD is connected, it can determine in real time, for example, the ratios R₀ and Ri and therefore, due to the relations (13, and (14), obtain the noise ε, considering that the angle Θ is a prede^¬ termined parameter also known to the receiver 300.

In particular, the specific curves obtainable from the equations (13) and (14) can be treated with numerical techniques. For example, it is possible to prepare in advance a matrix of numbers that place in correspondence each value of the noise variable ε with each pair of values of the ratios Ro, Ri by means of the equations (13) and (14).

This matrix is stored, for example, in a memory of the further management and control unit 5B of the receiver 300 and can be consulted by the calculation module 60. During transmission, the calculation module 60 accesses the matrix and recovers the value of the noise variable ε associated with values R₀ and Ri which are closest to those measured after decoding performed by the decoder SSD in real time.

According to an alternative and particularly advantageous method, the curves associated with the relations (13) and (14) can be linearized in a neighborhood of the noise value ε=0, so as to describe low noise implementations. The two curves represented in FIG. 4 will then be substituted by two straight lines, given by the approximation to the first order of the equations (13) and (14), thus being easily invertible.

In any case, the calculation module 60 determines the value, for example expressed as an angle, of the noise variable ε_ and supplies it to the control module 70. The control module 70 applies an appropriate control law so as to control the actuator device 90, which introduces on the input quantum states to the receiver 300 a modification of the electromagnetic radiation parameters aimed at compensating the valued noise ε_ .

In the case of the first embodiment, which provides for encoding of the logic values in the phase of the radiation, the valued noise ε_ represents an undesirable phase shift between the transmission modulator 82 and the receiving modulator 85 and therefore, for example, between the two corresponding Mach-Zehnder interferometers located remotely to one another .

In this case, the actuator device 90, (for example, by means of a further phase modulator or by means of the same phase modulator used to determine the measurement base) introduces in the quantum states received a phase shift Δφ substantially equal in absolute value and of opposite sign to that of the valued noise _ε: Δφ= - ε_.

In particular, in the first embodiment, the actuator device 90 acts on the phase Φ, describable as the longitude angle of the Poincare sphere. In the case of the second embodiment, which provides for encoding of the logic values in the radiation polarization, the valued noise _ represents an undesirable polarization rotation. In this case, the actuator device 90, produced by means of a polarization controller, introduces in the quantum states received a polarization rotation substantially equal in absolute value and of opposite sign to that of the valued noise ε__;_

In greater detail, the polarization compensation is obtained by controlling two parameters, Φ and Θ, which correspond respectively to the longitude and latitude angles of the Poincare sphere (FIG. 8). The case of phase alignment relative to the first embodiment can therefore be seen as a particular case of polarization alignment, which is obtained by fixing the latitude angle Θ of the Poincare sphere at zero. Phase encoding is therefore intrinsically simpler to treat than polarization encoding.

Moreover, with reference to polarization encoding, it should be noted that the measurement performed by the receiver 300 provides that a part of the signals sent by the transmitter 200 are used to complete noise analysis and therefore enable feedback through the actuator device 90, without however being able to be decoded for the purpose of obtaining the logic values associated with the string that defines the cryptography key.

It is assumed, for example, that the signal states |φ₀) and |φ,) are polarization encoded by the transmitter 200 choosing them so that they lie on the equatorial plane of the Poincare sphere in FIG. 8, that is, on the plane identified by the axes X and Z of this sphere. Through the technique described above, the receiver 300 will be capable of maintaining the two states aligned with respect to the axes X and Z, that is, of fixing the angle Φ of FIG. 8.

However, the receiver 300 does not have the tools to prevent the states from detaching from the equatorial plane of the Poincare sphere; that is, in the presence of noise and in the absence of further measures by the receiver 300, the signal states will tend to vary their relative angle with the axis of the Ys (that is, the angle Φ of FIG. 8) .

To prevent this from happening the receiver 300 must, from time to time, measure the radiation coming from the transmitter 200 along the axis Y of the Poincare sphere. In the absence of noise the measurement will be Θ=0; consequently, 50% of the photons measured by the receiver 300 will be aligned in the W positive direction of the axis Y (Θ=90^°), and the remaining 50% in- the negative direction (Θ=-90^°).

In the presence of noise, the angle Θ will differ from zero and the percentages of the quantum states measured by the receiver will change. In particular, the receiver 300 will find photons aligned in the positive direction of the Y axis with probability proportional to (1+sin Θ) /2, and photons aligned in the negative direction of the Y axis with probabilities proportional to (1-sin Θ)/2. This will allow it, with a technique analogous to the one described above, to detect the value of the angle Θ and compensate the noise of the channel OFl . It should be noted that the transmitter 200 can continue to polarization encode the same states that it would have phase encoded .

For the receiver 300 this additional measurement phase is not very significant, as the percentage of measurement along the axis Y will be considerably lower than that of the other measurements. In terms of QKD, the results of the Y measurements, which cannot be used to generate a quantum key, will be in much lower percentages than that can be used for this purpose, and therefore will not in general determine a decrease in the relevant efficiency of the process. It should be clarified that after quantum transmission the receiver 300 must communicate to the transmitter 200 the instances in which it performed Y measurements to allow Alice to eliminate these instances from the final key.

FIG. 5 schematically represents an example of the control law applied by the control module 70 to perform noise compensation. According to the control diagram of FIG. 5, the noise ε represents a disturbing quantity that alters the controlled quantity R, that is, the ratio between inconclusive and conclusive counts for a given base, defined by the equations (11) . The value R* represents the reference value given by the relation (12), relative to the condition of no noise.

The alteration introduced by the noise ε makes the controlled quantity R differs from its reference value R* . According to a simplified process, which however does not include all possible noise processes that can be produced, the action of the noise can be described as follows:

R = R*+B (ε - e_F) (15) where B is a gain factor and ε_Γ is the controlling, quantity. Different control algorithms from those of the formula (15) are also possible. By way of explanatory example, FIG. 6 reports graphs relative to a situation of no noise ε=0 to be compared with two situations with noise (in the examples of FIG. 6, an angle of 9=60° was considered between the two initial states). These graphs refer to the first embodiment that provides for phase encoding of the electromagnetic radiation. Graphically, the noise of the first transmission fiber OF1 can be described as a rotation of an angle ε of the two initial states about the axis passing through the centre of the circle and exiting from the plane of the figure .

Rotation can take place both counter-clockwise (positive angle ε) and clockwise (negative angle ε); the expressions (13) and (14) allow determination both of the absolute value and of the sign of this angle. In relation to the case of no noise (ε=0), FIG. 6 shows the diagram of the states and the histogram already reported in FIG. 3, for which R=1.667, as in the equation (12) . With reference to the case of ε= 45°, it can be noted that the ratio between the counts Dine and D_con is no longer equal to 5/3=1.667, but rather is equal to approximately 2.87.

This means that the receiver 300, in the instant in which it detects this value of R, can immediately return to the value of ε= 45°, and apply, through the actuator device 90, an angle of compensation equal to -45° to realign the receiver 300 with the transmitter 200.

In the case of noise ε= 135° the ratio R between Dine and D_con is still greater than 1, but the value of R becomes equal to 1.3. Also in this case the receiver 300, in the instant in which it detects this value of R, can immediately return to the noise value of ε= 135°, and apply, through the actuator device, an angle of compensation equal to -135° to realign with the transmitter 200.

Numerical simulation

The Applicant has also performed a numerical simulation aimed at evaluating the effectiveness of the solution described. FIG. 7 relates to this numerical simulation, based on a finite number of acquisitions. The time elapsing, in seconds, is reported on the abscissae, for a total of 100 seconds. The angles, in radians, corresponding to the various cases of interest, are reported on the ordinates. The curve NS of FIG. 7 represents the noise. This curve NS is, in fact, approximable to a square wave composed of jumps of +/- 2 radians which has random noise oscillations between one jump and the next. Each point of the curves of FIG. 7 corresponds to the acquisition · of 10³ total events by the detectors of the receiver 300, and it is assumed that in 1 second the detectors acquire 10⁴ events. An average misalignment of 0.1 rad/sec has also been assumed, in line with recent experiments reported in the literature .

The simulation refers to an encoding of the quantum states performed by the transmitter 200 as in FIG. 3. The receiver 300 starts to measure D_con and Dine , and through these obtains the ratios Ro and Ri, and compensates the noise present along the communications channel. The angle of misalignment between the receiver 300 and the transmitter 200 is indicated by the curve DA in FIG. 7.

It is seen that in the first 10 seconds the receiver 300 remains well aligned with the transmitter 200, as the angle of misalignment DA oscillates around zero, closely following the trace RF, which represents the situation of the best feedback (ideal situation) possible by the receiver, limited only and exclusively by the finite acquisition statistic. The average value for the misalignment angle between receiver and transmitter with feedback activated is equal to: -0.23 ° ± 2.54°. This value is fully compatible with the use of delicate cryptographic protocols such as B92, which requires low noise values. In this regard, it should be noted that the version of B92 described in the Italian patent application MI-A-2009/000894 provides for using parameters very close to those given in the present description. For example, the angle 9=60^° is almost optimum both for the active feedback described here and for the B92 described in MI-A-2009/000894, which has an optimal angle of utilization equal to 55.4°.

After the first 10 seconds, the noise of the channels jumps by 2 radians. The compensation mechanism implemented by the receiver 300 works very well also in this case as, after a first strong initial oscillation, its apparatus realigns rapidly with that of the transmitter 200 and remains well aligned for the next 10 seconds, after which there is a new jump of -2 radians, followed by a new compensation.

As already mentioned, in the numerical simulation the receiver 300 accumulated a statistic of 10³ counts per point, that is, to value the parameters Ro and R± through which to produce the feedback.

Aspects relative to synchronization

With reference to the example of FIG. 1, there have been described components (i.e., the further radiation source^' 83, the second optical fiber OF2 and the photodetector 84), relative to synchronization signals of the receiver 300 with the transmitter 200.

According to an alternative embodiment, these components dedicated to synchronization could be omitted or reduced in number. It should be remembered that synchronization consists in correct identification of the send times of the packets on the first optical fiber OF1. For example, if the transmitter 200 encodes the bit ^λ0' at time t_0/ the bit at time ti, and bit ^λ0' again at time t₃, the receiver 300 must be capable of correctly decoding the time sequence t_x, t₂ and t₃ to reconstruct the sequence ^λ010' sent by the transmitter.

Advantageously, it is possible to use the same quantum states used to transmit the key and to compensate the noise, also to synchronize the two remote systems. The Applicant was able to observe that the noise of one channel is rarely subject to fluctuations as large as those shown in FIG. 7; it is usually subject to fluctuations that are smaller orders of magnitudes. Therefore, Alice could intentionally produce large fluctuations in order to use them to send the additional information required to synchro- nize the receiver and the transmitter on the same quantum channel^' used for transmitting the key.

This type of communication, based on the active compensation of large fluctuations, is advantageously performed at lower velocity with respect to the QKD communications relative to transmission of the key, as if there are too many fluctuations of this types the subjacent QKD, which requires very low noise, would be impeded.

Therefore, two types of communication that take place on the same quantum channel can be defined as "fast" (the type associated with the QKD) and "slow" (the type associated with synchronization) . Fast information generating the quantum key and the feedback and slow information created by the transmitter 200 by means of large noise fluctuations can be transmitted together, using a decoder that indicates to the calculation module 60 and/or to the control module 70 which of the two compensation mechanisms to use, either the continuous one (low noise thresholds) or the stepped one (high noise thresholds) . The use for synchronization of quantum signals avoids the drawbacks related to the use of classical optical pulses on the same transmission channel used for sending relative quantum states to the key. It is clear from the description that, with the exception of what is indicated with reference to the additional measurement to be performed in relation to polarization encoding, the alignment technique described does not require interruption of the QKD transmission session. In fact, the signals used by the receiver for feedback are exactly the same signals used by it for distillation of a secure quantum key .

Moreover, it is specified that in order to implement the solution described it is not necessary to make use of intense light signals. In fact, the statistic performed by the receiver collecting packets of quantum signals is more than sufficient, in the majority of operating situations, to align (and for example, also to synchronize) transmitter and receiver .

It should also be observed that in order to implement the technique described it is not necessary to use bidirectional Plug-and-Play type setups, which have passive compensation of channel noise based on the use of bidirectional channels and intense light signals. In fact the channel noise is actively com^¬ pensated by the feedback mechanism described herein and it is also applicable in the absence of intense light pulses.

The embodiment that provides for alignment for compensation of noise together with synchronization, thus producing "stabilization" of the communications system, is particularly advantageous.

Claims

1. Quantum signal receiver (300) comprising:

an input connectable to a transmission channel (OF1) for receiving signal quantum states having an asymmetric distribution and associated to a cryptographic key to be determined,

a decoding and managing block (SSD, 60, 70) responsive to the quantum states distribution symmetry for measuring the signal quantum states and returning a quantity representing a channel introduced error, an actuator (90) associated with the decoding and managing block adapted to vary physical parameters of signal quantum states coming to the receiver for reducing the channel introduced error.

2. Receiver (300) according to claim 1, wherein the decoding and managing block comprises:

a decoding device responsive to the quantum states symmetry (SSD) adapted to measure the quantum states received for obtaining information about the key and determining a first conclusive results number and a second non conclusive results number;

a computing module (60) configured to determine said error from the first and second number,

a command module (70) adapted to output actuator command signals depending on said determined error.

3. Receiver (300) according to claim 2, wherein the computing .module (60) is configured to determine said error from at least one mathematical relation relating the error, the first number, and the second number to an angle between vectors representative of two asymmetric distribution quantum states sent to the receiver.

4. Receiver (300) according to claim 3, comprising a memory readable by the computing modulus and containing data associating a plurality of precalcu- lated values of said error to values of said first and second number and to said angle.

5. Receiver (300) according to at least one of the preceding claims, wherein:

the decoding and managing block (SSD, 60, 70) is configured to perform a decoding of quantum states having information codified in the phase of the associated electromagnetic radiation, and

said actuator device (90) is structured to vary the electromagnetic radiation phase associated to said quantum states .

6. Receiver (300) according to at least one of claims 1-4, wherein:

the decoding and managing block (SSD, 60, 70) is configured to perform a decoding of quantum states having information coded in the polarization of the associated electromagnetic radiation, and

said actuator device (90) is configured to change the electromagnetic radiation polarization associated with said quantum states .

7. Receiver (300) according at least to claim 3, wherein :

the decoding device (SSD) is configured to perform measurements of said signal quantum states with respect to a first measurement base and a second measurement base;

said error can be expressed as an angular value and said computing module (60) is adapted to determine the error absolute value and angular value sign based on said at least one mathematical relation comprising: a first mathematical relation related to measurements associated with the first base and a second mathematical relation related to measurements associated with the second base.

8. Quantum cryptography communications system (100) comprising:

a transmitter (200) adapted to generate signal quantum states having an asymmetrical distribution; a quantum channel (OF1) for allowing signal quantum states transmission; a receiver (300) of said signal quantum states connected to the quantum channel;

characterized by the fact that the receiver is configured according to at least one of the preceding claims .

9. Communications system according to claim 8, wherein said quantum channel (OF1) comprises an optical fiber.

10. Communications system according to at least one of claims 8 and 9, wherein the system is configured to operate according to protocol B92.

11. Communications system according to at least one of the preceding claims 8-10, wherein:

the transmitter is configured to send on the quantum channel (OF1) further signals associated with the synchronization of the receiver and -transmitter; the decoding and managing block (SSD, 60, 70) and actuator being capable of reducing by compensation a further noise associated to said synchronization signals .

12. Quantum cryptography method comprising:

generating by a transmitter (200) signal quantum states having an asymmetric distribution;

transmitting said signal quantum states on a quantum channel (OF1); measuring the signal quantum states in a receiver (300) responsive to the asymmetric distribution;

determining by said measurements a quantity representative of an error introduced on the channel;

changing physical parameters of signal quantum states reaching the receiver by reducing the channel introduced error based on said determined quantity.