Quantum Communication Part 1: Basics - Télécom ParisTech · Quantum Communication Part 1: Basics...

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Fiber Optics: Light in Action from Science to Technology

Sept/Oct2015U. Fort Hare, Alice/Hogsback, South Africa

Philippe Gallion

Telecom ParisTech, Ecole Nationale Supérieure des TélecommunicationsCentre Nationale de la Recherche Scientifique, LTCI

Paris, France

Quantum CommunicationPart 1: Basics

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TELECOM ParisTech Ecole Nationale Supérieure des Télécommunications

(ENST)✔  Since 1878…✔  Belong to

❏  Ministry of Economy and Industry❏  ParisTech (Paris Institute of Technology)❏  Institut Télécom

✔  Graduate level electrical curriculum❏  Engineering and computer science❏  600 students toward engineering degree❏  Selective national wide admission contest❏  Acceptance upon degree

✔  PhD Program❏  350 students toward PhD Degree❏  Doctoral School : EDITE (Computer Science,

Telecommunications and Electronics)

2Quantum Communication Part 1: Basics Philippe Gallion!

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TELECOM ParisTech, in few words

✔  The leading Engineering School in ICT, in France✔  Top 3rd to 7th in national rankings of Engineering Schools inFrance (among 250)✔  Highly international

❏  42% international students❏  64 partner Universities

✔  High professor/student Ratio : 1/7✔  205 full-time professors for 1500 students✔  Innovation

❏  130 sustainable companies ❏  1000 jobs created since 1999

✔  Strong links with industry

3Quantum Communication Part 1: Basics Philippe Gallion

Where are we located ?

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Télécom ParisTech

Quantum Communication Part 1: Basics Philippe Gallion

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Four Teaching and Research Departments

✔  Communication & Electronics (Mine)❏  Digital Communications❏  Optoelectronics and Optical Communications❏  Microwaves and Radio Communications❏  Digital and analog electronics integration

✔  Computer Science & Networking

✔  Signal and Image processing

✔  Economics, Management & Social Sciences


Outline …(for today)

✔ 1. Introduction✔ 2. Classical Cryptography (CC)✔ 3. Basics Concepts for Quantum Cryptography (QC)✔ 4. Quantum Cryptography Protocols✔ 5. Security and Attack Strategies✔ 6. Conclusion

Tomorrow : …. Quantum receiver and implementation


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What optical signal at quantum levels for ? 1 Space communications

✔  Low photon numbers ❏  Low transmitted power ❏  Strong attenuation

✔  Applications ❏  Airborne❏  Satellite-to-ground ❏  Inter satellite❏  Space to ground❏  Planetary, deep space

V. Chan, IEEE-JLT 2003


What optical signal at quantum levels for ? 2 - Quantum cryptography

✔  Information is supported by single photon

❏  The quantum aspect appears

❏ Measurement destroys the quantum system

❏  Non duplication ( non cloning) of unknown state

Security !


5

Alice is the message sender

Bob is the receiverEve is the

eavesdropper

Alice in wonderland

(Lewis Caroll)Bob Marley the Reagee Music

StarCranach’ the Elder (Germain Painting)

Traditional Cryptography Starring


Outline

✔  1. Introduction✔ 2. Classical Cryptography (CC)✔ 3. Basics Concepts for Quantum Cryptography (QC)✔ 4. Quantum Cryptography Protocols✔ 5. Security and Attack Strategies✔ 6. Conclusion


6

Encryption

Decryption

Cryptosystem (or Ciphers)

Message + Key Cryptogram

Cryptogram + Key Message

✔ Decryption without the key is:❏  « Impossible » (nothing is)❏  Difficult (growing exponentially with the key length)❏  « Easy » (growing polynomially with the key length )

✔ Key is a secret shared by Alice & Bob


Secret Key (Symmetrical) Cipher - 1

✔ Principles❏  Single key for encryption and decryption ❏  Alice encrypts with this secret key ❏  Bob decrypts with this secret key

✔ Only Computational Security❏ More secure than asymmetrical ciphers at a given Key

length✔ Data Encryption Standard (DES)

❏  56 bits❏  Used for e-commerce….


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TEXT Q CASCII 1 0 0 0 1 0 1 1 1 0 0 0 0 1+ KEY 0 1 1 0 0 1 1 0 1 1 0 0 1 1

ENCODEDMESSAGE

1 1 1 0 1 1 0 1 0 1 0 0 1 0

+KEY 0 1 1 0 0 1 1 0 1 1 0 0 1 1ASCII 1 0 0 0 1 0 1 1 1 0 0 0 0 1TEXT Q C

0 10 0 11 1 0

Secret Key (Symmetrical) Cipher - 2

✔ Exclusive OR or XOR, or addition modulo 2

✔ Two consecutive additions return to the initial message


“One Time Pad” Necessity (Vernan Code)

✔ Eve’s recording of allows to start knowledge of the message

✔ The addition of 2 messages scrambled with the same key is the sum of the 2 messages

✔ Perfectly secure for “One Time Pad” (OTP)

✔ Key of the same length than the message


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✔ Security secret key systems using one time pad (OTP)❏  Key distribution problem

Quantum Key Distribution (QKD)

✔ Teleportation (secure faithful copy transfer) possible

❏  System state description possible with arbitrary precision❏  Secure transmission with a key with appropriated length❏  Bob can in principle reconstruct a copy of the system❏  Original system is not affected❏ Many copies are possible

Hello Dolly !

Classical Systems


Outline



9

“…and it may well be doubted

whether human ingenuity can construct an enigma of

the kind which human ingenuity may

not, by proper application,

resolve.”

Edgar Allan PoeThe Gold-Bug, Tales of Mystery

and Ratiocination, 1843


Classical Physics Basics

✔ Determinism❏  The state of a system at a given time is totally

determinate by previous state❏  Statistical imprecision is a consequence of the formalism❏  Not of the system nature

✔ Objectivity❏  Reality exists independently from its observation

✔ Completeness❏  Each element of the reality have a theoretical counterpart

Only based on macroscopic experience !18Quantum Communication Part 1: Basics Philippe Gallion

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Quantum Physics Basics

✔  Principle of indetermination (Heisenberg)❏  Indeterminism inherent to the nature

✔  The wave nature (de Broglie, Schrödinger)❏  Describing probabilities

✔  Principle of complementary (Bohr)❏  Wave and corpuscular nature are two (classical) perspectives ❏  Its is a duality❏  It is NOT dualism❏  But not a classical particle or a particle wave !

✔  Principle of correspondence (Ehrenfest)❏  Quantum mechanics and classical one agree as the quantum

nature disappears❏  Classical mechanics is a limit


What is a Quantum Communication System ?

✔  There is no information without physical representation✔  Information is supported by a quantum system

❏  Individual photon❏  Coherent state…

✔  Ensemble systems are (semi) classical and do not work✔  Information is encoded by using

❏  Polarization❏  Phase ❏  Frequency❏  …

✔  Other parts of the communications channel are classical❏  Optical fiber❏  Free space


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✔  A Quantum state is the superposition of its eigenstates

✔  A measurement converts a quantum state into one of its eigenstate

❏  The measurement result is the corresponding eigenvalue❏  The probability of this result is IαiI 2

✔  Only eigenstates are kept unchanged✔  Consequences

❏  Except for the eigenstates, measurement destroys the system❏  Simultaneous and precise measurements may be impossible❏  Duplication of unknown quantum state impossible

Quantum Physics Basics

21

A = a1 A1 + a2 A2 with ai = Ai A

A Ai


Schrödinger’s Cat Paradox✔  When does

❏  The wave function collapse ?❏  The superposition disappears ?❏  The cats dies ?

✔  At the (eventually microscopic) killing process ❏  Linked to physic ?

✔  At the box opening ❏  linked to observation and conscience ?

✔  Problem with Cartesian dualism : mental/physical✔  Monism : reduction to a single “substance” (Spinoza)

❏  Idealism (Platon, Leibnitz, Hegel …)•  The superposition disappears with the conscience intervention

❏  Materialism (Epicure, Marx…)•  The superposition disappears with the microscopic killing process

✔  Idealism/materalism “Superposition” interpretation 22Quantum Communication Part 1: Basics Philippe Gallion

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Quantum FishAndré Breton 1924

✔ The fish is melted out in the water✔  It is only defined by a concentration✔ When does the quantum fish appear?

❏ When biting the bait? Materialism ?❏ When seen by the fisher man ? Idealism ?


1

0

1

0

Classical Bit v.s. Quantum BitClassical Bit :

Any macroscopic 2-state system

✔  Exclusive states : 0 or 1 at a given time

✔  States exist independently of measurement

✔  p(1) + p(0) = 1

✔  Measurement keeps the system unchanged

Quantum Bit (QB)Any 2-level quantum system

✔  State superposition: 0 and 1 at the same time : IQB > =α I0> + β I1>

✔  One of the 2 eigenstates is obtained after a measurement

✔  IαI2 is the probability to obtain I0>IαI2+ IβI2= 1

✔  Measurement destroys the superposition

Ensemble average


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Quantum Bit (QB)

✔  2 dimensions✔  n qubits belongs to a 2n

dimension space

I 0 >

IQB>

I 1 >continuousvariable

0 1

Classical Bit v.s. Quantum Bit - 2/2

Classical Bit (CB)

✔  1 dimension✔  Areas selected for bit

value representation✔  2 possibilities✔  n bits belongs to an n

dimension space


Qbit Communication System using Simple Eigenstate Encoding

✔  Polarization (i.e. spin) is an example❏  Any 2-level system acts in the same way

✔  Q bit are used : IQB > =α I0> + β I1>✔  Simple eigenstate information encoding

❏  α =1 for bit 0 and β =0 ❏  α =0 for bit 1 and β =1

✔  Bit discrimination by simple polarization splitting❏  A 2 detector arrangement is mandatory❏  Correct detection probabilities are equal to 1❏  Error free transmission

I 0 >

IQB>

I 1 >

Detector 1

Detector 2

Polarizingsplitter I0>

I1>

€

! = 0" = 1

Where is the Rub ?✔  Eve can

❏  Intercept, ❏  Detect the same way….and get the key❏  Resend to Bob

✔  Simple eigenstate encoding not relevant✔  Protocol required for QKD


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✔  2 qubits lead to a 4 dimension basisI00>, I01>, I10>, I11>

✔  Some combinations are combination of separated (non correlated) qubit states

I01>+ I11>= (I0>+ I1>) I1>

✔  Some of them are notI00>+ I11>

✔  n qubits belongs to a 2n dimensions space❏  Enhanced complexity❏  n classical bits only belongs to a n dimensions space

✔  Great interest for quantum information processing€

00 + 01 = 0 ⊗ 0 + 1( )00 + 10 = 0 + 1( ) ⊗ 0

00 + 11 = 2 entangled states

01 + 10 = 2 entangled states

01 + 11 = 0 + 01( ) ⊗ 1

10 + 11 = 1 ⊗ 0 + 1( )

Qbit Association


✔  Entanglement, Verschränkung (i.e.“arm in arm”)✔  Quantum « super correlation » ≠ classical correlation✔  2 photon parametric generation

✔  The individual photon have a random polarization

✔  When both measured they have always orthogonal polarization

ψ AB =12

↑A→ B + → A ↑

B[ ]

ψ =12

↑ + →[ ]

Photon A,ω

Photon B,ω

2ω χ(2)

Entangled States - 1


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Entangled States - 2

✔  Measurement on one instantaneously influences the other❏  No matter how far apart they may be❏  In opposition with the theory of relativity❏  Einstein, Podolski, Rosen (EPR) paradox (1935)❏  Hidden variables❏  Bell’s inequality violation (1964)❏  Demonstrated experimentally by Clauser (1976) & Aspect (1981)❏  Quantum mechanics completeness (no hidden variable)

✔  They have interacted in the past❏  They constitute a single quantum object

✔  They have some locally inaccessible information in common ❏  This information cannot be accessed in any experiment performed on

either of them alone


✔  4 quantum states , forming 2 basis are used

✔  Binary value attributions on each basis✔  Simultaneous observations using the 2 basis are

impossible✔  Alice and Bob can select any basis✔  Basis coincidence allows correct bit detection✔  Basis anti-coincidence

p(0) = p(1)= 1/2 whatever is the transmitted bit✔  A second detection is impossible (quantum demolition)

I0>

I90> I45>I135>

Using 2 Conjugated Basis

€

! =12

" + #( ) and $ =12

" − #( )


16

€

0 00 11 11 0

"

#

$ $ $ $

%

&

' ' ' '

=

1 0 0 00 1 0 00 0 0 10 0 1 0

(

)

* * * *

+

,

- - - -

0 00 11 01 1

"

#

$ $ $ $

%

&

' ' ' '

€

x b → x x

€

α 0 + β 1( ) b →α 0 0 + β 1 1 ≠ α 0 + β 1( ) α 0 + β 1( )

Original

CopyCloning

✔  Classical state cloning (controlled XOR gate)

✔  Cloning impossibility for superposed states

✔  Except for α = β = 1/2


Quantum Teleportation

✔  Under complete system knowledge, the problem turns to a classical one❏  Knowledge implies Ψ collapse❏  Duplication of unknown quantum state impossible

✔  Quantum state transfer from Alice to Bob requires:❏  The sharing many entangled qubits ❏  A classical communication channel❏  A quantum teleportation protocol

✔  The quantum teleportation result is:❏  Alice’s system destruction without knowledge for herself❏  Bob get an isomorphic, but unknown, system


17

Outline

✔ 1. Introduction✔ 2. Classical Cryptography (CC)✔ 3. Basics Concepts for Quantum Cryptography (QC)✔ 4. Quantum Cryptography Protocols

❏  BB84 Protocol❏  Eckert Protocol❏  Continuous variable protocol❏ ….

✔ 5. Security and Attack Strategies✔ 6. Conclusion


I0>

I90> I45>I135>

Bennett-Brassard 1984 (BB84)✔  4 quantum states , forming 2 basis for the same space

✔  Simultaneous observations using the 2 basis are impossible

✔  Binary value attributions

✔  Alice and Bob can select any basis✔  Basis coincidence allows correct bit detection✔  Basis anti-coincidence leads to p(0) = p(1)= 1/2 after detection,

whatever is the transmitted bit✔  A second detection Eve is impossible (quantum demolition)


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BB84/QKD : Initial Alice to Bob Transmission on the Quantum Channel

✔  1 - Alice chooses a random series of bits✔  2 - Alice sends each bit with a random bases choice✔  3 - Bob detects each bit using another random choice of the basesResulting BER is 25%

1 0 1 0 0 1 1 1

Quantum Chanel

0

011


BB84/QKD : Reconciliation on the Public Classical Channel

✔  4 - Bob publicly announces his series of bases choices (not the measurement result!)

✔  5 - Alice publicly announces the bases coincidences i.e. the bits correctly detected by Bob

✔  6 - Bob & Alice use this bit sequence as the key: ReconciliationTheoretical BER is 0%

1 0 1 0 1

Classical Chanel


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BB84/QKD : Key Distillation on the Public Classical Channel

✔  7 - Error correction✔  8 - Privacy amplificationAt the expense of the key length reduction !

Classical Chanel

But systems impairments and Eve intervention turn to BERAnd Eve have catch some information about the key


1 Qubit BB84 Protocol

✔ Neither Alice and Bob decide of the key✔ Key is a result of random basis choice

coincidences in a random series of bits✔ Security relies on

❏  Quantum demolition measurement❏  Non cloning

✔ Phase, frequency or polarization may be used


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✔  Classical systems average out the quantum character✔  Quantum system must be used✔  2 representations of the 2 binary symbols on 2 conjugated bases

Polarization Encoding Phase Encoding (QPSK)

Orthogonal states of polarization

(Discrimination by polarizer)

Antipodal state of Phase

(Discrimination by interferenceor homodyne detection)

Modulation bandwidth FSK(Discrimination by filters)

Frequency Encoding

Encoding Information on a Single Photon

0

0

1

1

1

0

0

1

0

1 0

1

1 Base 1

0 Base 2

0

1


Foiling Eavesdroppers

1 - Alice sends each bit with a first random basis choice2 - Eve detects each bit using a second random choice of the basis3 - Bob detects each bit using a third random choice of the basis4 - many error thanks to wrong basis5 - Bob and Alice are able to detect error by telling each other a

portion of it


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Using QC

✔ Retrospect security ❏  Useful for the message itself❏  Solves the key distribution problem because intercepted key

may be discarded✔ Conventional (classical channel) may be used in OTP


!

Common

Source

2 Qubit Ekert (EPR) Protocol

✔ The 2 qubits are in the same state chosen randomly among the 4 states of the BB84 protocol❏  The source announces the base❏  Alice & Bob only consider compatible basis measurements❏  Equivalent to BB84 protocol❏  But the source may be controlled by Eve!

✔ The 2 qubits are emitted as an entangled state❏  Reduction of basis coincidence probability

!!

! !


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Outline

✔ 1. Introduction✔ 2. Classical Cryptography (CC)✔ 3. Basics Concepts for Quantum Cryptography (QC)✔ 4. Quantum Cryptography Protocols✔ 5. Security and Attack Strategies

❏  QBER Amplification❏  Privacy Amplication

✔ 6. Conclusion


Improving the QBER of Sifted Keys

✔ The shared key contains error thanks to❏  Technical imperfections❏  Eve’s intervention

✔ QBER is usually in the few percent range❏  Corrected with a posteriori classical error correction❏  Implying a key reduction❏  Public channel is used to distill a shorter key without error

✔ QBER differs from BER usually in the 10-9 range❏  Corrected a priori using FEC and over heading of the signal


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✔  Alice randomly chooses pairs of bits✔  Alice compute their sum modulo 2 (XOR)✔  Alice publicly announces:

❏  The result❏  The rank of the 2 bits

✔  For same results Alice and Bob❏  Keep the first bit❏  Throw away the second to avoid Eve information improvement

✔  For wrong results they throw away the 2 bits

€

p→ p2

p2 + (1− p)2 < p pour p < 12

Both wrong Both OK

A Simple, but low Performance, Algorithmfor QBER Improvement


Basics of Communication Theory

Sourceentropy H

H(X/Y)

0

MutualInformation I

I(X,Y)

0

Sourceentropy H

H(X)

0

MutualInformation I

0

(a) Non operating channel

(b) Operating channel

H(X)

ü  No communication channel

q  The available information H (entropy) remains stored in the right vessel (a)

ü  Moving down the vessel of the right side

q  Reduces the available source information

q  Increases of the same amount the mutual information (b).

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I(X,Y ) = H (X)−H (X /Y )


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!

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

bina

ry e

ntro

py

probabity

Basics of Communication Theory

No communicationp =1/2

Source Entropy H(X)

Mutual Information I(X,Y)

Conditional Entropy H(X/Y)

Probability after communication

€

H 2( p) = − p log2 p − (1− p)log2(1− p)Binary entropy :


!

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

bina

ry e

ntro

py

probabity

✔  Improving mutual information difference

✔  Public channel is used✔  Key is shortened

€

I(Alice,Eve)

€

I(Alice,Bob) ≈1€

ΔI

Bit probability for Eve

€

ΔI = I(Alice,Bob) − I(Alice,Eve)

Privacy Amplification


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A Simple, but low Performance, Algorithmfor Privacy Amplification

✔  Alice randomly chooses pairs of bits✔  Alice compute their sum modulo 2 (XOR)✔  Alice does not announce the result✔  Alice only announces the rank of the 2 bits✔  Alice and Bob replace the 2 bits by their sum modulo 2

✔  Heuristic demonstration:❏  Eve only knows one of the two bits: Its sum contains no more

information p = 1/2❏  Eve knows the two bits with probability p : the probability turn to p2 +(1-p)2< p for p >1/2

✔  Performant when p for Eve is close to 1/2


Eve knows Nothing

Privacy vanishes out

Perfect security

€

34log2 3−1

Quantum Unconditional Security✔  Security rely on

❏  Non cloning❏  Quantum demolition

✔  Ensemble systems❏  Average the quantum behavior❏  Are (semi) classical

✔  Information must supported by quantum system❏  Individual photon❏  Coherent state

A Simple Attack Strategy: Random interception with probability ω and resend

Eve tries to attack the QDK

!

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

IAEIABIAB-IAE

Mut

ual i

nfor

mat

ion

Interception Probability


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Some Other Attack Strategies

✔ Selection of an other base✔ Using teleportation✔ Photon number splitting (PNS)✔ Performances limited by

❏  the channel imperfection❏  the available time for a key sharing❏  Eve resources


Outline



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About Unconditional Security

✔  QKD is the only protocol to provide it

✔  Considered as limited only by the principles of physics ❏  What about the resources that could realistically have Eve on a

given time scale ❏  Where is the better emergence probability

•  For the technologies usually evoked in unconditional security? •  For technology that would collapse the traditional security systems ?

✔  Quantum layer approach is not sufficient to achieve an end-to-end security up to the application layer ❏  Attack on the quantum layer is an heavy strategy mistake for Eve ❏  Conventional integrated electronics circuits are very vulnerable to

the so-called side-channel attacks

✔  Needs very limited v.s. security on demand✔  Confining into academics or thought experiments ?


From the promises of physics of the last century…toward quantum security engineering✔  Implementation of the physical layer is demonstrated

❏  1550nm wavelength operation without photon counter❏  Standard optical fiber❏  One way system, in single optical fiber❏  Off-the shelf and low cost optoelectronics components

✔  End to end approach already started❏  True Random Number Generators (for symbols & bases)

•  100 to 1000 time faster than the application data rate•  Robust against attacks

❏  Raw key processing•  Electronics interface•  Buffering for key material management•  Secured electronics processing

❏  Application interface•  Key distillation using public channel•  Key management•  Upper layer interface

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Will be discussedWednesday


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Thank You for the Quality

of Your Listening


Quantum Acception

✔ Latin : how much✔  Kant :

❏  Things as they appear (phenomenal) may differ from things as they are and persist in being (noumenal)

❏  Quantum is the invariant of the substance and the object of measurement

✔  Classical Physic : Corpuscular❏  Determinism inherent to the nature, ❏  Indeterminism only inherent to the formalism (statistics)

✔  Planck, Einstein, Bohr, Sommerfeld : defined quantities✔  Quantum Physics

❏  Indeterminism inherent to the nature❏  Principle of indetermination


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Goals of Cryptography✔ Making communications interaction as safe as

perfectly secret personal meeting

✔ Confidentiality✔  Identification

❏  Credit card number❏  PINs

✔ Authentication, Digital Signature

✔ Non-repudiation


Quantum information

Quantum Information

✔ Quantum information processing❏  Quantum Processors❏  Quantum Algorithm

✔ Quantum Communication❏  Teleportation❏  Quantum Cryptography


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Basics of Communication Theory 1/3Intrinsic source

propertiesSource perception

through thecommunication channel

Communication channeleffect

Signal probability Probability

€

p(x)

Conditional Probability

€

p(x / y)

€

p(x / y) > p(x)

Improved probability

For a given occurrence xof the input

InformationI(x) = − log2 p(x)

Low probability resultsin large information

Conditional information

€

I(x / y) = −log2 p(x / y)

Residual sourceinformation

Mutual information

€

I(x,y) = I(x) − I(x /y)

Decay of the sourceinformation

Ensemble averageoverall the possible

occurrences

Source entropy

€

H(X) = I(x) x

Potential informationavailable from the

source

Conditional entropy

€

H(X /Y ) = I(x / y) x,y

Residual sourceinformation after

communication process

Average mutualinformation

I(X,Y) = H(X) − H(X /Y)

Decay of the sourceentropy provided by thecommunication process


✔  Information is encoded as CW modulation of two optical field quadrature (EX, EY) or (A, ϕ)

✔  Security relies on❏  Non simultaneous precision

measurements❏  Non cloning

✔  Conversion into digital signal for❏  Privacy amplification❏  Error Correction

✔  Squeezing or EPR correlation are not required

✔  Chaos cryptography is an other way

Continuous Variable Protocol


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