Calculation Algorithm for Finding the Mini-Max Value in Quantum Hypothesis Testing

Calculation Algorithm for Finding the Mini-Max Value in Quantum Hypothesis Testing

加藤研太郎 / Kentaro Kato

國立清華大学　電機工程学系

加藤大崎 相馬 臼田 吾妻

富田

佐々木広田

BennettFuchs Josza

Holevo

@Tamagawa University, Japan

@Oiso, Kanagawa, Japan

加藤

大崎

広田

臼田Van Enk相馬

宇佐見

SmolinFuchs

LutkenhausSchmecher

Bennett山崎

南部

OUTLINE

• Background

• Quantum Hypothesis Testing

• Bayes Strategy

• Mini-max Strategy

• Calculation Algorithm

• Example

• Conclusion

EncryptionPlaintext

Eve, the eavesdropper

Decryption

Alice, the sender

Bob, the receiver

Alice, Bob, and Eve

cipher text Plaintext

Classification of Quantum Cryptographyby functions

BB84 Key Distribution

B92 Key Distribution

YK Key Distribution

Y-00 Direct EncryptionCoherent

Single photon

Function

4 states

2 states

2 states

M states (M>100)

Source

Coherent states[Def.] Coherent state of light (with complex amplitude )

Control technique Signal Modulation

Example)

Y-00 protocol

The coherent-state quantum cryptosystem by Y-00 protocol is called quantum stream cipher (in JAPAN) or alpha-eta scheme (in USA).

--- high-speed (up to internet level; ~ Gbps)--- long-distance (over 100km) --- and secure

台北ー高雄＞３００ｋｍ東京ー大阪＞６００ｋｍBackbone >2.5Gbps

Basic Model of Y-00

Alice: Sender

Bob: Receiver

Plaintext

Secret KeyPRNG

Pseudo-Random Number Generator

Signal

Secret KeyPRNG

Pseudo-Random Number Generator

Plaintext

Signal

Multi-ary Signal Modulator

Detector

(it is not single photon!)

System Requirements for Y-00

(2) Multi-ary Signal Modulation (Alice)

(1) Secret Key and PRNG (Alice and Bob)Legitimate users, Alice and Bob, share the secret key.Enemy, Eve, has no key.The secret key is used for driving Pseudo-Random Number Generators (PRNG).

(3) Binary Detector (Bob)

Signal Modulator is controlled by output sequences of the PRNG and Plaintext at Alice’s side.That is, emitted signals are determined by outputs of the PRNG and Plaintext.

So far, there are two major implementation schemes: A. Phase Shift Keying (PSK) -based quantum stream cipher (Northwestern University) B. Intensity Modulation -based quantum stream cipher (Tamagawa University)

Bob’s receiver is controlled by the output sequences of the PRNG.The output of the PRNG determines measurement basis, so that Bob’s task is to distinguish the binary signal belonging the basis.

Basic Model: Multi-ary signal modulator

(3’) Signal constellation and mapping rule:

(Example)PSK# of bases M= 7# of signals 2M=14

(basis)

(bit)

Running key

Plaintext Signal distance >> 1

Signal distance <<1

Pseudo Random Number Generator

True random number

Pseudo random number

M-sequence (LFSR), Kasami-sequence （嵩） , etc,

It is given by some deterministic function, butIt seems to be random: - 0 and 1 are equiprobable, - Long period, No correlation, etc,

Nobody can guess what is next result deterministically.

Linear Feedback Shift Resister

AND AND AND AND

＋ ＋ ＋OR OR OR

Lc 1Lc 2c 1c

Output

ir

1ir 2ir i Lr 1i Lr

1 2 3, , ,r r r

1 1 1 1 0, , , ,L Lr r r r Initial values

1 2 2 1 1i L i L L i L i ir c r c r c r c r Output

1 2 1, , , ,L Lc c c c Connection coefficients

Given by primitive polynomial

Basis #3 Basis #7 Basis #2

0 1 0

PRNG

Mod.

Plain Text

Secret key

Signal

2

7

3

0 01

Running

key

Alice

Signal

PRNGSecret key

2

7

3

0 01

Running

keyPlain Text

Receiver

Bob

Encoding/Decoding Procedures - 1/3

X. Setup:

X-1: Legitimate two users, Alice and Bob, share the secret key .X-2: They also have the same type PRNG.X-3: Alice and Bob know the signal partitioning rule for signaling bases;

X-4: Alice and Bob know the bit assignment rule for each signals;

Signaling Basis = a set of two signals

16 PSK Basis#0 Basis#1 Basis#2 Basis#7

Basis#0 Basis#1 Basis#2 Basis#7

01

0

10

1

0

1

0

111

111

1 1

0

0

0

0

0

0

0

One signal is assigned to 0 and another is to 1 in each basis.

Encoding/Decoding Procedures - 2/3

A. Encoding Procedures:

A-1: By using the secret key , Alice ganerates pseudo-random numbers. This output sequence of PRNG is called a running key .A-2: From the running key , Alice determines the signaling bases for each slot.

001 011 000 010 101 110 110 100 111 100 101 …

#1 #3 #0 #2 #5 #6 #6 #4 #7 #4 #5Basis#

A-3: If a plaintext bit is 0, Alice sends the signal assigned to 0 in the basis determined by PRNG, and vice versa.

#1 #3 #0 #2 #5 #6 #6 #4 #7 #4 #5Basis# 0 1 1 0 1 1 1 0 0 0 1 …

Signal

Encoding/Decoding Procedures - 3/3

B. Decoding Procedures:

B-1: By using the secret key , Bob generates running key and determines the signaling bases for signal detection.

#1 #3 #0 #2 #5 #6 #6 #4 #7 #4 #5Basis#

001 011 000 010 101 110 110 100 111 100 101 …

B-2: For each slot, binary detection is done by using information of the bases.

B-3: Thus, Bob can get the plaintext.

If signaling basis is #2, the decision region is given as follows:

0

1

Received signalby Bob

0

1Emitted signalby Alice

Error Free

Quantum Stream Cipher as a random cipher

・・・

Ｙ－

００

Nobody can get the true ciphertext without the initial shared key.

Mesurement results are probabilistic by virtue of quantum noise

There are so many resulting patterns and each of themcontains error bitsPattern#1 Pattern#2 Pattern#X

Ciphertext signal can be measured only once.(Quantum No-cloning Theorem)

Yellow block stands for error bit Ordinary attacks do not work anymore

Keyword: Random cipher

Implementation Schemes for Y-00

PSK - based quantum stream cipher, (NWU)

Intensity Modulation - based quantum stream cipher (Tamagawa)

QAM - based quantum stream cipher, (KK)

Optical QAM

Target: High-speed

Target: Long-distance

PSK

Intensity Level

(Eye pattern)Close Open

MotivationWe wish to evaluate the security level of the cryptosystem:

What is the best receiver for an eavesdropper?

Quantum Signal Detection Theory

“Mini-max strategy”

Key words

HistoryTheory of Games Hypothesis Testing

Decision Function

RADAR system

1928 von NeumannMini-max theorem

1933 Neyman and Pearson

“Ideal Receiver”1940-1945, MIT RadLab

1953 Middleton,Analysis of signal detection process by statistical hypothesis testing

1939 A.Wald

Signal Detection Theory

“Cost”“Risk”

1960 年， C.W.Helstrom ， Statistical Theory of Signal Detection1960 年， D.Middleton ， An Introduction to Statistical Communication Theory

Two-parson game

Nature v.s. Observer

1944 von NeumannTheory of Games

1955 Middleton,

Formulation of Signal Detection problemsbased on Decision Function

1954 PetersonReceiver design by likelihood ratio

Generalization of Neyman-Pearson Theory

? - 1940, UK

Pioneering works

C.W.Helstrom, Information and Control 10, 254 (1967)

H.P.Yuen, R.S.Kennedy, M.Lax, Proc.IEEE 58, 1770 (1970)

E.B.Davies, J.T.Lewis, Commun.Math.Phys. 17, 239 (1970)

A.S.Holevo, J.Multivari.Anal. 3, 337 (1973)

O.Hirota, S.Ikehara, Trans.IECE Japan E65, 627 (1982)

In 1967, Helstrom : first example of quantum signal detection problem

Yuen et al. : Necessary and Sufficient conditions (conjecture)

Davies and Lewis established

a generalized quantum measurement theory

(POVM theory) beyond von Neumann theory.

In 1973, Holevo : the quantum Bayes strategy

In 1982, Hirota : the quantum Minimax strategy .

Quantum Hypothesis Testing量子仮説検定

Quantum System

???

We wish to determine the state of the system with small error

Quantum Signal Detection Theory 量子信号検出理論

Quantum Communication System

???

We wish to determine which signal was transmitted with small error.

Let be a subspace (or subset) of the K-dim vector space ,i.e. . Then convex region is defined as follows:

[Definition] Convex region (or Convex set)

Convex Region

ExampleConvex regions (2-dim case)

(1. ellipse)

(2. oval )(3. trigon)

(4. hexagon)(5. tetragon)

ExampleNon-convex regions (2-dim case)

ExampleConvex region / Non-convex region (2-dim case)

Straight line = Convex region

Curved line = Non-Convex region

[Probability vector] (= Vector representation of probability distribution)

Set of Probability Vectors

where

[Set of probability vectors]

[Lemma]

For any and any such that , the nextrelation holds:

Set of Probability Vectors

The set of probability vectors is a convex set.

(Proof)

[Lemma]

Set of Probability Vectors

The set of probability vectors is bounded and closed

(Proof)See textbook

[Definition] Convex function

Convex FunctionLet be a real-valued function defined on a convex region

Convex = Convex upward = - convex

[Graphical image of convex function]

Convex Function

[Remark]

Any convex function is defined on a convex region.

[Definition] Concave function

Concave Function

Concave = Convex downward = - convex

Let be a real-valued function defined on a convex region

[Graphical image of concave function]

Concave Function

[Remark]

Any concave function is defined on a convex region.

ExampleConvex functions

Concave functions

[Lemma]

Lemma

Let be a concave function of over the regionAssume that the partial derivatives, are defined and continuous over the region with the possible exception that .

Then the necessary and sufficient conditions on a probability to maximize the function over the region are given by

with some

Quantum Hypothesis Testing量子仮説検定

• Suppose that there are hypotheses about the states of a quantum system.

• The -the hypothesis is the proposition that its density operator is .

• We wish to determine the state of the system through measurement.

Hypothesis Testing

Positive Operator-Valued Measure (POVM)正作用素値測度

• [Decision Operators: 決定作用素 ]

• [POVM]

Positive Operator-Valued Measure (POVM)正作用素値測度

• The probability of choosing when is true:

Positive Operator-Valued Measure (POVM)正作用素値測度

• Lemma:Let be the set of all POVMs.

is a compact convex set.

A.S.Holevo, J.Multivar. Anals., 3, 337-394 (1973)

Bayes Costsベイズコスト（損失係数）

• Bayes costs: If we made a wrong decision, we must pay a penalty Penalty = Cost It can be denoted by a real number

• In general,

Bayes Costsベイズコスト（損失係数）

• Example: Radar system

The average Bayes cost平均ベイズコスト（平均損失）

• Let be the prior probability of .Suppose that is employed for our decision.

Then the average Bayes cost is given by

where

The average Bayes cost平均ベイズコスト（平均損失）

[Check] Joint probability:

The average probability of error平均誤り確率

If , then the average Bayes cost becomes the average probability of decision errors.

Bayes Strategyベイズ戦略

• A strategy minimizing the average Bayes cost for any assignment of cost.

• Prior probabilities are known. Under this condition we wish to minimize the average Bayes cost.

• Bayes Problem:Find such that

Bayes Strategyベイズ戦略

• Lemma:The optimal POVM of the Bayes problem exists.

It exists because

(1) is compact (2) is continuous

Necessary and Sufficient Conditions for Bayes strategy

• Theorem (Holevo 1973):

where

[A]

[B]

A.S.Holevo, J.Multivar. Anals., 3, 337-394 (1973)

Necessary and Sufficient Conditions for Bayes strategy

• Remark:The following three conditions are equivalent.

• By this theorem,

[A]

[A’]

[A”]

Necessary and Sufficient Conditions for Bayes strategy

• Outline of the proof Perturbation of the average Bayes cost　　（摂動計算）

“ Minimum”

Concavity of the minimum Bayes cost

• [Lemma]The minimum Bayes cost is a concave function of .

Concavity of the minimum Bayes cost

• [proof] Consider and

Let

Then

Concavity of the minimum Bayes cost

• [proof] Let , where , i.e.

and let

Then

Concavity of the minimum Bayes cost

• [proof] It is arranged to the following form:

Concavity of the minimum Bayes cost

• [proof] Observe that

Hence

Concavity of the minimum Bayes cost

• [proof] Hence we have

Bayes Cost Reduction Algorithm (by Helstrom)

• Finding the closed-form expression of the minimum Bayes cost difficult

• But, we can find the minimum Bayes cost by using a numerical computing algorithm. Helstrom’s algorithm Eldar’s algorithm

Helstrom’s iterative algorithm for finding the minimum Bayes cost

• Let be a POVM (not necessary to be optimal)

• Choose a pair of indices , where• Then we can find a new POVM

such that

• Therefore,

• Repeating this procedure,

new old

Disadvantage of Bayes strategy

• In Bayes strategy, we have assumed that

• But, it is difficult to specify the probabilities in advance. [Example] Eavesdropping a cryptosystem

• What kind of strategy should he/she use when the true prior probabilities are unknown? Mini-max Strategy

Mini-max Problem

• Find such that

• is called the mini-max value

Mini-max Theorem in Quantum Hypothesis Testing

• Theorem (Hirota & Ikehara; 広田修 & 池原止戈夫 ):

Mini-max Theorem in Quantum Hypothesis Testing

• Theorem (Generalized version):

Mini-max Theorem ミニマックス定理

• Mini-max Theorem (von Neumann):Let and be convex compact sets, and let and .

If is (a) an upper semi-continuous concave function of for fixed , and(b) a lower semi-continuous convex function of for fixed , then there exist and such that

Mini-max Strategyミニマックス戦略

• Theorem (Hirota & Ikehara): Necessary and sufficient conditions for mini-max strategy (Error probability)

where we have assumed that all signals are non-orthogonal

Mini-max Strategyミニマックス戦略

• Theorem (General): Necessary and sufficient conditions for mini-max strategy

Property

• Lemma: Let be the solution to the mini-max problem,and let be the mini-max value. That is,

Then

Concavity of the minimum Bayes cost

• Image

A key inequality

• Suppose that is an optimal POVM for a given prior probability distribution .

• Choose indices • From the concavity of the solution set to the minimum

average Bayes cost, we have

where

A key inequality

• Inequality

one-parameter maximization concave function Easy to find the maximum e.g. Golden Section Search

(W.H.Press, et al, “Numerical Recipes”, Cambridge, 2007)

Bisection Search二分探索法

• Fact:

If and , thenthe function has a maximum in the interval

Bisection Search二分探索法

• Choose such that

In this case,

has a maximum in the interval

Calculation algorithm for finding the mini-max value

START

Initialization

A

Find such that

B

A

Loop A start :

Loop B start :

F

Find such that

B

Renewal of Data

Loop B end :

Loop A end :

C

F

E

C

Renewal of Data

D

NO

YES

E

D

Check:

Necessary and sufficient conditions must be satisfied.

END

Display and Store the result

Example:Application to Optical Communications

• Mini-max Receiver for Optical Communication System

To evaluate the system performance, we wish to knowthe mini-max value.

Signal Measurement results

Mini-max receiver

Ternary Amplitude Shift-Keying (3ASK)

• Alphabet:• Signal set: “Coherent state of light”

prior distribution:• Receiver:

• Find the solution to the problem

The mini-max value for 3ASK system(closed-form expression)

• Mini-max value

• Optimal distribution in mini-max strategy

The mini-max value for 3ASK system(by closed-form expression)

The mini-max value for 3ASK system(by closed-form expression)

Numerical computation by the algorithm

Numerical computation by the algorithm

Conclusion

• The mini-max theorem in Quantum Hypothesis Testing was considered

• Calculation algorithm for finding the mini-max value was shown

• Example: 3ASK

future tasks

• Tuning up the algorithm

• Application to the quantum stream cipher