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Calculation Algorithm for Finding the Mini-Max Value in Quantum Hypothesis Testing 加加加加加 / Kentaro Kato 加加加加加加 加加加加加加

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

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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. Bennett. Holevo. Fuchs. Josza. 佐々木. 広田. 富田. 相馬. 臼田. 大崎. 吾妻. 加藤. @Tamagawa University, Japan. 臼田. 相馬. Van Enk. Lutkenhaus. 大崎. Schmecher. 南部. 宇佐見. 山崎. - PowerPoint PPT Presentation

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Page 1: 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

Page 2: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

加藤大崎 相馬 臼田 吾妻

富田

佐々木広田

BennettFuchs Josza

Holevo

@Tamagawa University, Japan

Page 3: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

@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

Page 4: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

OUTLINE

• Background

• Quantum Hypothesis Testing

• Bayes Strategy

• Mini-max Strategy

• Calculation Algorithm

• Example

• Conclusion

Page 5: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

EncryptionPlaintext

Eve, the eavesdropper

Decryption

Alice, the sender

Bob, the receiver

Alice, Bob, and Eve

cipher text Plaintext

Page 6: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 7: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Control technique Signal Modulation

Example)

Page 8: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 9: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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!)

Page 10: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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.

Page 11: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 12: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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.

Page 13: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 14: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Basis #3 Basis #7 Basis #2

0 1 0

PRNG

Mod.

Plain Text

Secret key

Signal

2

7

3

0 01

Running

key

Alice

Page 15: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Signal

PRNGSecret key

2

7

3

0 01

Running

keyPlain Text

Receiver

Bob

Page 16: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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.

Page 17: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 18: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 19: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 20: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 21: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

(Eye pattern)Close Open

Page 22: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 23: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 24: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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 .

Page 25: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Quantum Hypothesis Testing量子仮説検定

Quantum System

???

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

Page 26: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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.

Page 27: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 28: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

ExampleConvex regions (2-dim case)

(1. ellipse)

(2. oval )(3. trigon)

(4. hexagon)(5. tetragon)

Page 29: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

ExampleNon-convex regions (2-dim case)

Page 30: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Straight line = Convex region

Curved line = Non-Convex region

Page 31: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Set of Probability Vectors

where

[Set of probability vectors]

Page 32: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

[Lemma]

For any and any such that , the nextrelation holds:

Set of Probability Vectors

The set of probability vectors is a convex set.

(Proof)

Page 33: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

[Lemma]

Set of Probability Vectors

The set of probability vectors is bounded and closed

(Proof)See textbook

Page 34: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

[Definition] Convex function

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

Convex = Convex upward = - convex

Page 35: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

[Graphical image of convex function]

Convex Function

[Remark]

Any convex function is defined on a convex region.

Page 36: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

[Definition] Concave function

Concave Function

Concave = Convex downward = - convex

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

Page 37: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

[Graphical image of concave function]

Concave Function

[Remark]

Any concave function is defined on a convex region.

Page 38: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

ExampleConvex functions

Concave functions

Page 39: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

[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

Page 40: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 41: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

• [Decision Operators: 決定作用素 ]

• [POVM]

Page 42: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

• The probability of choosing when is true:

Page 43: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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)

Page 44: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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,

Page 45: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Bayes Costsベイズコスト(損失係数)

• Example: Radar system

Page 46: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 47: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

The average Bayes cost平均ベイズコスト(平均損失)

[Check] Joint probability:

Page 48: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

The average probability of error平均誤り確率

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

Page 49: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 50: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 51: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Necessary and Sufficient Conditions for Bayes strategy

• Theorem (Holevo 1973):

where

[A]

[B]

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

Page 52: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Necessary and Sufficient Conditions for Bayes strategy

• Remark:The following three conditions are equivalent.

• By this theorem,

[A]

[A’]

[A”]

Page 53: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Necessary and Sufficient Conditions for Bayes strategy

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

“ Minimum”

Page 54: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Concavity of the minimum Bayes cost

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

Page 55: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Concavity of the minimum Bayes cost

• [proof] Consider and

Let

Then

Page 56: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Concavity of the minimum Bayes cost

• [proof] Let , where , i.e.

and let

Then

Page 57: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Concavity of the minimum Bayes cost

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

Page 58: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Concavity of the minimum Bayes cost

• [proof] Observe that

Hence

Page 59: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Concavity of the minimum Bayes cost

• [proof] Hence we have

Page 60: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 61: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 62: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 63: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Mini-max Problem

• Find such that

• is called the mini-max value

Page 64: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Mini-max Theorem in Quantum Hypothesis Testing

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

Page 65: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Mini-max Theorem in Quantum Hypothesis Testing

• Theorem (Generalized version):

Page 66: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 67: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 68: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Mini-max Strategyミニマックス戦略

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

Page 69: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Property

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

Then

Page 70: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Concavity of the minimum Bayes cost

• Image

Page 71: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 72: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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)

Page 73: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Bisection Search二分探索法

• Fact:

If and , thenthe function has a maximum in the interval

Page 74: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Bisection Search二分探索法

• Choose such that

In this case,

has a maximum in the interval

Page 75: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Calculation algorithm for finding the mini-max value

START

Initialization

A

Page 76: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Find such that

B

A

Loop A start :

Loop B start :

F

Page 77: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Find such that

B

Renewal of Data

Loop B end :

Loop A end :

C

F

E

Page 78: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

C

Renewal of Data

D

NO

YES

E

Page 79: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

D

Check:

Necessary and sufficient conditions must be satisfied.

END

Display and Store the result

Page 80: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 81: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Ternary Amplitude Shift-Keying (3ASK)

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

prior distribution:• Receiver:

• Find the solution to the problem

Page 82: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

• Mini-max value

• Optimal distribution in mini-max strategy

Page 83: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 84: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

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

Page 85: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Numerical computation by the algorithm

Page 86: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Numerical computation by the algorithm

Page 87: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

Conclusion

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

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

• Example: 3ASK

Page 88: Calculation Algorithm for  Finding the Mini-Max Value in Quantum Hypothesis Testing

future tasks

• Tuning up the algorithm

• Application to the quantum stream cipher