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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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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