* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Calculation Algorithm for Finding the Mini
Quantum entanglement wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Coherent states wikipedia , lookup
Density matrix wikipedia , lookup
Scalar field theory wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum group wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum computing wikipedia , lookup
Renormalization group wikipedia , lookup
History of quantum field theory wikipedia , lookup
Canonical quantization wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Bell's theorem wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
EPR paradox wikipedia , lookup
Quantum state wikipedia , lookup
Quantum teleportation wikipedia , lookup
Hidden variable theory wikipedia , lookup
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 Fuchs Holevo Josza 佐々木 広田 富田 加藤 大崎 相馬 臼田 吾妻 @Tamagawa University, Japan Schmecher 臼田 相馬 Lutkenhaus Van Enk 大崎 南部 宇佐見 山崎 Bennett Fuchs 広田 Smolin 加藤 @Oiso, Kanagawa, Japan OUTLINE • • • • • • • Background Quantum Hypothesis Testing Bayes Strategy Mini-max Strategy Calculation Algorithm Example Conclusion Alice, Bob, and Eve Alice, the sender cipher text Encryption Plaintext Bob, the receiver Plaintext Decryption Eve, the eavesdropper Classification of Quantum Cryptography by functions Function Source BB84 4 states Key Distribution B92 2 states Key Distribution YK 2 states Key Distribution Y-00 M states (M>100) Direct Encryption Single photon Coherent 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 Backbone >2.5Gbps 台北ー高雄>300km 東京ー大阪>600km Basic Model of Y-00 Alice: Sender Pseudo-Random Number Generator PRNG Secret Key Signal Multi-ary Signal Modulator (it is not single photon!) Plaintext Pseudo-Random Number Generator Bob: Receiver PRNG Secret Key Signal Detector Plaintext System Requirements for Y-00 (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). (2) Multi-ary Signal Modulation (Alice) 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) (3) Binary Detector (Bob) 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: (basis) (bit) Running key Plaintext Signal distance >> 1 (Example) PSK # of bases M= 7 # of signals 2M=14 Signal distance <<1 Pseudo Random Number Generator True random number Nobody can guess what is next result deterministically. Pseudo random number It is given by some deterministic function, but It seems to be random: - 0 and 1 are equiprobable, - Long period, No correlation, etc, M-sequence (LFSR), Kasami-sequence(嵩), etc, Linear Feedback Shift Resister Initial values ri L1 riL cL cL1 r1L , r1L1 , , r1 , r0 ri 2 ri 1 c1 c2 AND AND ri AND AND + + + OR OR OR Connection coefficients cL , cL1 , , c2 , c1 Given by primitive polynomial Output ri cL riL cL1ri L c2 ri 2 c1ri 1 Output r1 , r2 , r3 , Alice PRNG Secret key 7 2 Plain Text Running key 3 Signal Mod. 0 1 0 Basis #3 0 Basis #7 1 Basis #2 0 Bob 3 7 2 Running key PRNG Secret key Receiver Signal 0 1 0 Plain Text 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; Signaling Basis = a set of two signals 16 PSK Basis#0 Basis#1 Basis#2 Basis#7 X-4: Alice and Bob know the bit assignment rule for each signals; 1 0 1 0 0 1 1 0 Basis#0 Basis#1 Basis#2 Basis#7 One signal is assigned to 0 and another is to 1 in each basis. 0 1 1 0 1 1 01 01 0 0 1 0 0 1 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 … Basis# #1 #3 #0 #2 #5 #6 #6 #4 #7 #4 #5 A-3: If a plaintext bit is 0, Alice sends the signal assigned to 0 in the basis determined by PRNG, and vice versa. Basis# Signal 0 1 1 0 1 1 1 0 0 0 1 #1 #3 #0 #2 #5 #6 #6 #4 #7 #4 #5 … Encoding/Decoding Procedures - 3/3 B. Decoding Procedures: B-1: By using the secret key , Bob generates running key determines the signaling bases for signal detection. and 001 011 000 010 101 110 110 100 111 100 101 … Basis# #1 #3 #2 #0 #5 #6 #6 #4 #7 #4 #5 B-2: For each slot, binary detection is done by using information of the bases. 0 0 If signaling basis is #2, the decision region is given as follows: Received signal by Bob Error Free 1 B-3: Thus, Bob can get the plaintext. 1 Emitted signal by Alice Quantum Stream Cipher as a random cipher Keyword: Random cipher Nobody can get the true ciphertext without the initial shared key. Mesurement results are probabilistic by virtue of quantum noise Y-00 Ciphertext signal can be measured only once. (Quantum No-cloning Theorem) ・・・ Pattern#1 Pattern#2 Pattern#X Yellow block stands for error bit There are so many resulting patterns and each of them contains error bits Ordinary attacks do not work anymore Implementation Schemes for Y-00 PSK - based quantum stream cipher, (NWU) Target: Long-distance PSK Intensity Modulation - based quantum stream cipher (Tamagawa) Target: High-speed QAM - based quantum stream cipher, (KK) Optical QAM Intensity Level Close (Eye pattern) Open Motivation We wish to evaluate the security level of the cryptosystem: What is the best receiver for an eavesdropper? Key words Quantum Signal Detection Theory “Mini-max strategy” History Theory of Games Hypothesis Testing RADAR system 1928 von Neumann Mini-max theorem 1944 von Neumann Theory of Games 1933 Neyman and Pearson ? - 1940, UK Two-parson game “Cost” “Risk” Decision Function Nature v.s. Observer 1940-1945, MIT RadLab “Ideal Receiver” 1939 A.Wald Generalization of Neyman-Pearson Theory 1953 Middleton, Analysis of signal detection process by statistical hypothesis testing 1954 Peterson Receiver design by likelihood ratio 1955 Middleton, Formulation of Signal Detection problems based on Decision Function Signal Detection Theory 1960年,C.W.Helstrom,Statistical Theory of Signal Detection 1960年,D.Middleton,An Introduction to Statistical Communication Theory Pioneering works In 1967, Helstrom : first example of quantum signal detection problem C.W.Helstrom, Information and Control 10, 254 (1967) Yuen et al. : Necessary and Sufficient conditions (conjecture) H.P.Yuen, R.S.Kennedy, M.Lax, Proc.IEEE 58, 1770 (1970) Davies and Lewis established a generalized quantum measurement theory (POVM theory) beyond von Neumann theory. E.B.Davies, J.T.Lewis, Commun.Math.Phys. 17, 239 (1970) In 1973, Holevo : the quantum Bayes strategy A.S.Holevo, J.Multivari.Anal. 3, 337 (1973) In 1982, Hirota : the quantum Minimax strategy . O.Hirota, S.Ikehara, Trans.IECE Japan E65, 627 (1982) 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. Convex Region [Definition] Convex region (or Convex set) Let i.e. be a subspace (or subset) of the K-dim vector space . Then convex region is defined as follows: , Example Convex regions (2-dim case) (2. oval ) (3. trigon) (1. ellipse) (4. hexagon) (5. tetragon) Example Non-convex regions (2-dim case) Example Convex region / Non-convex region (2-dim case) Straight line = Convex region Curved line = Non-Convex region Set of Probability Vectors [Probability vector] (= Vector representation of probability distribution) where [Set of probability vectors] Set of Probability Vectors [Lemma] The set of probability vectors is a convex set. (Proof) For any relation holds: and any such that , the next Set of Probability Vectors [Lemma] The set of probability vectors is bounded and closed (Proof) See textbook Convex Function Let be a real-valued function defined on a convex region [Definition] Convex function Convex = Convex upward = - convex Convex Function [Graphical image of convex function] [Remark] Any convex function is defined on a convex region. Concave Function Let be a real-valued function defined on a convex region [Definition] Concave function Concave = Convex downward = - convex Concave Function [Graphical image of concave function] [Remark] Any concave function is defined on a convex region. Convex functions Concave functions Example Lemma [Lemma] Let be a concave function of over the region Assume 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 a quantum system. • The -the hypothesis operator is . hypotheses about the states of is the proposition that its density • 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 decision. . is employed for our 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) (2) is compact is continuous Necessary and Sufficient Conditions for Bayes strategy • Theorem (Holevo 1973): [A] [B] where A.S.Holevo, J.Multivar. Anals., 3, 337-394 (1973) Necessary and Sufficient Conditions for Bayes strategy • Remark: The following three conditions are equivalent. [A] [A’] [A”] • By this theorem, 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 of . is a concave function Concavity of the minimum Bayes cost • [proof] Consider Let Then and Concavity of the minimum Bayes cost • [proof] Let and let Then , where , i.e. 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 new • Therefore, • Repeating this procedure, 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 and let Then be the solution to the mini-max problem, be the mini-max value. That is, 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 , then the 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 A Find such that Loop A start : Loop B start : B F F B Find such that Renewal of Data Loop B end : Loop A end : E C E C YES NO Renewal of Data D D Check: Necessary and sufficient conditions must be satisfied. Display and Store the result END Example: Application to Optical Communications • Mini-max Receiver for Optical Communication System Mini-max receiver Signal Measurement results To evaluate the system performance, we wish to know the mini-max value. Ternary Amplitude Shift-Keying (3ASK) • Alphabet: • Signal set: prior distribution: • Receiver: “Coherent state of light” • 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 minimax value was shown • Example: 3ASK future tasks • Tuning up the algorithm • Application to the quantum stream cipher