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

1. 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

2. Bennett Holevo Fuchs Josza 佐々木 広田 富田 相馬 臼田 大崎 吾妻 加藤 @Tamagawa University, Japan

3. 臼田 相馬 Van Enk Lutkenhaus 大崎 Schmecher 南部 宇佐見 山崎 Bennett Smolin 広田 Fuchs 加藤 @Oiso, Kanagawa, Japan

4. OUTLINE • Background • Quantum Hypothesis Testing • Bayes Strategy • Mini-max Strategy • Calculation Algorithm • Example • Conclusion

5. Alice, Bob, and Eve Alice, the sender Plaintext cipher text Bob, the receiver Encryption Plaintext Decryption Eve, the eavesdropper

6. Classification of Quantum Cryptographyby functions Function Source Single photon BB84 4 states Key Distribution B92 2 states Key Distribution Coherent YK 2 states Key Distribution Y-00 M states (M>100) Direct Encryption

7. Coherent states [Def.] Coherent state of light (with complex amplitude ) Control technique  Signal Modulation Example)

8. 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＞６００ｋｍ Backbone >2.5Gbps

9. Pseudo-Random Number Generator Pseudo-Random Number Generator PRNG PRNG Secret Key Secret Key Plaintext Plaintext Basic Model of Y-00 Alice: Sender Signal Multi-ary Signal Modulator (it is not single photon!) Bob: Receiver Signal Detector

10. 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.

11. Basic Model: Multi-ary signal modulator (3’) Signal constellation and mapping rule: (basis) (bit) Running key Signal distance >> 1 Plaintext (Example) PSK # of bases M= 7 # of signals 2M=14 Signal distance <<1

12. 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,

13. Linear Feedback Shift Resister Initial values AND AND AND AND Output ＋ ＋ ＋ OR OR OR Connection coefficients Given by primitive polynomial Output

14. Alice Secret key PRNG 3 Running key 7 2 Plain Text Signal Mod. 0 1 0 Basis #3 Basis #7 Basis #2 0 1 0

15. Bob Secret key PRNG 3 Running key 7 2 Receiver 0 1 0 Signal Plain Text

16. 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 0 1 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 1 Basis#0 Basis#1 Basis#2 Basis#7 One signal is assigned to 0 and another is to 1 in each basis.

17. 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 #2 #5 #6 #4 #7 #4 #0 #6 #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. 0 1 1 0 1 1 1 0 0 0 1 … Basis# #1 #3 #2 #5 #6 #4 #7 #4 #0 #6 #5 Signal

18. 0 1 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. 001 011 000 010 101 110 110 100 111 100 101 … Basis# #1 #3 #2 #5 #6 #4 #7 #4 #0 #6 #5 B-2: For each slot, binary detection is done by using information of the bases. 0 Received signalby Bob If signaling basis is #2, the decision region is given as follows: Error Free Emitted signalby Alice 1 B-3: Thus, Bob can get the plaintext.

19. 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 Ciphertext signal can be measured only once. (Quantum No-cloning Theorem) Ｙ－００ ・・・ There are so many resulting patterns and each of themcontains error bits Pattern#1 Pattern#2 Pattern#X Ordinary attacks do not work anymore Yellow block stands for error bit

20. Implementation Schemes for Y-00 PSK - based quantum stream cipher, (NWU) Target: Long-distance PSK Intensity Modulation - based quantum stream cipher (Tamagawa) Intensity Level Target: High-speed QAM - based quantum stream cipher, (KK) Optical QAM

21. Open (Eye pattern) Close

22. 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”

23. History Theory of Games Hypothesis Testing RADAR system ? - 1940, UK 1928 von Neumann 1933 Neyman and Pearson Mini-max theorem 1940-1945, MIT RadLab 1944 von Neumann “Ideal Receiver” “Cost” “Risk” Theory of Games Two-parson game Decision Function 1953 Middleton, Analysis of signal detection process by statistical hypothesis testing Nature v.s. Observer 1939 A.Wald Generalization of Neyman-Pearson Theory 1954 Peterson Receiver design by likelihood ratio 1955 Middleton, Formulation of Signal Detection problemsbased 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

24. 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)

25. Quantum Hypothesis Testing量子仮説検定 ??? Quantum System We wish to determine the state of the system with small error

26. 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.

27. Convex Region [Definition] Convex region (or Convex set) Let be a subspace (or subset) of the K-dim vector space ,i.e. . Then convex region is defined as follows:

28. Example Convex regions (2-dim case) (2. oval ) (3. trigon) (1. ellipse) (5. tetragon) (4. hexagon)

29. Example Non-convex regions (2-dim case)

30. Example Convex region / Non-convex region (2-dim case) Straight line = Convex region Curved line = Non-Convex region

31. Set of Probability Vectors [Probability vector] (= Vector representation of probability distribution) where [Set of probability vectors]

32. Set of Probability Vectors [Lemma] The set of probability vectors is a convex set. (Proof) For any and any such that , the nextrelation holds:

33. Set of Probability Vectors [Lemma] The set of probability vectors is bounded and closed (Proof) See textbook

34. Convex Function Let be a real-valued function defined on a convex region [Definition] Convex function Convex = Convex upward = - convex

35. Convex Function [Graphical image of convex function] [Remark] Any convex function is defined on a convex region.

36. Concave Function Let be a real-valued function defined on a convex region [Definition] Concave function Concave = Convex downward = - convex

37. Concave Function [Graphical image of concave function] [Remark] Any concave function is defined on a convex region.

38. Example Convex functions Concave functions

39. 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

40. 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

41. Positive Operator-Valued Measure (POVM)正作用素値測度 • [Decision Operators:決定作用素] • [POVM]

42. Positive Operator-Valued Measure (POVM)正作用素値測度 • The probability of choosing when is true:

43. 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)

44. 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,

45. Bayes Costsベイズコスト（損失係数） • Example: Radar system

46. 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 bywhere

47. The average Bayes cost平均ベイズコスト（平均損失） [Check] Joint probability:

48. The average probability of error平均誤り確率 If , then the average Bayes cost becomes the average probability of decision errors.

49. 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

50. Bayes Strategyベイズ戦略 • Lemma:The optimal POVM of the Bayes problem exists. It exists because • is compact • (2) is continuous