Quantum Minimax Theorem

Transcription

1 This presentation is shown in RIS conference, Kyoto, JAPAN Quantum inimax Theorem in Statistical ti ti Decision i Theory November, 4 Public Version 田中冬彦 Fuyuhiko TANAKA Graduate School of Engineering Science, Osaka University If you are interested in details, please see quant-ph ; citations are also welcome!

2 Prologue: What is Bayesian Statistics? Non-technical Short Version *It is for non-statisticians esp. researchers in quantum information.

3 ission of Statistics ission of Statistics To give a desirable and plausible method according to statistical models, purpose of inference point estimation, prediction, hypothesis testing, model selection, etc., and loss functions. *Precise evaluation of estimation error is secondary. Difference from Pure ath Statistics is required to meet various needs in the world Significant problems in statistics have changed according to the times This trivial fact does not seem to be known so much except for statisticians.

4 What s Bayesian Statistics? / Bayesian Statistics Statistics i admitting i the usage of prior distributions ib i prior distribution= prob. dist. on the unknown parameter We do not care about the interpretation of this prob. *Interpretation heavily depends on individual researcher.

5 What s Bayesian Statistics? / Relation to frequentist s statistics traditional statistics. Bayesian statistics has enlarged the framework and satisfies various needs. Not an alternative of frequentists statistics!. Sticking to frequentists methods like LEs is the preference to a specific prior distribution narrow-minded! Indeed, LEs often fail; their independence of loss is unstuaible 3. Frequentists results give good starting points to Bayesian analysis For details, see, e.g., C. Robert, Bayesian Choice Chap. Frequentists Bayesian Statistics Traditional statistics

6 isunderstanding Since priors are not objective, Bayesian methods should not be used for scientific result. Objective priors Noninformative priors are proposed for such usage. Bayesian statistics do not deny frequentists way and statistical inference based on an objective prior often agrees with frequentists one. ex: LE = AP estimate under the uniform prior It seems unsolved which noninformative prior should be used. The essence of this problem is how to choose a good statistical method with small sample. It is NOT inherent to Bayesian statistics, but has been a central issue in traditional statistics. Frequentists and mathematicians avoid this and only consider mathematically tractable models and/or asymptotic theory it s large-sample approximation! In Bayesian statistics, this problem is transformed into the choice of noninformative prior

7 Further Reference Preface in RIS Kokyuroku 834 Theory and applications of statistical inference in quantum theory POINT - Point out some misunderstanding of Statistics for non-experts esp. Physicists - Define the word quantum statistical inference as an academic term Sorry but it is written in Japanese.

8 CONTENTS. Notation. Introduction. Quantum Statistical odels and easurements 3. Quantum Statistical Decision Problems 4. Risk Functions and Optimality Criteria 5. ain Result 6. inimax POVs and Least Favorable Priors 7. Summary

9 . Notation

10 Description of Probability in Q Quantum echanics Q: described by operators in a Hilbert space Probability: bilit described d by density op. and measurement Density operator easurementpov d x Probability distributions dx Tr d x easurements dx x dx a x Statistical processing of data

11 Density operators density matrices, state Density Operators S : { all d.o.s on a Hilbert space } { L Tr, } When dim H n p U U p n * p i Tr n j p j

12 easurements in Q / athematical model of measurements = Positive Operator Valued easure POV POV def. for finite sets Sample space POV { j } j,, k = Possible values for a measurement { x, x,, x k = a family of linear ops. satisfying j } j : { x j } j j I j O Positive operator positive semidefinite matrix

13 easurements in Q / How to construct a probability space Axiom of Q do d.o. sample space S POV easurement { x, x,, x k } { j } j,, k Then the prob. obtaining a measurement outcome x j p j : Tr j * The above prob. satisfies conditions positive and summing to one due to both def.s of d.o.s and POVs. Tr and j j j j O I

14 . Introduction

15 Quantum Bayesian Hypothesis Testing/. Alice chooses one alphabet denoted by an integer and sends one corresponding quantum state described d by a density operator,,..., k {,,..., k } S H. Bob receives the quantum state and perform a measurement and guesses the alphabet Alice really sent to him. The whole process is described by a POV { j } j,,, k 3. The proportion of alphabets is given by a distribution called a prior distribution k,,, k

16 Quantum Bayesian Hypothesis Testing/ * The probability that Bob guesses j when Alice sends i. j i i A j B p Tr, j i w * Bob s loss estimation error when he guesses j while Alice sends i., j i w k * Bob s risk for the i-th alphabet k j i A j B p j i w i R, k i R * Bob s average risk his task is to minimize by using a good POV i i i R

17 ,,..., k Summary of Q-BHT Chooses one alphabet and sends one quantum state {,,..., k } S Proportion of each alphabet prior distribution k, k Guess the alphabet by choosing a measurement POV { j } j,,, k H Average risk k r k, : i R i R i w i, j Tr i Bayes POV w.r.t. = POV minimizing the average risk j i j

18 inimax POV.Chooses one alphabet and sends one quantum state,,..., k {,,..., k } S H Guess the alphabet by choosing a measurement POV { j } j,,, k Bob has no prior information inimize the worst-case risk The worst-case risk r * : sup R i i inimax POV = POV minimizing the worst-case risk

19 Quantum inimax Theorem Simple Ver..Chooses one alphabet and sends one quantum state,,..., k {,,..., k } S H Guess the alphabet by choosing a measurement POV { j } j,,, k Theorem Hirota and Ikehara, 98 inf sup R i sup inf k i i R i i inimax POV agrees with the Bayes POV w.r.t. the worst case prior. *The worst-case prior to Bob is called a least favorable prior.

20 Example / Quantum states / / Quantum states, / /,. 3 - loss ij j i w, inimax POV ij j,, / / / /, / / / /. 3 3 *This is a counterexample of Theorem in Hirota and Ikehara 98, where their proof seems not mathematically rigorous.

21 Quantum states Example / / /, / /, 3. w i, j - loss ij LFP LF /, 3 LF LF inimax POV is constructed as a Bayes POV w.r.t. LFP. Important Implication When completely unknown, it is not necessarily wise to find an optimal POV with the uniform prior. Although Statisticians already know this fact long decades ago

22 .Nature Nt Rewritten in Technical Terms Parameter space,,..., k Quantum Statistical odel { } S H Experimenter and Statistician Decision space U,,..., k POV on U { u } uu Loss function w, u Theorem inf sup R Risk function R uu w, u Tr sup inf R u

23 ain Result Brief Summary Quantum inimax Theorem inf sup R Po P sup inf R d Po where R : U w, u Tr du cf inf sup R D P D sup inf R d Conditions, assumptions, and essence of proof are explained.

24 Previous Works Wald 95 Statistical Decision Theory Holevo 973 Quantum Statistical ti ti ldecision i Theory Recent results and applications many! Kumagai and Hayashi 3 Guta and Jencova 7 Hayashi 998 Le Cam 964 Classical inimax Theorem in statistical decision theory First ver. is given by Wald 95 we show Quantum inimax Theorem in quantum statistical decision theory

25 . Quantum Statistical odels and easurements

26 Formal Definition of Statistical odels naïve Quantum statistical model = A parametric family of density operators 3 i Ex. 3 3 i i 3 R,, 3 ;, s r s r s r R Basic Idea Holevo, 976 A quantum statistical model is defined by a map : H L Next, we see the required conditions for this map

27 Regular Operator-Valued Function / Definition Locally compact metric space K compact set K : {, K K : d, } L H For a map trace-class operators on a Hilbert space T : L H T K : inf X, we define : X T T X,, K Definition An operator-valued function limt K T is regular K

28 Remark Regular Operator-Valued Function / limt K K Uniformly continuous w.r.t. trace norm on every compact set sup T T, K as Converse does not hold if dim H Remark The regularity assures that the following operator-valued integral is well-defined for every f f T d C and P T K : inf X : X T T X,, K

29 Definition Quantum Statistical odels S H : { X L H : Tr X, X } : S H is called a quantum statistical model if it satisfies the conditions below. Conditions. Identifiability one-to-one map,. Regularity limt K K Necessary for our arguments

30 A U U easurements POVs Decision space locally compact space Borel sets the smallest sigma algebra containing all open sets Dfiii Definition Positive-operator valued function is called a POV if it satisfies B B AU B j B j B, B, A U B i B j, i j j j U I Po U All POVs over U

31 Axiom of Quantum echanics Born Rule For the quantum system described by and a measurement described by a POV dx measurement outcome is distributed ib t daccording to x ~ Tr d x d x POV d x x dx

32 3. Quantum Statistical Decision Problems

33 Situationti Basic Setting For the quantum system specified with the unknown parameter experimenters extract tinformation for some purposes. POV dx x d x a x Typical sequence of task. To choose a measurement POV. To perform the measurement over the quantum system 3Totake an action ax among choices based on the measurement 3. To take an action ax among choices based on the measurement outcome x formally called a decision function

34 Decision Functions Example - Estimate the unknown parameter x x x a n Estimate the unknown parameter - Estimate the unknown d.o. rather than parameter n x a x a x a 3 3 * x a x a x a x a x a x a Constr ct confidence region credible region - Validate the entanglement/separability, x a ] [ - Construct confidence region credible region ], [ x a x a R L

35 Remarks Remarks for Non-statisticians Ifh. the quantum state is completely l known, then the distribution ib i of measurement outcome is also known and the best action is chosen.. Action has to be made from finite data. 3. Precise estimation of the parameter is only a typical example. POV dx x dx a x

36 Loss Functions Performance of decision functions is compared by adopting the loss functions smaller is better. Ex: Parameter Estimation Quantum Statistical odel { Action space : R Loss function squared error w, a a m } Later we see the formal definition of the loss function.

37 easurements Over Decisions Basic Idea From the beginning, g, we only consider POVs over the action space. Put together POV dx x d x a x POV N da a da

38 Quantum Statistical Decision Problems Formal Definitions Parameter space locally compact space U Quantum statistical model *Decision space locally compact space w : U R { } The triplet t, U, w Loss function lower semi continuous**; bounded below is called a quantum statistical ldecision problem. cf statistical decision problem p, U, w p dx U w Statistical model Decision space : U R { } p Loss function *we follow Holevo s terminology instead of saying action space. ** we impose a slightly stronger condition on the loss than LeCam, Holevo.

39 4. Risk Functions and Optimality Criteria

40 Comparison of POVs The loss e.g. squared error depends on both unknown parameter and our decision u, which is a random variable. u ~ d u Tr d u In order to compare two POVs, we focus on the average of the loss wrt w.r.t. the distribution of u. POV du u du E[ w, u ] Compared at the same N d u ' POV u ' E'[ w, u' ] du '

41 Definition iti Risk Functions The risk function of U Po U R : w, u Tr d u E w, u u ~ Tr d u Remarks for Non-statisticians. Smaller risk is better.. Generally there exists no POV achieving the uniformly smallest risk among POVs. Since R depends on the unknown parameter, we need additional optimality criteria.

42 Optimality Criteria / Definition A POV is said to be minimax if it minimizes the supremum of the risk function, i.e., sup R * inf Po U sup R Historical Remark Bogomolov showed the existence of a minimax POV Bogomolov 98 in a more general framework. Description of quantum states and measurements is different from recent one.

43 P Optimality Criteria / All probability distributions defined on Borel sets In Bayesian statistics, P is called a prior distribution. Definition The average risk of Po U w.r.t. r, : R d A POV is said to be Bayes if it minimizes the average risk, i.e., r r, inf, Po U Historical Remark Holevo showed the existence of a Bayes POV Holevo 976; see also Ozawa, 98 in a more general framework.

44 Ex. of Loss Functions and Risk Functions Parameter Estimation } R : { m U, u u w } R : { U u u R d Tr ] [ E u Construction of Predictive Density Operator H m S U u D u w m } R : { p n S H U, u D u w U m m u u D R d Tr U

45 5. ain Result

46 ain Result Quantum inimax Theorem Theorem * U Let and be a compact metric space. Then the following equality holds for every quantum statistical decision problem. inf Po U sup R sup inf R d P Po U where the risk function is given by U R : w, u Tr d u Corollary For every closed convex subset Q Po U the above assertion holds. *see quant-ph for proof

47 Key Lemmas For Theorem Compactness of the POVs Holevo 976 If the decision space U is compact, then Po UU is also compact. Equicontinuity of risk functions Lemma by FT w : U R Loss function If w is bounded and continuous, then F : { R : Po U } C is uniformly equicontinuous. The equicontinuity implies the compactness of F C under the uniform convergence topology We show main theorem by using Le Cam s result with both lemmas. However, not a consequence of their previous old results.

48 6. inimax POVs and Least Favorable Priors

49 Previous Works LFPs Wald 95 Statistical Decision Theory Jeffreys 96 Jeffreys prior Bernardo 979, 5 Reference prior; reference analysis Komaki Latent information prior Tanaka DP prior Reference prior for pure-states model Objective priors in Bayesian analysis are indeed least j p y y favorable priors.

50 ain Result Existence Theorem of LFP Definition If a prior achieves the supremum, i.e., the following holds inf R d LF Po U P Po U sup inf R d The prior LF is called a least favorable priorlfp. where R : w, u Tr du U Theorem Let and U be a compact metric space. w: continuous loss function. Then, for every decision problem, there exists a LFP.

51 Pathological Example Remark Even on a compact space, a bounded lower semicontinuous and not continuous loss function does not necessarily admit a LFP. Example [,] R L, u R sup R sup R d [,] But for every prior, [,] R d [,]

52 inimax POV is constructed using LFP Corollary U Let and be a compact metric space. If a LFP exists for a quantum statistical decision problem, Every minimax POV is a Bayes POV with respect to the LFP. In particular, if the Bayes POV w.r.t. the LFP is unique, then it is minimax. Corollary For every closed convex subset Q Po U the above assertion holds. uch of theoretical results are derived from quantum minimax theorem.

53 7. Summary

54 Discussion. We show quantum minimax theorem, which gives theoretical basis in quantum statistical decision theory, in particular in objective Bayesian analysis.. For every closed convex subset of POVs, all of assertions still hold. Our result has also practical meanings for experimenters. Future Works - Show other decision-theoretic results and previous known results by using quantum minimax theorem - Propose a practical algorithm of finding a least favorable prior - Define geometrical structure in the quantum statistical decision problem If you are interested in details, please see quant-ph ; citations are also welcome!