Asymptotic equivalence of pure quantum state estimation and Gaussian white noise M. Nussbaum Cornell University joint work with C. Butucea and M. Guta High-dimensional Problems and Quantum Physics Marne-la-Vallée, June 2015 ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 1 / 16
Information contained in additional observations � � P � n ϑ , ϑ 2 Θ Le Cam, 1974: consider product experiments: E n = , Θ � R k , regularity (LAN) I compare E n and E n + m with regard to the information they contain about ϑ ( m additional observations ) I shown: if ∆ ( � , � ) is the de…ciency distance between experiments then �r m � ∆ ( E n , E n + m ) = O . n I Recall Le Cam’s de…ciency distance: I De…nition. (Le Cam ∆ -distance). Suppose that P = f P ϑ , ϑ 2 Θ g and Q = f Q ϑ , ϑ 2 Θ g are two experiments indexed by the same parameter set Θ , but with possibly di¤erent sample space. The de…ciency of P with respect to Q is δ ( P , Q ) = inf K sup k Q ϑ � KP ϑ k TV ϑ 2 Θ ( inf over all Markov kernels) ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 2 / 16
Characterization in terms risk functions: I δ ( P , Q ) � ε i¤: for every decision problem with loss function L ( d , ϑ ) � 1 we have: I for every decision rule in Q there is a decision rule in P such that risk is increased by ε (at most), uniformly over ϑ 2 Θ . I the symmetric version (i.e. ∆ - distance) is ∆ ( P , Q ) = max ( δ ( P , Q ) , δ ( Q , P ) I Sequences P n , Q n are asymptotically equivalent if ∆ ( P n , Q n ) ! 0 . Quantum analog ∆ q of de…ciency distance: Guta, 06 I experiments: families of quantum states P = f ρ ϑ , ϑ 2 Θ g ; space Θ arbitrary I replace total variation distance k�k TV by trace norm distance k ρ � σ k = Tr j ρ � σ j I replace Markov kernel by quantum analog (tp-cp maps) I for two quantum experiments P and Q , on di¤erent spaces but same Θ , ∆ q ( P , Q ) is de…ned ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 3 / 16
Equivalent and asymptotically equivalent experiments Experiments P , Q are called equivalent if ∆ ( P , Q ) = 0. We then write P $ Q . I if P , Q live on the same sample space ( Ω , F ) then a one-to-one data transformation generates equivalence. I if they live on di¤erent sample spaces ( Ω , F ) and ( Ω 0 , F 0 ) then a su¢cient statistic T : ( Ω , F ) ! ( Ω 0 , F 0 ) generates equivalence (if Q ϑ = L ( T j P ϑ ) , ϑ 2 Θ ). I primary example : Gaussian shift X i � N ( ϑ , 1 ) , i = 1 , . . . , n ; equivalent � ϑ , n � 1 � experiment: sample mean ¯ X n � N , ϑ 2 R I P is called more informative than Q if δ ( P , Q ) = 0 (one sided de…ciency of P wrt Q ). We write P � Q . I P � Q typically happens if Q ϑ = L ( T j P ϑ ) and the map T is not su¢cient (loses information) Asymptotic versions: sequences P n , Q n are called asy. equivalent if ∆ ( P n , Q n ) ! 0. We write P n ' Q n . I P n is called asy. more informative than Q n if δ ( P n , Q n ) ! 0 . We write P % Q Quantum versions: straighforward by using q-Markov kernels. Connection to q-su¢ciency has been worked out (Jenµ cova, Petz, 07). ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 4 / 16
Additional observations in nonparametric case (classical) starting point: white noise experiment dY t = f 1 / 2 ( t ) dt + n � 1 / 2 dW t , t 2 [ 0 , 1 ] , f 2 F ( α , M ) I background: N (96), f probability density for n i.i.d. observations, F ( α , M ) function class of smoothness α I n replaces observation number of i.i.d. case; I below we will replace f 1 / 2 by f (for shortness) I model is more ‡exible than i.i.d. model; allows easy formulation of equivalent (exactly) experiments: ∆ ( � , � ) = 0 I multiply observations by n 1 / 2 : equivalent experiment (up to f 1 / 2 6 = f ) dY t = n 1 / 2 f ( t ) dt + dW t , t 2 [ 0 , 1 ] , f 2 . . . . I Call this experiment G n , compare with G n + m ( m "additional observations") I …rst consider local versions: for a …xed function f 0 de…ne neighborhood Σ n ( f 0 ) = f f : k f � f 0 k � γ n g where k�k is L 2 -norm ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 5 / 16
I subtract the "known" function f 0 from the data, multiply by n 1 / 2 : equivalent local experiment dY t = n 1 / 2 ( f ( t ) � f 0 ( t )) dt + dW t , t 2 [ 0 , 1 ] , f 2 Σ n ( f 0 ) \ F ( α , M ) I do the same in G n + m ; compare the above with dY t = ( n + m ) 1 / 2 ( f ( t ) � f 0 ( t )) dt + dW t , t 2 [ 0 , 1 ] , f 2 . . . I squared Hellinger distance between the two measures (drift functions g 1 , g 2 , say ) bounded by � ( n + m ) 1 / 2 � n 1 / 2 � 2 k g 1 � g 2 k 2 k f � f 0 k 2 = � � 2 ( 1 + m / n ) 1 / 2 � 1 n k f � f 0 k 2 = I a Taylor expansion, assuming m / n = o ( 1 ) gives � m � 2 n k f � f 0 k 2 � C m 2 n γ 2 � C n . n I smoothness assumption f 0 2 F ( α , M ) means that f 0 can be estimated with rate γ n = n � α / ( 2 α + 1 ) (in terms of k�k ) So assuming γ n of that rate will allow us to "get into the neighborhoods" Σ n ( f 0 ) later. ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 6 / 16
I we need m 2 n � 1 γ 2 n = o ( 1 ) for Hellinger distance to be o ( 1 ) . That gives a condition 1 = n 1 / 2 n α / ( 2 α + 1 ) = n 1 � s ( α ) , s ( α ) = m � n 1 / 2 γ � 1 n 4 α + 2 for asymptotic equivalence of the two local experiments, with f 2 Σ n ( f 0 ) , n 1 / 2 f ( t ) dt + dW t , dY t = ( n + m ) 1 / 2 f ( t ) dt + dW t dY t = I extreme cases: a) α = ∞ (…nite dimension): s ( α ) = 0, hence m � n su¢ces (Le Cam) I b) α > 0 , condition is m � n 1 / 2 + ε phenomenon: the larger the parameter space, the smaller the allowed m for asy. equivalence (= the higher the informational content of additional observations) I (holds for local equivalence, but for global too: see below) ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 7 / 16
Globalization: get rid of f 2 Σ n ( f 0 ) , keep only smoothness f 2 F ( α , M ) . I idea: sample splitting; start with global experiment G n with observations dY t = f ( t ) dt + n � 1 / 2 dW t , f 2 F ( α , M ) I this is equivalent (by su¢ciency argument) to indep. observations � Y ( 1 ) , Y ( 2 ) � where dY ( i ) = f ( t ) dt + ( n / 2 ) � 1 / 2 dW ( i ) , i = 1 , 2 . t t I use local equivalence under f 2 Σ n ( f 0 ) of Y ( 2 ) and an estimator ˆ f 0 of f 0 � Y ( 2 ) � based on Y ( 1 ) , to establish global equivalence to Y ( 1 ) , ˜ where = f ( t ) dt + (( n + m ) / 2 ) � 1 / 2 dW ( 2 ) Y ( 2 ) d ˜ . t t I Repeat, with roles of Y ( 1 ) and ˜ Y ( 2 ) reversed, to obtain global equivalence to � Y ( 2 ) � Y ( 1 ) , ˜ ˜ where = f ( t ) dt + (( n + m ) / 2 ) � 1 / 2 dW ( 1 ) Y ( 1 ) d ˜ . t t Y ( 2 ) (su¢ciency argument) to obtain I take average of ˜ Y ( 1 ) , ˜ Y t = f ( t ) dt + ( n + m ) � 1 / 2 dW t , d ˜ f 2 F ( α , M ) . ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 8 / 16
Result: the Gaussian experiments G n and G n + m are asy. equivalent, globally under f 2 F ( α , M ) if 1 m � n 1 � s ( α ) , s ( α ) = ( 4 α + 2 ) (informational content of "additional observations"). similar method used before for i.i.d. experiment (N 96): I recall experiment E n : i.i.d. observations, root density f 2 F ( α , M ) (=density f 2 , on unit interval etc.) I we have ∆ ( E n , G n ) ! 0 if G n is the Gaussian white noise experiment with drift function f I method applied for proof: …rst localization around f 0 , rate γ n � n � 1 / 4 ; I consider local experiments E n ( f 0 ) , G n ( f 0 ) , approximation ∆ ( E n ( f 0 ) , G n ( f 0 )) ! 0 I globalization by sample splitting: E n $ E n / 2 � E n / 2 , (symbol $ :exact equivalence) I then use estimate of f 0 based on E n / 2 to replace 2nd E n / 2 by G n / 2 I obtain E n / 2 � G n / 2 then (reverse) obtain G n / 2 � G n / 2 $ G n I fact ∆ ( E n , G n ) ! 0 implies ∆ ( E n , E n + m ) ! 0 (carry over results about additional observations from Gaussian to i.i.d. case) ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 9 / 16
Quantum experiments pure quantum state: unit vector φ 2 H (complex) Hilbert space I random variables are created by observables A : self-adjoint operators. I A discrete spectrum: A = ∑ k j = 1 a j Π j spectral decomposition (all real � � k eigenvalues di¤erent, possibly 0, ∑ k j = 1 Π j = I ). Then Π j j = 1 is a � � = p j = φ � Π j φ . measurement, and X A is de…ned by P X A = a j I expectations: EX A = ∑ k j = 1 a j φ � Π j φ = φ � A φ = tr A φφ � . I ( A continuous spectrum: X A continuous r.v., e.g. Gaussian) quantum analog of i.i.d. observations: tensor product state φ � n I consider nonparametric quantum i.i.d. experiment. I assume smoothness class for φ : Hilbert space H - in…nite dimensional, complex, exists ONB � � j = 1 j 2 α � � � � 2 φ : ∑ ∞ I de…ne F ( α , M ) = � φ j � � M : Sobolev-type smoothness class � � I thus the i.i.d. experiment is E n = φ � n , φ 2 F ( α , M ) ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 10 / 16
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