On a Mathematical Theory of Repeated Quantum Measurements Vojkan Jaksic McGill University Based on joint works with T. Benoist, N. Cuneo, Y. Pautrat, C-A. Pillet, A. Shirikyan October 3, 2018
REPEATED QUANTUM MESUREMENTS • Hilbert space H , dim H < ∞ . B ( H ) , � X, Y � = tr( X ∗ Y ) . • Outcomes indexed by finite alphabet A = { a 1 , · · · , a n } . • Quantum instrument in the Heisenberg picture: { Φ a } a ∈A , where Φ a : B ( H ) → B ( H ) are completely pos- itive, and Φ = � a Φ a is unital, Φ( 1 ) = 1 . Ex: Φ a ( X ) = V a XV ∗ a • Density matrix ρ > 0 such that Φ ∗ ( ρ ) = ρ . 1
• Generalized (Kraus) repeated quantum measurement pro- cess: the probability of observing the sequence of outcomes ( ω 1 , · · · , ω L ) is P L ( ω 1 , · · · , ω L ) = tr( ρ Φ ω 1 ◦ · · · ◦ Φ ω L ( 1 )) . • The family { P L } L ≥ 1 uniquely extends to a probability mea- sure P on Ω = A N invariant under the shift map φ ( ω 1 , ω 2 , · · · ) = ( ω 2 , ω 3 , · · · ) . • Dynamical system (Ω , P , φ ) . • Assumption: Φ is irreducible ⇒ (Ω , P , φ ) is ergodic. 2
KRAUS MEASUREMENTS At time t = 1 , when the system is in the state ρ , a measurement is performed. The outcome ω 1 ∈ A is observed with probability tr(Φ ∗ ω 1 ( ρ )) , and after the measurement the system is in the state Φ ∗ ω 1 ( ρ ) ρ ω 1 = ω 1 ( ρ )) . tr(Φ ∗ A further measurement at time t = 2 gives the outcome ω 2 with probability ω 2 ( ρ ω 1 )) = tr((Φ ∗ ω 2 ◦ Φ ∗ ω 1 )( ρ )) tr(Φ ∗ , tr(Φ ∗ ω 1 ( ρ )) and the joint probability for the occurence of the sequence of outcomes ( ω 1 , ω 2 ) is tr((Φ ∗ ω 2 ◦ Φ ∗ ω 1 )( ρ )) = tr( ρ (Φ ω 1 ◦ Φ ω 2 )( 1 )) . 3
Example 1. von Neumann measurements. a P a = 1 . • A ⊂ R , { P a } a ∈A projections, � Observable A = � a ∈A aP a . • Dynamics: unitary U : H → H (unit time propagator). • The instrument Φ a ( X ) = V a XV ∗ a , V a = U ∗ P a . Projective von Neumann measurement of observable A . • If ρ t is the state at time t , then a measurement of A at time t + 1 yields ω with the probability tr( P ω Uρ t U ∗ P ω ) and ρ ω = P ω Uρ t U ∗ P ω / tr( P ω Uρ t U ∗ P ω ) is the state after the measurement. 4
Example 2. Ancila measurements. • Probes described by H p and ρ p . Coupled system H ⊗ H p , ρ ⊗ ρ p , U : H ⊗ H p → H ⊗ H p . • { P a } ω ∈A projections on H p , � a P a = 1 , A = � a ∈A aP a . • Quantum instrum: reduced projective measurement of A � ( 1 ⊗ P ω ) U ( ρ ⊗ ρ p ) U ∗ � . Φ ∗ ω ( ρ ) = tr H p (1) • Any instrument arises in this way: given { Φ a } a ∈A , one can find H p , ρ p , U , and { P a } a ∈A so that (1) holds for all density matrices ρ on H . 5
MOTIVATION • Haroche’s non-demolition measurements of photons (Nobel Prize 2012). • Finitely correlated states (Fannes, Nachtergaele, Werner 1992) • Novel class of dynamical/spin systems (Ω , P , φ ) with some surprising properties. 6
OBJECT OF INTEREST: ENTROPY • S ( P L ) = − � ω ∈A L P L ( ω ) log P L ( ω ) . • S ( P ) = lim L →∞ 1 L S ( P L ) . • S L ( ω ) = − log P L ( ω ) , S ( P L ) = � A L S L d P L . Shannon-McMillan-Breiman (SMB) 1 LS L ( ω ) → S ( P ) P − a.s. and in L 1 (d P ) . • GOAL: Refinement of the SMB theorem. 7
THREE ROUTES • LDP– Fluctuations that accompany the SMB theorem � 1 � ∼ e − LI ( s ) LS L ( ω ) ∼ s P L • Dimension theory/multi-fractal formalism: topological struc- ture and the fractal dimensions of the level sets 1 � � L s = ω ∈ Ω | lim LS L ( ω ) = s . L →∞ • Declaring E ± L ( ω 1 , · · · , ω L ) = ± S L ( ω 1 , · · · , ω L ) to be the energy of the spin configuration ( ω 1 , · · · , ω L ) , develop sta- tistical mechanics of the resulting ”spin system”. 8
PRESSURE • e − βE ± F ± L ( ω ) . � L ( β ) = log ω ∈ supp P L Theorem 1 For all β > 0 , the following limit exists 1 L F ± F ± ( β ) = lim L ( β ) L →∞ F + is finite and differentiable on ]0 , ∞ [ , the Gibbs varia- tional principle holds and the equilibrium measure is unique. Proof : Development of sub-aditive thermodynamic formal- ism. 9
SUBADITIVITY • Upper decoupling: with λ 0 = min sp( ρ ) , for any ω, ν ∈ Ω fin , P ( ων ) ≤ λ − 1 0 P ( ω ) P ( ν ) . Suffices for the development of thermodynamical formalism in the ” + ” regime, but not for differentiabilty. • Lower decoupling: There exists τ > 0 and C > 0 such that for any ω, ν ∈ Ω fin one can find ζ ∈ Ω fin , | ζ | ≤ τ , such that P ( ωζν ) ≥ C P ( ω ) P ( ν ) . Follows from the irreducibility of Φ and yields uniqueness of the equilibrium states and differentiability of F + . 10
CONSEQUENCES • Local LDP for the 1 L S L ( ω ) on ] ∂ F + (0) , ∂ F + ( ∞ )[ G¨ artner-Ellis. • Local multi-fractal formalism (Billingsley): formula for dim H L s in the terms of the local LDP rate function. L s is an uncount- able dense subset of supp P . • What about Global LDP and full multi-fractal formalism? They would hold if the above results also hold for F − ( β ) . 11
HOWEVER It may happen that F − ( β ) = ∞ for all β > 0 k V k XV ∗ • Number theoretic flavour: Φ a ( X ) = � k , and all matrix elements of V k are algebraic numbers, then F − ( β ) is finite. • Rotational instrument. H = C 2 , ρ = 1 2 1 , A = { 0 , 1 , 2 } , Φ 0 ( X ) = 1 Φ i ( X ) = 1 2 R θ XR − θ , 2 P i XP i . Theorem 2. (1) For a.e. θ ∈ [0 , 2 π ] one has F − ( β ) < ∞ . (2) For a dense set of θ ’s, F − ( β ) = ∞ . 12
MAIN RESULTS • Except in special cases, there is no thermodynamic formal- ism in the ” − ” case. • Global LDP holds with ( αs − F sgn( − α ) ( | α | )) . I ( s ) = s + sup α ∈ R Proof : Lanford-Ruelle function. Level III LDP also holds. Complete multifractal formalism holds: 1 dim H L s = log n ( I ( s ) + s ) 13
SOME EXAMPLES • Farey fractions instr. H = C 2 , A = { 0 , 1 } , θ ∈ ]0 , 2[ , � � 1 x 11 + θx 22 0 Φ 0 ( X ) = , 0 θx 22 2 + θ � � 1 x 11 0 Φ 1 ( X ) = , 0 (2 − θ ) x 11 + θx 22 2 + θ ρ = 1 2 [2 − θ, θ ] . � � �� 1 1 P L ( ω ) = (2 + θ ) L ρ · M ω 1 · · · M ω L , 1 � � � � 1 1 0 θ M 0 = M 1 = , . 0 2 − θ θ θ 14
• For θ � = 1 , F − ( β ) real analytic, P θ ∼ spin system with exponentially decaying interactions. • θ = 1 , P θ is weak Gibbs with continuous potential, F − ( β ) is real analytic and strictly convex for β > − 2 , and F − ( β ) = F ( − 2) + c ( β + 2) for β ≤ − 2 . Second order order phase transition at β cr = − 2 . • Number theoretic spin-chains, extensively studied in 1990- 2010 (Knauf, Kleban-Ozluk, many others) 15
os instrument. H = C 2 , A = { 0 , 1 , 2 } , ρ = [1 / 2 , 1 / 2] • Erd¨ � � �� P L ( ω ) = 1 1 5 L ρ · M ω 1 · · · M ω L , 1 � � � � � � 1 1 1 0 1 1 M 0 = , M 1 = , M 2 = . 0 1 1 1 1 1 • P is weak Gibbs with continuous potential, F − ( β ) is real analytic and strictly convex except for a first order phase transition at − 3 < β cr < − 2 . • P is weak Gibbs with the same potential as the well-known self-similar Bernoulli convolution at golden mean (Erd¨ os mea- sure, related to work of Feng and collaborators). 16
• Host of exactly solvable instruments • Two times measurement instruments with applications to thermal probes (thermodynamics), various spin-spin instru- ments. • Keep-Switch instrument for which the pressure exhibits sec- ond order phase transition at β = 0 with an anomalous central limit theorem. 17
FURTHER DEVELOPMENTS • Theory of two instruments. � � { ˆ Same H , same A , ( { Φ ω } , ρ ) , Φ ω } , ˆ ρ . ( P , ˆ P ) . Entropy replaced with relative entropy. Applications: Hypothesis testing, Gallavotti-Cohen Fluctua- tion Theorem. • New view on quantum detailed balance for completely pos- itive maps. • Parameter estimation (under development). 18
REFERENCES (1) Benoist, J., Pautrat, Pillet: On entropy production of repeated quantum measurements I. General theory. Comm. Math. Phys, 2017. (2) Cuneo, J. Pillet, Shirikyan: Large deviations and fluctuation theorem for selectively decoupled measures on shift spaces. Submitted. (3)-(4) Benoist, Cuneo, J., Pillet: On entropy production of re- peated quantum measurements II and III. Examples. (5) Benoist, Cuneo, J., Pillet: On entropy production of repeated quantum measurements IV: Multifractal formalism. (6) Benoist, J., Pautrat, Pillet: On entropy production of repeated quantum measurements V: quantum detailed balance. (7) Benoist, Cuneo, J., Pillet: Parameter estimation for repeated quantum measurements. 19
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