Quantum trajectories: invariant measure uniqueness and mixing - PowerPoint PPT Presentation

Quantum trajectories: invariant measure uniqueness and mixing arXiv:1703.10773 with M. Fraas, Y. Pautrat and C. Pellegrini Tristan Benoist Venice, August 2017 IMT, Universit´ e Paul Sabatier 1 / 24

A canonical experiment (S. Haroche’s group) Images: LKB ENS 2 / 24

State space Definition (Quantum states) Density matrices : D := { ρ ∈ M d ( C ) | ρ ≥ 0 , tr ρ = 1 } . Definition (Pure states) Pure states are the extreme points of D . Namely, ρ ∈ D is a pure state iff. ∃ x ∈ C d \ { 0 } s.t. ρ = P x := | x �� x | . Definition (Metric) Unitary invariant norm distance: d( ρ, σ ) = � ρ − σ � . Remark d ( U ρ U ∗ , U σ U ∗ ) = d ( ρ, σ ) . For U ∈ U ( d ) , 3 / 24

System evolution without conditioning on measurements Definition (Completely positive trace preserving (CPTP) maps) Without conditioning on measurement results the system evolution is given by a CPTP map: Φ : D → D ℓ � V j ρ V ∗ ρ �→ j j =1 with Kraus operators V j ∈ M d ( C ) for all j = 1 , . . . , ℓ s.t. � ℓ j =1 V ∗ j V j = Id d . Remark Seeing Φ as arising from the interaction of the system with an auxiliary system (probe), Kraus operators V j = � e i | U Ψ � := � ℓ j =1 U ij � e j | Ψ � with: • The initial state of the probe | Ψ �� Ψ | • The system–probe interaction U • The observable measured on the probe J := � ℓ j =1 j | e j �� e j | Different observables on the probe give different V j but same Φ . ℓ � Φ( ρ ) := tr probe ( U ρ ⊗ P Ψ U ∗ ) = � e j | U Ψ � ρ � Ψ | U ∗ e j � . j =1 4 / 24

Indirect measurement Initial state: ρ ∈ D • Evolution unconditioned on the measurement: ρ �→ Φ( ρ ). 5 / 24

Indirect measurement Initial state: ρ ∈ D • Evolution unconditioned on the measurement: ρ �→ Φ( ρ ). • Conditioning on the measurement of J : V j ρ V ∗ ρ �→ ρ ′ = j with prob. tr( V ∗ j V j ρ ) , j V j ρ ) tr( V ∗ 5 / 24

Indirect measurement Initial state: ρ ∈ D • Evolution unconditioned on the measurement: ρ �→ Φ( ρ ). • Conditioning on the measurement of J : V j ρ V ∗ ρ �→ ρ ′ = j with prob. tr( V ∗ j V j ρ ) , j V j ρ ) tr( V ∗ Remark that: E ( ρ ′ | ρ ) = Φ( ρ ) . 5 / 24

Repeated interactions Probes : U Sys: ρ n = Φ ◦ n ( ρ ). • Without conditioning on the measurement, after n interactions: ¯ 6 / 24

Repeated interactions Measurements Probes : U Sys: j n ; j n − 1 ; : : : ; j 2 ; j 1 ρ n = Φ ◦ n ( ρ ). • Without conditioning on the measurement, after n interactions: ¯ • Given the state after n − 1 measurements of J is ρ n − 1 , after n measurements of J : V j ρ n − 1 V ∗ j with prob. tr( V ∗ ρ n := j V j ρ n − 1 ) , j V j ρ n − 1 ) . tr( V ∗ Equivalently, given ρ 0 = ρ , after n measurements of J producing result sequence j 1 , . . . , j n : V j n . . . V j 1 ρ V ∗ j 1 . . . V ∗ j n with prob. tr( V ∗ j 1 . . . V ∗ ρ n := j n V j n . . . V j 1 ρ ) , j n V j n . . . V j 1 ρ ) . tr( V ∗ j 1 . . . V ∗ 6 / 24

Quantum trajectories as Markov chains Definition (Quantum trajectory) j =1 s.t. � ℓ Given a finite set of d × d matrices { V j } ℓ j =1 V ∗ j V j = Id d , a quantum trajectory is a realization of the Markov chain of kernel: ℓ � V j ρ V ∗ � j � tr( V ∗ Π( ρ, A ) := 1 A j V j ρ ) tr( V ∗ j V j ρ ) j =1 for any A ⊂ D mesurable. 7 / 24

Preliminary results: Perron–Frobenius Theorem for CPTP maps Definition (Irreducibility) The CPTP map Φ is said irreducible if the only non null orthogonal projector P such that Φ( PM d ( C ) P ) ⊂ PM d ( C ) P is P = Id d . Theorem (Evans, Høegh-Krohn ’78) A CPTP map Φ : M d ( C ) → M d ( C ) is irreducible iff. ∃ ! ρ inv . ∈ D s.t. ρ inv . > 0 and Φ( ρ inv . ) = ρ inv . . Moreover, if Φ is irreducible, its modulus 1 eigenvalues are simple and form a finite sub group of U (1) . The sub group size m ∈ { 1 , . . . , d } is equal to the period of Φ and ∃ 0 < λ < 1 and C > 0 s.t. ∀ ρ ∈ D , m − 1 � � 1 � � � Φ ◦ ( mn + r ) ( ρ ) − ρ inv . � ≤ C λ n . � � � m � � r =0 8 / 24

Preliminary results: Strong law of large numbers for the state Theorem (K¨ ummerer, Maassen ’04) Let ( ρ n ) n be a quantum trajectory. Then, n 1 � lim ρ k = ρ ∞ a . s . n →∞ n k =1 with Φ( ρ ∞ ) = ρ ∞ . Particularly, if Φ is irreducible, ρ ∞ = ρ inv . a.s. 9 / 24

Preliminary results: Purification For any ρ ∈ D let S ( ρ ) := − tr( ρ log ρ ) be its von Neumann entropy. Then: S ( ρ ) = 0 ⇐ ⇒ rank( ρ ) = 1 ⇐ ⇒ ρ is a pure state . Theorem (K¨ ummerer, Maassen ’06) The following statements are equivalent: 1. An orthogonal projector π s.t. π V ∗ j 1 · · · V ∗ j p V j p · · · V j 1 π ∝ π for all j 1 , . . . j p ∈ { 1 , . . . , ℓ } is of rank 1 , 2. For any ρ 0 ∈ D , n →∞ S ( ρ n ) = 0 lim a . s . Remark • If π is s.t. π V ∗ j 1 · · · V ∗ j p V j p · · · V j 1 π ∝ π for all j 1 , . . . j p ∈ { 1 , . . . , ℓ } , there exists unitary matrices U π j 1 ,..., j p s.t. V j p · · · V j 1 π ∝ U π j 1 ,..., j p π. • In dimension d = 2 , either lim n →∞ S ( ρ n ) = 0 a.s., or all the matrices V j are proportional to unitary matrices. 10 / 24

Uniqueness and convergence towards the invariant measure Theorem (B., Fraas, Pautrat, Pellegrini ’17) If the following two assumptions are verified, ( Φ -erg.) Φ is irreducible, (Pur.) Any orthogonal projector π s.t. π V ∗ j 1 · · · V ∗ j p V j p · · · V j 1 π ∝ π for all j 1 , . . . j p ∈ { 1 , . . . , ℓ } is of rank 1 , Π accepts a unique invariant probability measure ν inv . . Moreover, ∃ 0 < λ < 1 and C > 0 s.t. for any probability measure ν over D , � m − 1 � 1 � ν Π mn + r , ν inv . ≤ C λ n W 1 m r =0 with m ∈ { 1 , . . . , d } the period of Φ . 11 / 24

Previous similar results • Products of i.i.d. (Furstenberg, Guivarc’h, Kesten, Le Page, Raugi . . . ’60–’80, Books: Bougerol et Lacroix ’85, Carmona et Lacroix ’90) Markov kernel: ℓ � V j ρ V ∗ � j � Π 0 ( ρ, A ) = 1 A p j tr( V ∗ j V j ρ ) j =1 with ( p j ) ℓ j =1 a probability measure over { 1 , . . . , ℓ } . • Generalization (Guivarc’h, Le Page ’01–’16) Markov kernel: ℓ � V j ρ V ∗ � � � s j � Π s ( ρ, A ) = N ( s ) − 1 tr( V ∗ 1 A j V j ρ ) p j tr( V ∗ j V j ρ ) j =1 for s ≥ 0 (Q. Traj.: s = 1). j V ∗ No assumption that � j V j = Id d but the matrices V j need be invertible and a stronger irreducibility condition is assumed. • { V j } ℓ j =1 is strongly irreducible( i.e. no non trivial finite union of proper subspaces is preserved by the matrices V j . Then, strong irreducibility = ⇒ ( Φ -erg.) ), • The smallest closed sub semigroup of GL d ( C ) containing { V j } ℓ j =1 is contracting (equivalent to (Pur.) for a strongly irreducible family of invertible matrices). 12 / 24

Examples • Let p ∈ ]0 , 1[ and √ p � √ p � � � � � � � 0 0 0 0 0 0 V 1 = , V 2 = √ 1 − p , V 3 = , V 4 = √ 1 − p . 0 0 0 0 0 0 The family { V 1 , V 2 , V 3 , V 4 } verifies conditions ( Φ -erg.) and (Pur.) . • Let, � � � � 1 1 0 1 0 1 √ √ Z = and X = . 0 − 1 1 0 2 2 The family { Z , X } verifies ( Φ -erg.) but not (Pur.) . There exists uncountably many mutually singular Π-invariant probability measures concentrated on the pure states. • Let, � � � � e i 1 0 1 cos 1 i sin 1 Z = √ and X = √ . e − i 0 i sin 1 cos 1 2 2 The family { Z , X } verifies ( Φ -erg.) but not (Pur.) . Nevertheless Π accepts a unique invariant probability measure concentrated on pure states. 13 / 24

Non suitable methods Let, e 0 = (1 , 0) T , e 1 = (0 , 1) T and √ 1 − p � √ p � � � 0 0 V 1 = and V 2 = √ 1 − p √ p 0 0 with p ∈ ]0 , 1 / 2[. The family { V 1 , V 2 } defines a CPTP map Φ and verifies ( Φ -erg.) and (Pur.) . φ -irreducibility: Π n ( P e 0 / 1 , { P e 0 , P e 1 } ) = 1 for any n . Hence, if Π is φ -irreducible it is so only for φ ≪ 1 2 ( δ P e 0 + δ P e 1 ). Though Π n ( P e + , · ) is atomic and Π n ( P e + , { P e 0 , P e 1 } ) = 0 for any n with e + = 1 2 (1 , 1) T . Hence √ φ ( A ) > 0 = ⇒ P ( τ A < ∞| ρ 0 = P e + ) = 0 . Particularly, � δ P e 0 Π n − δ P e + Π n � � ∀ n ∈ N . TV = 1 , � � � Contractivity: For all n ∈ N and j 1 , . . . , j n ∈ { 1 , . . . , ℓ } , V j n . . . V j 1 P e 0 V ∗ j 1 . . . V ∗ V j n . . . V j 1 P e 1 V ∗ j 1 . . . V ∗ � � j n j n d j n V j n . . . V j 1 P e 0 ) , = 1 . tr( V ∗ j 1 . . . V ∗ tr( V ∗ j 1 . . . V ∗ j n V j n . . . V j 1 P e 1 ) 14 / 24

Proof of uniqueness structure 1. Assuming ( Φ -erg.) , for any Π-invariant probability measure, the distribution of the sequences ( j n ) n of J measurement results is the same, 2. Assuming (Pur.) , there exists a process ( σ n ) n taking value in D and depending only on ( j n ) n s.t. n →∞ d ( ρ n , σ n ) = 0 lim a . s . 15 / 24