On a Mathematical Theory of Repeated Quantum Measurements Vojkan - - PowerPoint PPT Presentation

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 = {a1, · · · , an}.
• Quantum instrument in the Heisenberg picture:

{Φa}a∈A, where Φa : B(H) → B(H) are completely pos- itive, and Φ =

a Φa is unital, Φ(1) = 1.

Ex: Φa(X) = VaXV ∗

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 (ω1, · · · , ωL) is PL(ω1, · · · , ωL) = tr(ρΦω1 ◦ · · · ◦ ΦωL(1)).

• The family {PL}L≥1 uniquely extends to a probability mea-

sure P on Ω = AN invariant under the shift map φ(ω1, ω2, · · · ) = (ω2, ω3, · · · ).

• 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 ω1 ∈ A is observed with probability tr(Φ∗

ω1(ρ)), and after the measurement the system is in the

state ρω1 = Φ∗

ω1(ρ)

tr(Φ∗

ω1(ρ)).

A further measurement at time t = 2 gives the outcome ω2 with probability tr(Φ∗

ω2(ρω1)) = tr((Φ∗ ω2 ◦ Φ∗ ω1)(ρ))

tr(Φ∗

ω1(ρ))

, and the joint probability for the occurence of the sequence of

• utcomes (ω1, ω2) is

tr((Φ∗

ω2 ◦ Φ∗ ω1)(ρ)) = tr(ρ (Φω1 ◦ Φω2)(1)).

3

Example 1. von Neumann measurements.

• A ⊂ R, {Pa}a∈A projections,

a Pa = 1.

Observable A =

a∈A aPa.

• Dynamics: unitary U : H → H (unit time propagator).
• The instrument Φa(X) = VaXV ∗

a , Va = U∗Pa.

Projective von Neumann measurement of observable A.

• If ρt is the state at time t, then a measurement of A at time

t + 1 yields ω with the probability tr(PωUρtU∗Pω) and ρω = PωUρtU∗Pω/tr(PωUρtU∗Pω) is the state after the measurement.

4

Example 2. Ancila measurements.

• Probes described by Hp and ρp.

Coupled system H ⊗ Hp, ρ ⊗ ρp, U : H ⊗ Hp → H ⊗ Hp.

• {Pa}ω∈A projections on Hp,

a Pa = 1, A = a∈A aPa.

• Quantum instrum: reduced projective measurement of A

Φ∗

ω(ρ) = trHp

(1 ⊗ Pω)U(ρ ⊗ ρp)U∗ .

(1)

• Any instrument arises in this way: given {Φa}a∈A, one can

find Hp, ρp, U, and {Pa}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(PL) = −

ω∈AL PL(ω) log PL(ω).

• S(P) = limL→∞ 1

LS(PL).

• SL(ω) = − log PL(ω), S(PL) =
• AL SLdPL.

Shannon-McMillan-Breiman (SMB) 1 LSL(ω) → S(P) P − a.s. and in L1(dP).

• GOAL: Refinement of the SMB theorem.

7

THREE ROUTES

• LDP– Fluctuations that accompany the SMB theorem

PL

1

LSL(ω) ∼ s

• ∼ e−LI(s)
• Dimension theory/multi-fractal formalism: topological struc-

ture and the fractal dimensions of the level sets Ls =

• ω ∈ Ω | lim

L→∞

1 LSL(ω) = s

• .
• Declaring E±

L (ω1, · · · , ωL) = ±SL(ω1, · · · , ωL) to be the

energy of the spin configuration (ω1, · · · , ωL), develop sta- tistical mechanics of the resulting ”spin system”.

8

PRESSURE

• F±

L (β) = log

 

• ω∈supp PL

e−βE±

L (ω)

  .

Theorem 1 For all β > 0, the following limit exists F±(β) = lim

L→∞

1 LF±

L (β)

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 λ0 = min sp(ρ), for any ω, ν ∈

Ωfin, 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 ∈ Ωfin one can find ζ ∈ Ωfin, |ζ| ≤ τ, such that P(ωζν) ≥ CP(ω)P(ν). Follows from the irreducibility of Φ and yields uniqueness

• f the equilibrium states and differentiability of F+.

10

CONSEQUENCES

• Local LDP for the 1

LSL(ω) on ]∂F+(0), ∂F+(∞)[

G¨ artner-Ellis.

• Local multi-fractal formalism (Billingsley): formula for dimH Ls

in the terms of the local LDP rate function. Ls 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

• Number theoretic flavour: Φa(X) =

k VkXV ∗ k , and all

matrix elements of Vk are algebraic numbers, then F−(β) is finite.

• Rotational instrument. H = C2, ρ = 1

21, A = {0, 1, 2},

Φ0(X) = 1 2RθXR−θ, Φi(X) = 1 2PiXPi. Theorem 2. (1) For a.e. θ ∈ [0, 2π] 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

I(s) = s + sup

α∈R

(αs − Fsgn(−α)(|α|)). Proof: Lanford-Ruelle function. Level III LDP also holds. Complete multifractal formalism holds: dimH Ls = 1 log n(I(s) + s)

13

SOME EXAMPLES

• Farey fractions instr. H = C2, A = {0, 1}, θ ∈]0, 2[,

Φ0(X) = 1 2 + θ

• x11 + θx22

θx22

• ,

Φ1(X) = 1 2 + θ

• x11

(2 − θ)x11 + θx22

• ,

ρ = 1

2[2 − θ, θ].

PL(ω) = 1 (2 + θ)Lρ ·

• Mω1 · · · MωL
• 1

1

• ,

M0 =

• 1

θ θ

• ,

M1 =

• 1

2 − θ θ

• .

14

• For θ = 1, F−(β) real analytic, Pθ ∼ spin system with

exponentially decaying interactions.

• θ = 1, Pθ 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

• rder phase transition at βcr = −2.
• Number theoretic spin-chains, extensively studied in 1990-

2010 (Knauf, Kleban-Ozluk, many others)

15

• Erd¨
• s instrument. H = C2, A = {0, 1, 2}, ρ = [1/2, 1/2]

PL(ω) = 1 5Lρ ·

• Mω1 · · · MωL
• 1

1

• ,

M0 =

• 1

1 1

• , M1 =
• 1

1 1

• , M2 =
• 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 < βcr < −2.

• P is weak Gibbs with the same potential as the well-known

self-similar Bernoulli convolution at golden mean (Erd¨

• s 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-
• nd order phase transition at β = 0 with an anomalous

central limit theorem.

17

FURTHER DEVELOPMENTS

• Theory of two instruments.

Same H, same A, ({Φω}, ρ),

• {ˆ

Φω}, 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