QUANTUM MARKOV SEMIGROUPS & DETAILED BALANCE Franco Fagnola - - PowerPoint PPT Presentation

Classical Detailed Balance Quantum Detailed Balance

QUANTUM MARKOV SEMIGROUPS & DETAILED BALANCE

Franco Fagnola

Politecnico di Milano

(joint work with V. Umanit` a and R. Rebolledo)

Quantissima in the Serenissima III August 20, 2019

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

1 Classical Detailed Balance 2 Quantum Detailed Balance

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Classical Detailed Balance (CDB)

T = (Tt)t≥0 Markov semigroup on L∞(E, E, µ) π invariant probability density

• E

(Ttf )π dµ =

• E

f π dµ ∀ t, f Definition Detailed balance (reversibility) for (T, π)

• E

g(Ttf )π dµ =

• E

(Ttg)f π dµ ∀ t ≥ 0, f , g ∈ L∞(E, E, µ).

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Classical Detailed Balance

If (Ttf )(x) =

• E

f (y)pt(x, y)µ(dy) classical detailed balance is equivalent to π(x)pt(x, y) = π(y)pt(y, x)

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Quantum Detailed Balance (QDB): definitions

h complex separable Hilbert space, T = (Tt)t≥0 semigroup of (completely) positive unital linear maps

• n B(h), ω invariant state

Definition Agarwal Z. Physik 258 (1973): principle of microreversibility or detailed balance for (T , ω) ω (Tt(x)y) = εxεy ω (Tt(y)x) , with εx, εy parities of x, y under time reversal.

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

QDB: definitions

Typical parity. θ : h → h antiunitary, e.g. conjugation w.r. basis (en)toan ≥ 0 θu =

• n≥0

unen x is even / odd if θx∗θ = x, εx = 1 / θx∗θ = −x, εx = −1 ω (Tt(x)y) = εxεy ω (Tt(y)x) ⇔ = ω (Tt(θy∗θ) θx∗θ)

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Parity: example

h = ℓ2(N) = Γ(C), c.o.n. basis (en)n≥0 θx∗θ = xT

T transpose

annihilation, creation, number a en = √n en−1, a†en = √ n + 1 en+1, N = a†a position and momentum q =

• a† + a
• /

√ 2 p = i

• a† − a
• /

√ 2 N = (p2 + q2 − 1

l)/2 even

• dd

even

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Quantum Markov (dynamical)semigroup

(Tt)t≥0 norm-continuous semigroup of unital CP maps on B(h), L generator Theorem (Gorini, Kossakowski, Sudarshan, Lindblad) GKSL L(x) = G ∗x + Φ(x) + xG, Φ(x) :=

• ℓ

L∗

ℓxLℓ

• 1. G, Lℓ ∈ B(h),

G ∗ +

ℓ L∗ ℓLℓ + G = 0,

2.

ℓ L∗ ℓLℓ strongly convergent.

G, Lℓ of a GKSL form are not unique!

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

GKSL generator

G := −1 2

• ℓ

L∗

ℓLℓ − iH,

H = H∗ L(x) = L0(x) + i[H, x]

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

GKSL generator

G := −1 2

• ℓ

L∗

ℓLℓ − iH,

H = H∗ L(x) = L0(x) + i[H, x] Fix ρ, choose Lℓ with tr(ρLℓ) = 0 and

1 l, L1, L2, . . . linearly independent (min)

If G ′, L′

ℓ also satisfy tr(ρL′ ℓ) = 0, (min) and

G ∗x +

• ℓL∗

ℓxLℓ + xG = L(x) = G ′∗x +

• ℓL′∗

ℓxL′ ℓ + xG ′

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

GKSL generator

⇒ ∃ a unitary (ujk) s.t. L′

j =

• k

ujkLk, H′ = H + c1

l,

c ∈ R. ⇒ unique L0 and unique H (up to c) in L(a) = L0(a) + i[H, a].

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Alicki QDB

T QMS on B(h) generated by L = L0 + i[H, ·], Definition L satisfies a quantum detailed balance condition w.r.t. a stationary state ρ if [H, ρ ] = 0,

tr (ρL0(x)y) = tr (ρxL0(y))

for all x, y ∈ B(h).

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Alicki QDB

T QMS on B(h) generated by L = L0 + i[H, ·], Definition L satisfies a quantum detailed balance condition w.r.t. a stationary state ρ if [H, ρ ] = 0,

tr (ρL0(x)y) = tr (ρxL0(y))

for all x, y ∈ B(h).

• L := L0 − i[H, ·]

⇔

tr (ρ(x)L(y)) = tr

• ρx

L(y)

• i.e., defining, Tt := etL,

Tt := et

L

tr (ρTt(x)y) = tr

• ρx

Tt(y)

• QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Agarwal QDB−θ w.r.t. ρ = ρT

tr (ρTt(x)y) = tr

• ρTt(yT)xT

= tr

• ρxTt(yT)T

Alicki QDB

tr (ρTt(x)y)

=

tr

• ρx

Tt(y)

• L(x) −

L(x) = −2i[H, x] Theorem If θHθ = H

• ⇒
• i[H, xT]

T = −i[H, x]

• then

QDB-θ = QDB

⇔

• L(x) = L(xT)T ⇔ L0(x) = L0(xT)T

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Duality

Both Agarwal QDB−θ and Alicki QDB imply

tr (ρTt(x)y)

=

tr

• ρx

Tt(y)

• Tt(y)

= T∗t(yρ)ρ−1 and ( Tt)t≥0 semigroup of Completely Positive maps

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Duality

Both Agarwal QDB−θ and Alicki QDB imply

tr (ρTt(x)y)

=

tr

• ρx

Tt(y)

• Tt(y)

= T∗t(yρ)ρ−1 and ( Tt)t≥0 semigroup of Completely Positive maps

• Tt is a ∗ map, i.e.

Tt(y∗) = Tt(y)∗ if and only if σt(a) := ρitaρ−it Tt ◦ σ−i = σ−i ◦ Tt

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Other dualities

0 ≤ s ≤ 1

tr

• ρsTt(x)ρ1−sy
• =

tr

• ρsxρ1−s

Tt(y)

• If s = 1/2 then (

Tt)t≥0 semigroup of CP maps

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Other dualities

0 ≤ s ≤ 1

tr

• ρsTt(x)ρ1−sy
• =

tr

• ρsxρ1−s

Tt(y)

• If s = 1/2 then (

Tt)t≥0 semigroup of CP maps If s ∈ [0, 1] − {1/2} Theorem (Majewski-Streater, J Phys A 1988) T is a QMS if and only if each Tt is a ∗-map. In this case Tt ◦ σz = σz ◦ Tt (|z| ≤ 1/2) and duals of T for s ∈ [0, 1] coincide.

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Standard QDB, no θ

Theorem A generator L satisfies SQDB L − L = 2i[K, ·] iff ∃ special representation of L by H, Lℓ s.t. ρ1/2L∗

ℓ =

• k

uℓkLkρ1/2

(♦)

for all ℓ, for some unitary (uℓk) symmetric i.e. uℓk = ukℓ.

• Rem. ρ invariant + (♦) ⇒ condition on G:

Gρ1/2 − ρ1/2G ∗ = i(2K + c)ρ1/2. Moreover, putting Cjk := tr

• ρL∗

j Lk

• , Bjk := tr
• ρ1/2L∗

j ρ1/2L∗ k

• .

SQDB holds iff CB = BC T

with

ujk =

• C −1B
• jk

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

Standard QDB-θ

Theorem L satisfies SQDB-θ

tr

• ρ1/2xρ1/2L(y)
• = tr
• ρ1/2Θ(L(Θ(x))ρ1/2y
• iff there exists a

special GKSL representation of L by G, Lℓ s.t. ρ1/2θG ∗θ = Gρ1/2 + irρ1/2 r ∈ R ρ1/2θL∗

ℓθ

=

• k

uℓkLkρ1/2 for all ℓ, for some unitary (uℓk) self-adjoint. Moreover, putting Cjk := tr

• ρL∗

j Lk

• , Rjk := tr
• ρ1/2L∗

j ρ1/2θL∗ kθ

• SQDB-θ holds iff

CR = RC

with

ujk =

• C −1R
• jk

QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance

The end

Thank you!

QMS & DETAILED BALANCE