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 = ( T t ) t ≥ 0 Markov semigroup on L ∞ ( E , E , µ ) π invariant probability density � � ( T t f ) π d µ = f π d µ ∀ t , f E E Definition Detailed balance ( reversibility ) for ( T , π ) � � ∀ t ≥ 0 , f , g ∈ L ∞ ( E , E , µ ) . g ( T t f ) π d µ = ( T t g ) f π d µ E E QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Classical Detailed Balance If � f ( y ) p t ( x , y ) µ ( d y ) ( T t f )( x ) = E classical detailed balance is equivalent to π ( x ) p t ( x , y ) = π ( y ) p t ( y , x ) QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Quantum Detailed Balance (QDB): definitions h complex separable Hilbert space, T = ( T t ) t ≥ 0 semigroup of (completely) positive unital linear maps on B (h), ω invariant state Definition Agarwal Z. Physik 258 (1973): principle of microreversibility or detailed balance for ( T , ω ) ω ( T t ( x ) y ) = ε x ε y ω ( T t ( y ) x ) , with ε x , ε y 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 ( e n ) t oan ≥ 0 � θ u = u n e n n ≥ 0 x is even / odd if θ x ∗ θ = x , θ x ∗ θ = − x , ε x = 1 / ε x = − 1 ω ( T t ( x ) y ) = ε x ε y ω ( T t ( y ) x ) ω ( T t ( θ y ∗ θ ) θ x ∗ θ ) ⇔ = QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Parity: example h = ℓ 2 ( N ) = Γ( C ), c.o.n. basis ( e n ) n ≥ 0 T transpose θ x ∗ θ = x T annihilation, creation, number √ a e n = √ n e n − 1 , a † e n = N = a † a n + 1 e n +1 , position and momentum � � √ � � √ a † + a a † − a N = ( p 2 + q 2 − 1 l ) / 2 q = / 2 p = i / 2 even odd even QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Quantum Markov (dynamical)semigroup ( T t ) 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 , L ∗ Φ ( x ) := ℓ xL ℓ ℓ G ∗ + � ℓ L ∗ 1. G , L ℓ ∈ B (h) , ℓ L ℓ + G = 0 , 2. � ℓ L ∗ ℓ L ℓ strongly convergent. G , L ℓ of a GKSL form are not unique! QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance GKSL generator � − 1 L ∗ H = H ∗ G := ℓ L ℓ − iH , 2 ℓ L ( x ) = L 0 ( x ) + i [ H , x ] QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance GKSL generator � − 1 L ∗ H = H ∗ G := ℓ L ℓ − iH , 2 ℓ L ( x ) = L 0 ( x ) + i [ H , x ] Fix ρ , choose L ℓ with tr ( ρ L ℓ ) = 0 and 1 l , L 1 , L 2 , . . . linearly independent (min) If G ′ , L ′ ℓ also satisfy tr ( ρ L ′ ℓ ) = 0, (min) and � � ℓ xL ℓ + xG = L ( x ) = G ′∗ x + ℓ L ′∗ G ∗ x + ℓ L ∗ ℓ xL ′ ℓ + xG ′ QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance GKSL generator ⇒ ∃ a unitary ( u jk ) s.t. � H ′ = H + c 1 L ′ j = u jk L k , l , c ∈ R . k ⇒ unique L 0 and unique H (up to c ) in L ( a ) = L 0 ( a ) + i [ H , a ] . QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Alicki QDB T QMS on B (h) generated by L = L 0 + i [ H , · ], Definition L satisfies a quantum detailed balance condition w.r.t. a stationary state ρ if tr ( ρ L 0 ( x ) y ) = tr ( ρ x L 0 ( y )) [ H , ρ ] = 0 , 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 = L 0 + i [ H , · ], Definition L satisfies a quantum detailed balance condition w.r.t. a stationary state ρ if tr ( ρ L 0 ( x ) y ) = tr ( ρ x L 0 ( y )) [ H , ρ ] = 0 , for all x , y ∈ B (h). � � � ρ x � tr ( ρ ( x ) L ( y )) = tr L := L 0 − i [ H , · ] ⇔ L ( y ) T t := e t � i.e., defining, T t := e t L , � L � � ρ x � tr ( ρ T t ( x ) y ) = tr T t ( y ) QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Agarwal QDB − θ w.r.t. ρ = ρ T � ρ T t ( y T ) x T � � ρ x T t ( y T ) T � tr ( ρ T t ( x ) y ) = tr = tr Alicki QDB � � ρ x � tr ( ρ T t ( x ) y ) tr = T t ( y ) L ( x ) − � L ( x ) = − 2 i [ H , x ] Theorem � � � � T = − i [ H , x ] i [ H , x T ] If θ H θ = H ⇒ then L ( x ) = L ( x T ) T ⇔ L 0 ( x ) = L 0 ( x T ) T � QDB- θ = QDB ⇔ QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Duality Both Agarwal QDB − θ and Alicki QDB imply � � ρ x � tr ( ρ T t ( x ) y ) = tr T t ( y ) � T ∗ t ( y ρ ) ρ − 1 T t ( y ) = and ( � T t ) 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 � � ρ x � tr ( ρ T t ( x ) y ) = tr T t ( y ) � T ∗ t ( y ρ ) ρ − 1 T t ( y ) = and ( � T t ) t ≥ 0 semigroup of Completely Positive maps T t is a ∗ map, i.e. � T t ( y ) ∗ if and only if � T t ( y ∗ ) = � ρ i t a ρ − i t σ t ( a ) := T t ◦ σ − i = σ − i ◦ T t QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Other dualities 0 ≤ s ≤ 1 � � � � ρ s x ρ 1 − s � ρ s T t ( x ) ρ 1 − s y tr tr = T t ( y ) If s = 1 / 2 then ( � T t ) t ≥ 0 semigroup of CP maps QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Other dualities 0 ≤ s ≤ 1 � � � � ρ s x ρ 1 − s � ρ s T t ( x ) ρ 1 − s y tr tr = T t ( y ) If s = 1 / 2 then ( � T t ) 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 � T t is a ∗ -map. In this case T t ◦ σ z = σ z ◦ T t ( | 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 = 2 i [ K , · ] iff ∃ special representation of L by H , L ℓ s.t. � ρ 1 / 2 L ∗ u ℓ k L k ρ 1 / 2 ℓ = ( ♦ ) k for all ℓ , for some unitary ( u ℓ k ) symmetric i.e. u ℓ k = u k ℓ . Rem. ρ invariant + ( ♦ ) ⇒ condition on G : G ρ 1 / 2 − ρ 1 / 2 G ∗ = i (2 K + c ) ρ 1 / 2 . � � � � ρ L ∗ ρ 1 / 2 L ∗ j ρ 1 / 2 L ∗ Moreover, putting C jk := tr , B jk := tr j L k . k SQDB holds iff � � CB = BC T C − 1 B with u jk = jk QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance Standard QDB- θ Theorem L satisfies SQDB- θ � � � � ρ 1 / 2 x ρ 1 / 2 L ( y ) ρ 1 / 2 Θ( L (Θ( x )) ρ 1 / 2 y tr = tr iff there exists a special GKSL representation of L by G , L ℓ s.t. G ρ 1 / 2 + i r ρ 1 / 2 ρ 1 / 2 θ G ∗ θ = r ∈ R � ρ 1 / 2 θ L ∗ u ℓ k L k ρ 1 / 2 ℓ θ = k for all ℓ , for some unitary ( u ℓ k ) self-adjoint. � � � � ρ 1 / 2 L ∗ j ρ 1 / 2 θ L ∗ Moreover, putting C jk := tr ρ L ∗ , R jk := tr j L k k θ SQDB- θ holds iff � � C − 1 R CR = RC with u jk = jk QMS & DETAILED BALANCE

Classical Detailed Balance Quantum Detailed Balance The end Thank you! QMS & DETAILED BALANCE