Pure state post-selection is universal Lajos Di osi Wigner Centre, - PowerPoint PPT Presentation

Pure state post-selection is universal Lajos Di´ osi Wigner Centre, Budapest 23 Aug 2017, Kolymbari Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 1 / 10

Pure state pre- and postselection 1 2-time state 2 General 2-time state 3 Preparing entangled 2-time pure state 4 Can we prepare any 2-time entangled pure state? 5 Mixed 2-time state 6 Inferring success rate without tomography 7 Summary 8 Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 2 / 10

Pure state pre- and postselection Pure state pre- and postselection Logical options: H i , H f , H are factors of H tot | Ψ � ∈ H tot total inital state | i � ∈ H i preselection state (projection by | i � � i | ) | f � ∈ H f postselection state (projection by | f � � f | ) H observed state space Special cases: 1) H tot = H i = H f = H 2) H tot = H i = H f = H ⊗ H ′ ( H ′ : ancilla space) SilvaGuryanovaBrunnerLindenShortPopescu PRA 89 ,012121(2014) Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 3 / 10

2-time state 2-time state � µ ˆ µ ˆ A µ = ˆ | i � , | f � ∈ H ; A † 1 = ⇒ p ( µ, succ ) � p ( µ, succ ) � p ( succ ) = p ( µ, succ ) , p ( µ | succ ) = µ p ( µ, succ ) µ A µ | i �| 2 = tr ( | f � |� f | ˆ � f | )(ˆ � i | ˆ A † p ( µ, succ ) = A µ | i � µ ) ≡ tr (ˆ A µ ⊗ ˆ A † µ ) � ρ if 2 − time density : ρ if = | i � � f | ⊗ | f � � i | � Ψ if ⊗ ˆ ˆ Ψ † ≡ if 2 − time pure state : ˆ Ψ if = | i � � f | So far ˆ ρ if = ˆ Ψ if ⊗ ˆ Ψ † Ψ if = | i � � f | is unentangled, � if is not mixed (pure). Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 4 / 10

General 2-time state General 2-time state p ( µ, succ ) = tr (ˆ A µ ⊗ ˆ A † µ ) � ρ if Success becomes independent of measurement if it is weak: p WM ( succ ) = tr � ρ if Definitive math conditions for � ρ if : 1) tr ( ˆ V ⊗ ˆ ∀ ˆ V † ) � ρ if ≥ 0 , V ; 2) tr � ρ if ≤ 1 Standard form: � ˆ if ⊗ ˆ tr ˆ if ˆ Ψ r † Ψ r † Ψ r Ψ s ρ if = � if , if = 0 ( r � = s ) r Can we prepare all � ρ if ? Yes, upto a prefactor since tr � ρ if = p WM ( succ ) Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 5 / 10

Preparing entangled 2-time pure state Preparing entangled 2-time pure state | i � , | f � ∈ H ⊗ H ′ � f | )(ˆ A µ ⊗ ˆ � i | (ˆ µ ⊗ ˆ 1 ′ ) ≡ tr (ˆ A µ ⊗ ˆ 1 ′ ) | i � A † A † p ( µ, succ ) = tr ( | f � µ ) � ρ if � i | ) ≡ ˆ Ψ if ⊗ ˆ Ψ † ( tr ′ | i � � f | ) ⊗ ( tr ′ | f � � ρ if = if ˆ tr ′ | i � Ψ if = � f | In coordinates: | i � = � | f � = � kr | k � ⊗ | r � ′ , kr | k � ⊗ | r � ′ k , r c i k , r c f � f | = tr ′ � kr | k � ⊗ | r � ′ � ˆ ls � l | ⊗ � s | ′ Ψ if = tr ′ | i � c i c f ⋆ k , r l , s � � k [ c i ][ c f ] † c if = l | k � � l | ≡ kl | k � � l | kl kl Can we reach any 2-time amplitudes c if kl ? Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 6 / 10

Can we prepare any 2-time entangled pure state? Can we prepare any 2-time entangled pure state? From previous slide: � � � kr | k �⊗| r � ′ , kr | k �⊗| r � ′ ⇒ ˆ k , r c i k , r c f kl c if | i � = | f � = = Ψ if = kl | k � � l | Ψ if | 2 = | tr c if | 2 p WM ( succ ) = | tr ˆ c if = c i c f † (recall: tr c i † c i = tr c f † c f =1) 2-time amplitude c if is subnormalized unless c i = c f (upto a phase). ∗ ∗ ∗ ∗ √ √ Ψ upto a factor: c f =1 d yields c if = c i / Silva et al. prepare any ˆ / d p WM ( succ ) = | tr c i | 2 / d can be suboptimal. � If c if ≥ 0, then we can choose c i = c f = c if / tr c if at p WM ( succ ) = 1. Open issue: To prepare a given ˆ Ψ what are the ‘closest’ | i � and | f � to guarantee the highest p WM ( succ )? Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 7 / 10

Mixed 2-time state Mixed 2-time state � � µ ˆ µ ˆ A µ = ˆ µ ˆ r ˆ r = ˆ ′ † | i � , | f � ∈ H ⊗ H ′ , A † A ′ 1 ′ 1 , A � f | )(ˆ A µ ⊗ ˆ � i | (ˆ µ ⊗ ˆ ′ † A ′ A † p ( µ, r , succ ) = tr ( | f � r ) | i � r ) A � p ( µ, r , succ ) ≡ tr (ˆ A µ ⊗ ˆ A † p ( µ, succ ) = µ ) � ρ if r �� � � �� � tr ′ (ˆ 1 ⊗ ˆ tr ′ (ˆ 1 ⊗ ˆ A ′ ′ † ρ if � = r ) | i � � f | ⊗ r ) | f � � i | A r � ˆ if ⊗ ˆ Ψ r † Ψ r ≡ if Example: r | i � = � � √ p r | i ; r �⊗| r � ′ , r | f ; r �⊗| r � ′ , ˆ r = | r � ′ � r | ′ 1 A ′ | f � = √ r d ′ � ρ if = 1 p r | i ; r � � f ; r | ⊗ | f ; r � � i ; r | � d ′ r Issue: Pure state pre/postselection has 1 / d ′ -times smaller p WM ( succ ) vs Silva et al. Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 8 / 10

Inferring success rate without tomography Inferring success rate without tomography Silva et al.: For mixed 2-states romography, projective and WM’s are insufficient, generalized measurements are needed, they constructed one. Assume pure 2-states! Both projective and WM’s remain insufficient. What WM’s are sufficient for? WMs yield p WM ( succ ) without tomography. Example: AAV spin weak value Ψ | 2 = 1 ˆ p WM ( succ ) = | tr ˆ � f | = | � n i �� � 2 (1 + � n i � Ψ = | i � n f | , n f ) σ W = Re tr ˆ σ ˆ = � n i + � � Ψ n f 1 σ W | 2 = � , | � tr ˆ 1 + � n i � p WM ( succ ) n f Ψ 1 p WM ( succ ) = ( σ xW ) 2 + ( σ xW ) 2 + ( σ zW ) 2 Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 9 / 10

Summary Summary ρ if introduced for p ( µ, succ ) = tr (ˆ A µ ⊗ ˆ A † 2-time density � µ ) � ρ if p WM ( succ ) = tr � ρ if All � ρ if are, upto normalization p WM ( succ ), preparable via pure state pre- and postselection. p WM ( succ ) is a figure of merit of preparation. Optimum preparartion protocols remain to be found. In the simple VVA case, weak measurements provide p WM ( succ ) without tomography: 1 p WM ( succ ) = ( σ xW ) 2 + ( σ xW ) 2 + ( σ zW ) 2 Lajos Di´ osi ( Wigner Centre, Budapest ) Pure state post-selection is universal 23 Aug 2017, Kolymbari 10 / 10