Estimation and discrimination of quantum networks Paolo Perinotti - PowerPoint PPT Presentation

Estimation and discrimination of quantum networks Paolo Perinotti in collaboration with G. Chiribella and G. M. D’Ariano DEX-SMI Workshop on Quantum Statistical Inference, 4 March 2009 NII, Tokyo

Summary Quantum combs: the theory of quantum networks Testers: measurements of network parameters Four results in quantum network estimation Optimal discrimination of two transformations Optimal covariant estimation of unitary channels Optimal tomography Analysis of Quantum Bit Commitment

Quantum channels A quantum channel is a linear trace-preserving CP map It is useful to represent quantum channels via their Choi operator � C := ( C ⊗ I )( | Ω �� Ω | ) , H out ⊗ H in ∋ | Ω � := | n � | n � n C ( ρ ) = Tr in [( I ⊗ ρ T ) C ] TRACE PRESERVATION CONDITION Tr out [ C ] = I in

Quantum networks We want to describe quantum networks What is the Choi operator of a network? We start from 2 channels: = C 1 C 2 C 3

Link product H 0 H 1 H 1 H 2 M N The definition of link product provides the Choi operator of the composed channel L = M ∗ N := Tr H 1 [( M ⊗ I 0 )( I 2 ⊗ N θ 1 )]

Link product H 0 H 1 H 1 H 2 M N The definition of link product provides the Choi operator of the composed channel L = M ∗ N := Tr H 1 [( M ⊗ I 0 )( I 2 ⊗ N θ 1 )]

Link product H 0 H 2 M ◦ N = L The definition of link product provides the Choi operator of the composed channel L = M ∗ N := Tr H 1 [( M ⊗ I 0 )( I 2 ⊗ N θ 1 )]

Link product � a c � A in A out A   a c b d A H in  H out f   b ⇒ d = � d  B � f g e B in B out B g e J = H d Choi-operator calculus A ∗ B = Tr J [ A θ J B ] ∈ B ( H out ⊗ H in ) AB := ( A a,b,c,d ⊗ I e,f,g )( I a,b,c ⊗ B d,e,f,g ) G. Chiribella, G. M. D'Ariano, and P. P., Phys. Rev. Lett. 101, 060401 (2008).

Networks as combs T 3 T 1 T 5 T 4 T 2 All networks can be sorted to form of a “comb network” T 1 T 2 T 3 T 4 T 5 R = T 1 ∗ T 2 ∗ T 3 ∗ T 4 ∗ T 5 G. Chiribella, G. M. D'Ariano, and P. P., Phys. Rev. Lett. 101, 060401 (2008).

The quantum comb We consider networks of this kind 0 1 2 3 2 N − 4 2 N − 3 2 N − 2 2 N − 1 V 0 V 1 V N − 2 V N − 1 One can prove that the Choi operator of the network satisfes Tr 2 n − 1 [ R ( n ) ] = I 2 n − 2 ⊗ R ( n − 1) , 1 ≤ n ≤ N R (0) = 1

Realisation theorem Also the converse is true: if R satisfies Tr 2 n − 1 [ R ( n ) ] = I 2 n − 2 ⊗ R ( n − 1) , 1 ≤ n ≤ N R (0) = 1 then it has a realisation scheme as a comb 1 U 0 3 0 1 2 3 4 5 6 7 V = 2 5 W A A A 1 2 3 4 7 T 6 G. Chiribella, G. M. D'Ariano, and P. P., Phys. Rev. Lett. 101, 060401 (2008). G. Chiribella, G. M. D'Ariano, and P. P., in preparation.

Testers We consider networks of this kind C 1 C 2 C 3 P i ρ Their Choi operator is T i and satisfies � T i = T = I 2 N − 2 ⊗ Ξ i Tr 2 n − 1 [ Ξ ( n ) ] = I 2 n − 2 ⊗ Ξ ( n − 1) , 1 ≤ n ≤ N − 1 Ξ ( N − 1) = Ξ , I 0 = 1 � R ∗ T i = Tr[ RT T i ] = p ( i | R ) , p ( i | R ) = 1 i

Realisation theorem Also the converse is true: if T i satisfies � T i = I 2 N − 2 ⊗ Ξ i Tr 2 n − 1 [ Ξ ( n ) ] = I 2 n − 2 ⊗ Ξ ( n − 1) , 1 ≤ n ≤ N − 1 Ξ ( N − 1) = Ξ , I 0 = 1 then for all R � R ∗ T i = Tr[ RT T i ] = p ( i | R ) , p ( i | R ) = 1 i and the operators T i correspond to a tester network

Decomposition of testers A particularly useful decomposition for testers is P i := ( I ⊗ Ξ − 1 2 ) T i ( I ⊗ Ξ − 1 2 ) R := ( I ⊗ Ξ T 1 2 ) R ( I ⊗ Ξ T 1 ˜ 2 ) i ] = Tr[ ˜ Tr[ RT T RP T i ] C 1 C 2 C 3 P i ρ

Discrimination of unitaries Problem: provided N uses of a black box which performs either U 1 or U 2 , discriminate the two cases Procedure 1: apply the N uses on a multipartite state and measure Procedure 2: apply the N uses in sequence on a single system, intercalated with fixed unitaries, and measure Procedure 3: insert the N uses in a quantum network and measure the output

Procedure 1 V = U † 1 U 2 U U U � π � N 1 = ∆ φ G. M. D’Ariano, P. Lo Presti, M. G. A. Paris, PRL 87, 270404 (2001); A. Acín, PRL 87, 177901(2001).

Procedure 2 U U U U V = U † 1 U 2 � π � N 2 = N 1 = ∆ φ R. Duan, Y. Feng, M. Ying, PRL 98, 100503 (2007)

Procedure 3 U U U U Question: what is the optimal disposition of unitaries for discrimination?

Spread lemma ∆ ( AB ) ≤ ∆ ( A ) + ∆ ( B ) A. M. Childs, J. Preskill, and J. Renes, J. Mod. Opt. 47, 155-176 (2000). U U U U W 3 W 2 W 1 ∆ [ W ( U ⊗ I ) W † ( U ⊗ I )] ≤ ∆ ( U ⊗ 2 ) The spread of the tester is not larger than that of U ⊗ N and U N The parallel and fully sequential scheme are both optimal No quantum memory or entanglement are required For optimal unambiguous discrimination only the POVM is different G. Chiribella, G. M. D’Ariano, and P. P., Phys. Rev. Lett. 101, 180501 (2008).

Discrimination of unitaries What happens for more than two unitaries? What happens for discrimination between sets of unitaries? Quantum computation (e.g. Grover, Deutsh-Jozsa, Simon) Oracle calls U U U U In general quantum memory is required C. Zalka, Phys. Rev. A 60, 2746 (1999)

Conditions for discrimination Discriminability of multiple use channels and more generally combs is determined by optimized testers What are conditions for perfect discriminability? Is optimal discrimination parallel? ★ perfect discriminability C 0 ( I 2 N − 1 ⊗ Ξ ) C 1 = 0 √ √ equivalently | ( I ⊗ C 1 ) | 2 ≥ | ( I ⊗ � � � C 0 | 2 , Ξ )( C 0 + λ Ξ ) ∀ λ ∈ C √ X † X | X | := G. Chiribella, G. M. D’Ariano, and P. P., Phys. Rev. Lett. 101, 180501 (2008).

Sequential discrimination ★ optimal discriminability for combs is not parallel Example: 0 1 2 3 d − 1 p,q | 3 , 2 ⊗ | p, q �� p, q | 1 � | W † � W † p,q � �� ⊗ I 0 , C 0 = d 2 p,q =0 C 1 = | 0 �� 0 | 3 ⊗ I 2 ⊗ I 1 d 2 ⊗ I 0

Operational network distance Existence of non parallel optimal discrimination schemes The proper distance for memory channels must be defined in terms of optimal discriminating testers � � �� � � � � I ⊗ Ξ ( N ) 1 I ⊗ Ξ ( N ) 1 D ( C ( N ) , D ( N ) ) := max ∆ � � � � 2 2 � � � � Ξ ( N ) 1 (17) ∆ := C − D CB-norm distance only accounts for parallel discrimination schemes G. Chiribella, G. M. D’Ariano, and P. P., Phys. Rev. Lett. 101, 180501 (2008).

Covariant estimation of unitaries Covariant unitary estimation problem: A group of unitaries, (Haar-distributed) | U g � �� � U g | A general tester for estimating the group element T h What is the optimal tester? One can prove that the optimal tester is covariant � T = d gT g G ⊗ I ) Θ ( U † ⊗ N T h = ( U ⊗ N [ T, U ⊗ N ⊗ I ) ⊗ I ] = 0 ⇒ h h h

Parallelization 1 1 1 1 2 ( | U g � 2 = ( U ⊗ N 2 | I � 2 ( U † ⊗ N � U g | ) ⊗ N T �� ⊗ I ) T �� � I | T ⊗ I ) T g g Any covariant tester prepares a set of covariant states Any covariant tester is equivalent to a parallel scheme U U U U U U ≃ U U G. Chiribella, G. M. D’Ariano, and P. P., Phys. Rev. Lett. 101, 180501 (2008).

Tomography <f > i P i ρ � � O � = f i ( O ) Tr[ P i ρ ] i The POVM must be informationally complete

Process tomography l ( T ) = � f [ T ] � C P i ρ Tester T i � Tr[ CX ] = f i [ X ] Tr[ T i C ] i The tester must be informationally complete

Optimization Tomogrphy - reconstruction of linear parameters Problem: how to achieve the minimum statistical error? In both cases f i is generally not unique What is the best processing for a fixed POVM/tester? Comparing POVMs/testers with optimal processing What is the optimal POVM/tester?

Optimal processing � P i → Λ : Λ c = c i P i i f [ X ] = Γ ( X ) , ΛΓΛ = Λ Statistical error: | f i [ X ] | 2 Tr[ P i ρ E ] − | � X � | 2 � ∆ ( X ) := E i � � ρ E := p E (d ρ ) ρ g ( ρ ) E := p E (d ρ ) g ( ρ )

Optimal processing � P i → Λ : Λ c = c i P i i f [ X ] = Γ ( X ) , ΛΓΛ = Λ Statistical error: | f i [ X ] | 2 Tr[ P i ρ E ] − | � X � | 2 � ∆ ( X ) := E i � � ρ E := p E (d ρ ) ρ g ( ρ ) E := p E (d ρ ) g ( ρ )

Optimal processing The only term depending on P i and Γ can be written as a norm � | 2 | | f [ X ] | π := f ∗ i [ X ] π ij f j [ X ] i The optimal f i must satisfy π ΓΛ = Λ † Γ † π Solution Γ = Λ ‡ − [( I − Λ ‡ Λ ) π ( I − Λ ‡ Λ )] ‡ π Λ ‡ π ij = δ ij Tr[ P i ρ E ] G. M. D’Ariano and P. P., Phys. Rev. Lett. 98, 020403 (2007).

Optimal process tomography � Tr[ CX ] = f i [ X ] Tr[ T i C ] i Problem: minimum statistical error reconstruction The problem is formally the same as for states the optimal processing can be found in the same way

Optimal tester Figure of merit: weighted sum of errors for a set of expectation values Assumption: the average channel/quantum operation of the ensemble is the totally depolarizing I � C E := p (d C ) C = d out � In this case g ( C ) E := p E (d C ) g ( C )