The Minimax Fidelity for Quantum Channels: Theory and Some Applications Maxim Raginsky University of I!inois V.P. Belavkin, G.M. D Ariano and M. Raginsky Operational distance and fidelity for quantum channels quant-ph/0408159; to appear in J. Math. Phys.
Quantum Systems, States, and Channels Hilbert spaces H, K,... States (density operators): S (H ) = {ρ L (H ) : ρ 0, Tr ρ = 1} Channels (operations): Φ : L (H ) L (K ) (1) linear Φ(αX + βy ) = αφ(x) + βφ(y ) α, β C; X, Y L (H ) (2) trace-preserving Tr Φ(X) = Tr X (3) completely positive (CP): n N, E with dim E = n L (H E ) A 0 = Φ id(a) 0 Channels map states to states, even when acting only on a subsystem: ρ S (H E ) = σ Φ id(ρ) S (K E )
Comparing Quantum States I: The Trace Norm Trace norm: ρ σ 1 = Tr ρ σ = λ i i λ 1, λ 2,... -- eigenvalues of ρ σ X = X X X L (H ) Useful formulas: ϕ ϕ ψ ψ 1 = 2 1 ϕ ψ 2 X 1 = max Tr(XU) U U (H ) Operational interpretation: binary quantum detector P e = 1 2 (1 p 1ρ 1 p 2 ρ 2 1 ) for equiprobable ρ 1, ρ 2 (p 1 = p 2 = 1/2) P e = 1 (1 12 ) 2 ρ 1 ρ 2 1 Pr(ρ 1 ) = p 1 Pr(ρ 2 ) = p 2 1 p 1 minimum average probability of error
Comparing Quantum States II: The Fidelity Fidelity: F (ρ, σ) = Tr ρ 1/2 σρ 1/2 [Uhlmann (1976), Jozsa (1994)] Uhlmann s Theorem: F (ρ, σ) = max ϕ ψ ϕ, ψ ρ = Tr E ϕ ϕ where ϕ, ψ H E are purifications of ρ, σ, i.e. σ = Tr E ψ ψ w.r.t. some auxiliary Hilbert space E (can always take E H ) Remark: equivalently, we may fix some purifications and instead write F (ρ, σ) = max ϕ 1 H U ψ U U (E ) ϕ, ψ H E The fidelity is equivalent to the trace norm: 1 F (ρ, σ) 1 2 ρ σ 1 1 F (ρ, σ) 2
Comparing Channels I: The CB-Norm C (H, K ) -- the set of all channels Φ : L (H ) L (K ) The norm of complete boundedness (CB-norm): Φ, Ψ C (H, K ) Φ Ψ cb = sup n N Properties of the CB-norm: sup ρ S (H C n ) Φ id(ρ) Ψ id(ρ) 1 (1) monotonicity Φ Ψ cb Φ cb Ψ cb (2) tensor-multiplicativity Φ Ψ cb = Φ cb Ψ cb The original definition can be simplified when dim H < : for any CP mapping Λ cb = sup ρ S (H H ) Λ : L (H ) L (K ) Λ id(ρ) 1
Comparing Channels II: The Fidelity? How to define a fidelity for channels analogous to the fidelity for states? The Choi-Jamiolkowski isomorphism: 1,..., d -- a fixed orthonormal basis in H ξ = 1 d n n -- the maximally entangled pure state in H H d n=1 Φ C (H, K ) σ Φ = Φ id( ξ ξ ) S (K H ) A candidate definition of fidelity: F (Φ, Ψ) = F (σ Φ, σ Ψ ) M. Raginsky, A fidelity measure for quantum channels Phys. Lett. A 290, 11-18 (2001)
Pros: Choi-Jamiolkowski Fidelity: Pros & Cons (1) easy to compute -- does not require solving an optimization problem (2) inherits many properties of the Jozsa-Uhlmann fidelity for states (3) operationally related to Quantum System Identification Cons: (1) makes sense only in finite dimension (2) much weaker comparison tool than the CB-norm: (a) 1 1 2 Φ id cb F (Φ, id) 1 1 4 Φ id 2 cb M. Raginsky, Quantum system identification, in Proc. PhysCon 2003, also quant-ph/0306008 (b) can find a sequence of channels Φ n, Ψ n : L (C n ) L (C n ), n 2 such that Φ n Ψ n cb 0 for all n, but lim F (Φ n, Ψ n ) = 0 n (let Φ n (ρ) = U n ρu n, Ψ n (ρ) = V n ρv n, where U n, V n are unitaries chosen so that lim ) n n 1 Tr U nv n = 0
The Minimax Approach To define a fidelity for channels, consider a trade-off between minimizing distinguishability of channels and maximizing distinguishability of their output states! Two Structure Theorems for quantum channels (1) Kraus: for any Φ C (H, K ) there exists a collection {F ν } N ν=1 of operators N N F satisfying F, such that Φ(ρ) = F ν ρf ν ν L (H, K ) ν F ν = 1 H. We can always choose ν=1 N dim H dim K. (2) Stinespring: for any Φ C (H, K ) there exist a Hilbert space E and an isometry F : H K E (i.e., F F = 1 ), such that Φ(ρ) = Tr E F ρf H. We can always choose E so that dim E dim H dim K. Connect the Kraus and the Stinespring representations: N F ψ = F ν ψ ν, ψ H { ν } N ν=1 a basis of E ν=1 ν=1
The Minimax Approach Two fixed channels Φ {F ν } N ν=1, Ψ {G µ } M µ=1 C (H, K ) assume w.l.o.g. that N = M = dim H dim K Two adversaries ALICE -- controls the input space and an isomorphic copy H M -- BOB controls the auxiliary space E (cf. Stinespring theorem) must choose a pure state must choose a unitary η H M to minimize U U (E ) to maximize J(U, η ) = η F µg ν 1 M η µ U ν µ,ν
The Minimax Approach We sha! define the fidelity between the channels Φ, Ψ as the valu% Claim 1: of the Alice-Bob game at the saddle point. inf η H M sup U U (E ) J(U, η ) = J = sup U U (E ) inf η H M J(U, η ) Proof (sketch): by adjusting U e iθ U can work with J(U, η ) = Re µ U ν Tr[(F µg ν 1 M ) η η ] µ,ν J(U, η ) is invariant under phase shifts of η : it is therefore a function of U, η η, and the domain of the 2nd argument can be extended to all density operators ρ S (H M ) : J(U, ρ) Re µ,ν µ U ν Tr[(F µg ν 1 M )ρ] J(U, ρ) : U (E ) S (H M ) R is defined on a product of closed convex sets and is affine in each argument, thus we can interchange the extrema by the von Neumann ninimax theorem. Q.E.D.
The Minimax Approach Claim 2: J = inf ρ S (H M ) F (Φ id(ρ), Ψ id(ρ)) Proof: By concavity of the fidelity for states, suffices to take infimum only over pure states. For a fixed η H M, define the vectors Ω Φ = µ (F µ 1 M ) η µ and Ω Ψ = µ (G µ 1 M ) η µ Then J(U, η ) = Ω Φ 1 K M U Ω Ψ. But Ω Φ, Ω Ψ are purifications of Φ id( η η ), Ψ id( η η ), respectively, so sup J(U, η ) F (Φ id( η η ), Ψ id( η η )) U U (E ) by Uhlmann s theorem. Taking the infimum over all η η concludes the proof. Q.E.D.
The Minimax Fidelity The saddle-point value on the channels J of the Alice-Bob game depends only Φ, Ψ, and can therefore be taken as the minimax fidelity F (Φ, Ψ) between Φ, Ψ. Some alternative forms of the minimax fidelity: F (Φ, Ψ) = F (Φ, Ψ) = inf ρ S (H ) sup {F ν },{G ν } Tr(F ν G ν ρ) = ν sup {F ν },{G ν } inf ρ S (H ) inf sup Tr(F ρg ) = sup inf Tr(F ρg ) ρ S (H ) F,G F,G ρ S (H ) Tr(F ν G ν ρ) ν F (Φ, Ψ) = inf ρ S (H ) Tr K F ρg 1 where: {F ν }, {G ν } are the Kraus (operator-sum) decompositions of Φ, Ψ with the number of terms equal to dim H dim K ; are the Stinespring isometries for with F, G Φ, Ψ dim E = dim H dim K
Comparison with CB-Norm The minimax fidelity is equivalent to the CB-norm distance! 1 F (Φ, Ψ) 1 2 Φ Ψ cb 1 F (Φ, Ψ) 2 Proof: 1 F (Φ id(ρ), Ψ id(ρ)) 1 2 Φ id(ρ) Ψ id(ρ) 1 1 F (Φ id(ρ), Ψ id(ρ)) 2 for all Φ, Ψ C (H, K ), ρ S (H M ). Take the supremum over ρ S (H M ) : 1 inf ρ Since M H, we have that sup Φ id(ρ) Ψ id(ρ) 1 Φ id Ψ id cb, which equals F (Φ id(ρ), Ψ id(ρ)) 1 2 sup ρ Φ Ψ cb ρ Φ (ρ) Ψ id(ρ) 1 1 inf ρ F (Φ id(ρ), Ψ id(ρ))2 by tensor-multiplicativity of the CB-norm. But we also have that inf ρ F (Φ id(ρ), Ψ id(ρ)) F (Φ, Ψ). Putting all these facts together, we obtain the desired statement of equivalence. Q.E.D.
Relation to Choi-Jamiolkowski Fix orthonormal bases { n } dim n=1 H, { α } dim α=1 K in H, K, respectively. Define M ψ = n,α α n ψ α n. Then for every channel Φ C (H, K ) operator, Z L (K H ), such that there exists a unique positive Φ(ρ) = Tr E [(1 K Z)MρM (1 K Z)]. (Belavkin & Staszewski, 1986). Furthermore, Z = (dim H ) σ T 2 Φ, where T 2 denotes the partial transpose w.r.t. the 2nd component of the tensor product: [M.Raginsky, J. Math. Phys. 44, 5003-5020 (2003)]. It is easy to show that F (Φ, Ψ) = (dim H ) Z = n,m Φ( n m ) m n inf ρ S (H ) σ T 2 Φ (1 K ρ) σ T 2 Ψ 1
Application: Quantum Bit Commitment ALICE vs. BOB (1) Commitment: ALICE commits to a choice of bit value b {0, 1} and must provide to BOB a piece of evidence certifying her choice of bit value. (2) Opening: ALICE reveals to BOB her choice b {0, 1}. (3) Verification: BOB checks the bit value announced by ALICE against the piece of evidence in his possession. The protocol must be concealing (i.e., BOB should not be able to determine b from the evidence prior to opening) and binding (i.e., ALICE should not be able to change the value of b after commitment).
Application: Quantum Bit Commitment A general single-step protocol for Quantum Bit Commitment (QBC): (C.0) BOB prepares a pure state η H M (the anonymous state) and hands the H subsystem to ALICE. (C.1) ALICE encodes her choice of bit b {0, 1} by acting onρ A = Tr M η η by a channel Φ b : L (H ) L (K ) and hands the K subsystem to BOB. The channels Φ b, b {0, 1} are assumed fixed via the Kraus decompositions {F ν (b) }, b {0, 1} by the protocol specification. Note: this is given from BOB s point of view as ALICE performs a pure operatio& ρ A F ν (b) ρ A F ν (b), but the actual label ν is unknown to BOB. (O.1) ALICE reveals to BOB the bit value b and the label ν. (V.1) BOB checks the state on state (F (b) 1 M ) η. ν K M in his possession against the claimed
Cheating: ALICE ALICE can try to change her choice of bit value after commitment and to force BOB to accept her new choice. Suppose ALICE initially committed b = 0. She can convince BOB that she committed b = 1 instead by means of modifying the Kraus decomposition {F ν (0) } of the channel Φ 0 with a suitably chosen unitary U on the auxiliary Hilbert space E, i.e. BOB will receive the state ] (F µ (0) 1 M ) η η (F ν (0) 1 M ) U µ ν U η η Tr E [ µ,ν ν ( F (0) ν (U) 1 M ) η η ( F (0) ν (U) 1 M ) where F (0) ν (U) = µ U νµ F (0) µ. The probability of success for this cheating scheme depends on U, η : P A c (U, η ) = ν η F (0) ν (U) F (1) ν 1 M η 2 η F (1) ν F (1) ν 1 M η
Since ALICE does not know the anonymous state η, chosen by BOB, she must make the minimax choice of U, resulting in the probability of cheating We can lower-bound ν Cheating: ALICE How should Alice choose the cheating transformation U? P A c sup U U (E ) P A c η F (0) ν (U) F (1) ν 1 M η 2 η F ν (1) F ν (1) = P A c sup U U (E ) sup U U (E ) inf P c A (U, η ) η H M by means of Jensen s inequality: ( η F ν (0) (U) F ν (1) 1 M η 1 M η ν ( ) 2 inf η F ν (0) (U) F ν (1) 1 M η η H M ν 2 inf η F µ (0) F ν (1) 1 M η µ U ν η H M µ,ν ) 2
Cheating: BOB BOB can cheat by trying to determine b from the state in his possession. Specifically, he can use a binary quantum detector to decide between the states ρ 0 Φ 0 id( η η ) and ρ 1 Φ 1 id( η η ), which are assumed to be equiprobable. BOB s probability of success in cheating is precisely the maximum average probability of correct decision by the detector: Pc B ( η ) = 1 (1 + 12 ) 2 Φ 0 id( η η ) Φ 1 id( η η ) 1 Optimal choice of the anonymous state P B c = 1 2 ( 1 + 1 2 sup η H M η is to maximize this: Φ 0 id( η η ) Φ 1 id( η η ) 1 )
Bad News From the expressions for ALICE s and BOB s optimal cheating probabilities: P c A F (Φ 0, Φ 1 ) 2 P c B = 1 (1 + 12 ) 2 Φ 0 Φ 1 cb Using the equivalence between the minimax fidelity and the CB-norm distance, we obtain the following inequality: P A c 4(1 P B c ) 2 That is -- a QBC protocol which is perfectly concealing (i.e., P c B = 1/2, BOB can do no better than pure guessing) is not binding with probability 1 : P c A = 1, (i.e., ALICE can cheat perfectly). However, this applies only to QBC protocols consisting of a single step and no) a!owing ALICE or BOB to abort. Security of more elaborate protocols is sti! being debated.
Acknowledgments & Thanks U.S. Army Research Office for support under Multiple Universities Research Initiative (MURI) Grant No. DAAD19-00-1-0177 Ministero Italiano dell Universita e della Ricerca (MIUR) for support under Cofinanziamento 2002 European Science Foundation for travel funds Quantum Information Theory Group (QuIT) at University of Pavia, Italy for kind hospitality Arnold and Mabel Beckman Foundation for current support under the Beckman Institute Fellowship Quantum Information and Computation Group at UC Berkeley for the invitation