JOURNAL OF MATHEMATICAL PHYSICS 46, 062106 ͑2005͒
Operational distance and fidelity for quantum channels ͒ Viacheslav P. Belavkina Department of Mathematics, University of Nottingham, NG7 2RD Nottingham, United Kingdom ͒ Giacomo Mauro D’Arianob QUIT Group, INFM Unità di Pavia, Università di Pavia, Dipartimento di Fisica “A. Volta,” via Bassi 6, I-27100 Pavia, Italy and Center for Photonic Communication and Computing, Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208-3118 ͒ Maxim Raginskyc Center for Photonic Communication and Computing, Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208-3118 ͑Received 4 November 2004; accepted 14 March 2005; published online 23 May 2005͒ We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras ͑explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski͒ and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, pre- cisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fi- delity ͑“generalized transition probability”͒ of Uhlmann, is topologically equivalent to the trace-norm distance. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1904510͔
I. INTRODUCTION Many problems in quantum information science,1,2 both in theory and in experiment, involve finding a set of quantum-mechanical states or channels that solve some sort of an optimization problem, typically formulated in terms of a numerical criterion that measures how close a given pair of states or operations are to each other. ͑Many such criteria have been proposed to date, each defined with specific theoretical or experimental considerations in mind; see Ref. 3 for a recent comprehensive survey.͒ Let us first consider the case of quantum states, i.e., density operators. Let h be a complex separable Hilbert space associated to a quantum-mechanical system. Given a pair of density operators ,, i.e., positive trace-class operators with unit trace, one can use either the fidelity4–7
͑ ͒ ͓͑1/21/2͒1/2͔͑͒ F , ª Tr 1 or the trace-norm (half-) distance
͒ a E-mail address: [email protected] ͒ b Author to whom correspondence should be addressed. E-mail address: [email protected] ͒ c E-mail address: [email protected]
0022-2488/2005/46͑6͒/062106/23/$22.5046, 062106-1 © 2005 American Institute of Physics
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D͑,͒ 1 ʈ − ʈ , ͑2͒ ª 2 T ʈʈ ͉͉ ͉͉ ͑†͒1/2 8,9 where TªTr and ª . Loosely speaking, two states and are close to each other if F͑,͒ is large, or if D͑,͒ is small. In fact, as follows from the key inequality5,10
1−F͑,͒ ഛ D͑,͒ ഛ ͱ1−F2͑,͒, ͑3͒ the fidelity and the trace-norm distance are equivalent in the sense that any two density operators that are close to one another in the sense of ͑1͒ are also close in the sense of ͑2͒, and vice versa. As for quantum channels, i.e., normal completely positive unital mappings from an operator algebra B=B͑h͒ into another algebra A=B͑g͒, where g and h are complex separable Hilbert spaces, things get somewhat complicated. Consider, for instance, the case when g is finite- ͕͉ ͖͘m ͉͘ −1/2͚m ͉ ͘ dimensional, and let mªdim g. Fix an orthonormal basis j j=1 of g, and let ªm j=1 j ͉j͘ be the normalized maximally entangled vector in the product space g g. Given two quan- tum channels ⌽,⌿:B→A, one can measure their closeness in terms of the fidelity of the states on B A, obtained from the maximally entangled state =͉͉͗͘ by applying the predual channels ⌽ ⌿ ͑ ͒ T and T cf. Sec. II for precise definitions to the first factor in the tensor product:
m m 1 ⌽ ͑͒ ⌽ ͉͑ ͗͘ ͉͒ ͉ ͗͘ ͉ϵ T id = ͚ ͚ T i k i k , m i=1 k=1
m m 1 ⌿ ͑͒ ⌿ ͉͑ ͗͘ ͉͒ ͉ ͗͘ ͉ϵ T id = ͚ ͚ T i k i k . m i=1 k=1 The fidelity F͑,͒, taken as the channel fidelity
F͑⌽ ⌿͒ ͑⌽ ͑͒ ⌿ ͑͒͒ ͑ ͒ , ª F T id , T id , 4 by Raginsky in Ref. 11, enjoys many properties parallel to those of the fidelity ͑1͒ for quantum states. Alternatively, one can adopt the ͑half-͒ distance1,12,13
D͑⌽,⌿͒ 1 ʈ⌽ − ⌿ʈ , ͑5͒ ª 2 CB ʈ ʈ ͑ where · CB denotes the so-called norm of complete boundedness or CB-norm for short; cf. Sec. II C for details͒. We note that the CB-norm half-distance ͑5͒ can be given in terms of the trace- norm distance ͑2͒ between density operators by means of the variational expression1,12,13
D͑⌽ ⌿͒ ͑⌽ ͑͒ ⌿ ͑͒͒ ͑ ͒ , = sup D T id , T id , 6
where the supremum is taken over all density operators on the tensor product space g g.By analogy with density operators of the states, we are tempted to say that two quantum channels, ⌽ and ⌿, are close either if F͑⌽,⌿͒ is large or if D͑⌽,⌿͒ is small. However, in addition to the finite-dimension restriction dim gϽϱ ͓the only case under which the definition ͑4͒ of the channel fidelity makes sense͔, we encounter the following difficulty. It turns out11 that, as a criterion of closeness, the CB-norm distance ͑5͒ is strictly stronger than the fidelity measure ͑4͒ in the sense that even when D͑⌽,⌿͒ is large, F͑⌽,⌿͒ may be quite large as well, and may even become equal to one in the limit dim g→ϱ. Consider, for instance, the case ⌿=id. Then one can show11 that
1−D͑⌽,id͒ ഛ F͑⌽,id͒ ഛ ͱ1−͑1/4͒D2͑⌽,id͒, ͑7͒ and we immediately see that when ⌽ is such that D͑⌽,id͒ attains its maximum value of unity, the fidelity F͑⌽,id͒ is still bounded between 0 and ͱ3/2. To make matters worse, the only bound on ͑5͒ in terms of ͑4͒ known so far is
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1 ജ D͑⌽,⌿͒ ജ 1−F͑⌽,⌿͒, ͑8͒ ͑ ͒ ͑ ͒ ͕⌽ ͖ ͕⌿ ͖ as follows readily from Eqs. 3 and 6 . Furthermore, one can easily find sequences m , m ⌽ ⌿ B͑ m͒→B͑ m͒ D͑⌽ ⌿ ͒Þ of channels m , m : C C , such that m , m 0 for all m, while F͑⌽ ⌿ ͒ lim m, m =1. m→ϱ Indeed, consider the unitarily implemented channels
⌽ ͑ ͒ † ⌿͑ ͒ † m B = UmBUm, B = VmBVm Þ with the unitaries Um ,Vm chosen in such a way that Um Vm but
1 ͑ † ͒ lim Tr UmVm =1. m→ϱ m Thus, the channel fidelity ͑4͒, apart from being applicable only in finite-dimensional settings, has the distinct disadvantage of not being equivalent to the CB-norm distance, in contrast to the case of the Uhlmann fidelity ͑1͒ and the trace-norm distance ͑2͒ on the state space of a quantum- mechanical system. The goal of this paper is to define and study a new fidelity criterion for quantum channels, which we term the minimax fidelity and which is a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Unlike the channel fidelity ͑4͒ of Ref. 11, the minimax fidelity is not only well defined for channels between finite- dimensional algebras, but also applies to a certain class of channels between infinite-dimensional algebras ͑explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski14͒ and is equivalent to the CB-norm distance, echoing the way the Uhlmann fidelity ͑1͒ for density operators is equivalent to the trace-norm distance ͑2͒. Apart from these technical features, the minimax fidelity f͑⌽,⌿͒ between two quantum channels ⌽,⌿ has a direct operational meaning: intuitively, it is defined as the minimum overlap ͑ ͒ ⌽ ⌿ of output states density operators of the predual channels T , T, when the operator-sum decompositions2 of the latter are chosen to be maximally overlapping; this is spelled out in precise terms in Sec. IV E. Our central result ͑Theorem 1͒ demonstrates that the minimax fidelity is independent of the order of these two optimizations. Furthermore, the equivalence of our minimax fidelity to the CB-norm distance, which is stated precisely in Sec. V in terms of dimension-free bounds, is a promising avenue for the study and characterization of dimension-free bounds ͑when- ever they exist͒ on other operationally meaningful distance measures for quantum operations3 in terms of the CB-norm distance. As pointed out in Ref. 15, such bounds are crucial for a successful generalization of the usual quantum capacity of a channel1,2 ͑i.e., with respect to the identity channel͒ to the case of comparing quantum channels to an arbitrary reference channel. We plan to pursue these matters further in a future publication. The paper is organized as follows. In Sec. II we fix the definitions and notation used through- out the paper. The minimax fidelity is then introduced in Sec. III. Section IV is devoted to the evaluation of the minimax fidelities in the various mathematical settings that arise in quantum information theory. Next, in Sec. V, we list key properties of the minimax fidelity. Finally, in Sec. VI we sketch some example applications of the minimax fidelity to several problems of quantum information theory.
II. PRELIMINARIES, DEFINITIONS, NOTATION A. Pairings, states, operations Let h be a complex separable Hilbert space; let B denote the Banach algebra B͑h͒ of all ʈ ʈ B bounded linear operators on h with the usual operator norm · ; and let T denote the Banach B ͑ ͒ ʈ ʈ B space T h of trace-class operators on h with the trace norm · T. The set of normal states on ,
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i.e., ultraweakly continuous positive unital linear functionals on B, will be denoted by S͑B͒ or, whenever we need to exhibit the underlying Hilbert space explicitly, by S͑h͒. Generic elements of S͑B͒ will be denoted by the stylized Greek letters ,%,. Note that the operator norm on B can be written as ʈBʈ=sup͕%͉͑B͉͒:%S͑B͖͒. We equip h ͑and shall equip all Hilbert spaces introduced in the sequel͒ with an isometric † 2 involution J=J , J =1h, having the properties of complex conjugation,
∀ J͚ j j = ͚ jJ j, j C, j h. j j
B ˜ † 16 We can thus define the transpose of any B as BªJB J, as well as introduce the trace pairing
͑ ͒ ͑ ˜͒ ͑˜ ͒ ∀ B B ͑ ͒ B, ª Tr B =Tr B , B , T 9 B B B of and T. Under this pairing, which differs from the usual one in that B is paired with the B B transpose of T rather than directly with , normal linear functionals on are in a one-to-one B correspondence with the elements of T. Thus to each normal state % we associate a unique positive trace-class operator with unit trace, denoted by the standard Greek letter and referred to as the density operator corresponding to %, via %͑B͒=͑B,͒ for all BB. Similarly, density operators corresponding to states denoted by and will be denoted by and , respectively. Apart from natural arguments from standard representation theory of operator algebras, one reason why we chose to pair B with the transposed operator ˜=J†J, rather than with ,istobe able to keep all notations conveniently parallel to the classical ͑commutative͒ case, as will be amply demonstrated throughout the paper. Note also that we can fix a complete orthonormal basis ͕͉j͖͘ of h and express the pairing ͑9͒ in terms of the matrix elements of B and as
͑ ͒ ͗ ͉ ͉ ͘ ͗ ͉͉ ͘ϵ jk B, = ͚ j B k · j k ͚ Bjk , j,k j,k where we have used the covariant indices for the matrix elements of bounded operators in B and B the contravariant indices for the matrix elements of trace-class operators in T, when the latter are identified via the pairing ͑9͒ with normal linear functionals on B. Yet another reason to opt for the pairing of B with the transposed operator ˜, further elaborated upon in Sec. II B, is that then the density operator of a normal state % will coincide with the operational density of %, understood as a quantum operation from B into the Abelian algebra C. A B͑ ͒ A B ͑ ͒ Introducing another Hilbert space g, the algebra ª g , and the trace class Tª T g , let us consider quantum operations, i.e., the completely positive normal linear mappings ⌽:B→A ⌽͑ ͒ഛ ⌽͑ ͒ ⌽ such that 1h 1g;if 1h =1g, then is referred to as a quantum channel. Any quantum ⌽ ⌽ A →B ⌽ operation possesses a unique predual T : T T, defined as the transpose of with respect to the trace pairing ͑9͒, i.e.,
͑⌽͑ ͒ ͒ ͑ ⌽ ͑͒͒ ∀ B A ͑ ͒ B , = B, T , B , T. 10 ⌽ A →B ⌽͑͒ Conversely, given a normal completely positive linear map : T T such that Trh ഛ A ͑ ͒ Trg for all T, we define its dual with respect to the trace pairing 9 as the unique mapping ⌽T :B→A for which
͑ ⌽͑͒͒ ͑⌽T͑ ͒ ͒ ∀ B A ͑ ͒ B, = B , , B , T. 11 ⌽T ⌽ Using these definitions, one readily obtains that T = for any normal completely positive map ⌽:B→A. Alternatively, one may define the predual of a normal completely positive map ⌽:B →A ⌽ A →B ⌽T ⌽ as the unique normal completely positive map T : T T such that T = . ⌽ 17 ⌽͑ ͒ ⌺ † If is given in the Kraus form B = Fj BFj, or more generally as an integral
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⌽͑B͒ = ͵ F͑z͒†BF͑z͒d͑z͒, ͑12͒ Z ͑ B ͒ with respect to a positive measure on a measurable space Z, Z , where the integration is 18 ⌽ understood in the sense of Bochner, then the predual map T has the transposed integral form
⌽ ͑͒ ͵ ͑ ͒† ͑ ͒ ͑ ͒ T = FT z FT z d z , Z ͒ ͑ ͒ ͑ ͉͗ۋ͉ ͑ ͒ ͗ۋ where g FT z are Hilbert-transposed to the operators h F z , that is FT z g =F͑z͒ for all zZ. Any normal state %S͑B͒ is automatically a quantum channel from B into the Abelian algebra C, and it is readily seen that the density operator of %, understood as acting on C on →B B ۋ the right, C , is precisely the predual %T :C T. Indeed, given B and C, we have ͑ ͑ ͒ ͒ ͑ ͒ ͑ ͑͒͒ % B , = B, = B,%T , T T which proves our claim that =%T. Thus we also have that %=%T = .
B. Operational densities In order to avoid technicalities involving unbounded operators, we shall henceforth assume that all quantum operations we deal with are completely majorized by the trace, considered as the ͑͒ B A B͑ ͒ 14 Ͼ map =1gTr of T into = g , in the sense that there exists a constant 0 such that the ⌽ B →A difference − is a completely positive map T . For example, this condition is satisfied by all quantum operations between finite-dimensional algebras.19 As was proven in Ref. 14, in this ⌽ H case there exists a unique positive operator on the Hilbert space ªg h, called the density of ⌽ with respect to the trace , such that
⌽͑ ͒ ͓͑ ˜ ͒⌽ ͔ ͑ ͒ B =Trh 1g B , 13 B͑H͒ where Trh Y, Y , denotes the partial trace of Y with respect to h,
͑ ͒ ͑ ͒ ∀ B ͑ ͒ TrhY, = Y, 1h , T g .
Moreover, ⌽ as a linear operator on H is bounded and majorized by :0ഛ⌽ ഛ1H, and the ⌽͑ ͒ ͓ ⌽͑ ͒ഛ ͔ ⌽ ͑ ⌽ ഛ ͒ operation is unital, 1h =1g contractive, 1h 1g if and only if Trh =1g Trh 1g . This ⌽ A →B ͑ ͒ ͑ ͒ is equivalent to saying that the predual map T : T T, which, using Eqs. 10 and 13 , can be written as
⌽ ͑͒ ͓⌽ ͑ ͔͒ ͑ ͒ T =Trg ˜ 1h , 14 is trace preserving ͑trace decreasing͒. As an example, consider a normal state % on B, which, being a quantum channel into C, satisfies the complete majorization condition with =ʈʈ, where is the density operator of %. Furthermore, it is easy to see that % =. Indeed, we can write
͑ ͒ ͑ ͒ ͑ ͒ ͑˜ ͒ ͓͑ ˜ ͔͒ % B = B, =Tr B˜ =Tr B =Trh 1C B , and the desired result follows upon comparing this with Eq. ͑13͒. This provides additional justi- ͑ ͒ fication for our definition of the trace pairing in Eq. 9 , since we then have that %T = =% for any normal state %. If the operation ⌽:B→A is given in the generalized Kraus form ͑12͒, we can write down its operational density ⌽ explicitly. To this end, suppose that all operators F͑z͒ are determined by generalized bra-vectors ⌫͑z͒=͑F͑z͉͒, densely defined as the linear functionals
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⌫͑z͉͒ ͘ = ͉͗F͑z͉͒͘ϵ͑F͑z͉͉͒ ͒͘
on the linear span of the ket-vectors ͉ ͘ϵ˜ in H=g h, where g is also treated as a bra-vector such that J=͉͗ and ͉͘=˜. Then the operational density ⌽ of ⌽ is given by the corresponding decomposition
⌽ ͵ ⌫͑ ͒†⌫͑ ͒ ͑ ͒ϵ⌫†⌫ ͑ ͒ = z z d z , 15
where the integral is, again, understood in the sense of Bochner.
C. Completely bounded maps Completely positive linear maps between operator algebras are a special case of completely 20 bounded maps. Consider, as before, the algebras B=B͑h͒ and A=B͑g͒. For each nN define M ͑B͒ӍB M M ϫ the nth matrix level n n, where n denotes the algebra of n n matrices with M ͑B͒ ϫ B complex entries. That is, n is the space of n n matrices with -valued entries, M ͑B͒ ͕͓ ͔ B ഛ ഛ ͖ n ª Bij :Bij ,1 i, j n . A M ͑A͒ Analogous construction can also be applied to to yield the matrix levels n . Each matrix M ͑B͒ B level n inherits a *-algebra structure from through
n ͓ ͔͓ ͔ ͚ͫ ͬ ͓ ͔† ͓ † ͔ Bij Cij ª BikCkj , Bij ª Bij . k=1 M ͑B͒ B͑ ͑n͒͒ In fact, by identifying n via a natural *-isomorphism with the algebra h of bounded ͑n͒ M ͑B͒ * linear operators on h , the direct sum of n copies of h, one can make n into a C -algebra. Thus, each matrix level of B possesses a unique C*-norm. ⌳ B→ ⌳͑n͒ ⌳ M ͑B͒ Now, for any n N a linear map : A induces the map ª idn from n into ͑ ͔͒ ͑⌳͓ۋ͔ ͓ M ͑A͒ ⌳͑n͒ n , defined by : Bij Bij . Let us define the norm of complete boundedness or ͒ ʈ⌳ʈ ͕ʈ⌳͑n͒ʈ ͖ CB-norm by CBªsup :n N , where ʈ⌳͑n͒ʈ sup ʈ⌳͑n͒͑B͒ʈ ª M ͑B͒ ʈ ʈഛ B n , B 1 is the usual operator norm of ⌳͑n͒. A linear map ⌳:B→A is called completely bounded if ʈ⌳ʈ Ͻϱ ⌽ B→A CB . Every completely positive map : is automatically completely bounded, with ʈ⌽ʈ ʈ⌽͑ ͒ʈ ⌳ ʈ⌳͑ ͒ʈഛʈ⌳ʈ CB= 1h . For a general completely bounded map , one has, by definition, 1h ഛʈ⌳ʈ CB. ⌳ A →B Passing to the predual map T : T T, we can similarly define induced maps ⌳͑n͒ M ͑A ͒→M ͑B ͒ T : n T n T , n N, and the predual CB-norm ʈ⌳ʈT ʈ⌳͑n͒ʈ CB ª sup T, nN where
ʈ⌳͑n͒ʈ ʈ⌳͑n͒͑͒ʈ T = sup T. ª M ͑A ͒ ʈʈ ഛ n T : T 1 ʈ⌳͑n͒ʈ ʈ⌳͑n͒ʈ ʈ⌳ʈ ʈ⌳ ʈT It is easy to see that = T T for all n N, so that CB= T CB. It is also straightforward ʈ ʈ ʈ ʈ ͑ ʈ⌳ ⌳ ʈ to see that the “unstabilized” norms · and · T are tensor supermultiplicative i.e., 1 2 ജʈ⌳ ʈʈ⌳ ʈ͒ ͑ ʈ⌳ ⌳ ʈ 1 2 , whereas the corresponding CB-norms are tensor multiplicative i.e., 1 2 CB ʈ⌳ ʈ ʈ⌳ ʈ ͒ = 1 CB 2 CB . There is also a useful nonvariational formula for the CB-norm of a map ⌳:B→A. Namely, let ᐉ2 denote the Hilbert space of square-summable infinite sequences of complex numbers, and let K͑ᐉ2͒ ᐉ2 ʈ⌳ʈ ʈ⌳ ʈ denote the space of compact operators on . Then CB= idK͑ᐉ2͒ . Since we have
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assumed that B=B͑h͒ with h a complex separable Hilbert space, and since all complex separable ᐉ2 ʈ⌳ʈ ʈ⌳ ʈ Hilbert spaces are canonically isomorphic to , we may also write CB= idK͑h͒ .
D. Miscellany Any positive operator BB͑h͒ has a unique positive square root, denoted by B1/2 and defined as the positive operator XB͑h͒ such that B=X2. This definition can be extended to any operator A that is similar to a positive operator ⌬B͑h͒, in the sense that there exists an operator S B͑h͒ such that A=S⌬S+, where S+ is the pseudoinverse of S, equal to S−1 on ran S and to 0 on ͱ ⌬1/2 + ker S. In that case, we may define AªS S . From now on, in order to distinguish this extended definition of the square root from the usual one, we shall always use the square root symbol ͱ· for this extended definition, and reserve the exponent notation ·1/2 for the usual defini- tion. Consider now two positive operators A,BB͑h͒. It is easy to see that their product AB is similar to A1/2BA1/2 with S=A1/2. Note that the operator AB is positive when restricted to the closure of ran A, when the latter is equipped with the weighted inner product ͉͗͘ ͗ −1/2͉ −1/2͘ A ª A A : ͉͗ ͘ ͗ −1/2͉ −1/2 ͘ ͉͗ ͘ ജ ∀ AB A = A A AB = B 0, ranA. ͱ ͑ 1/2 1/2͒ + 1/2 Thus we may define ABªS A BA S with S=A . This notation, again, allows for a convenient parallelism between the classical ͑commutative͒ formalism and the quantum ͑noncommutative͒ one. Indeed, consider two mutually commuting positive trace-class operators , , let ͕͉x͖͘ denote the set of their common eigenvectors, and let ϵ͗ ͉͉ ͘ ϵ͗ ͉͉ ͘ ͱ x x x , x x x denote the corresponding eigenvalues. Then is also trace-class, and
ͱ ͱ Tr = ͚ x x. x ͕ ͖ ͕ ͖ ͱ If Tr =1=Tr , then Pª x and Qª x are probability distributions, and Tr then gives the classical fidelity ͑also known as the Bhattacharyya coefficient͒10 F͑P,Q͒ between P and Q. Our main technical tool in this paper is given by the following: Lemma 1: Let H be a complex separable Hilbert space, and let R, SB͑H͒ be positive operators such that R1/2SR1/2 is trace class. Then the supremum
sup ͕Tr͑X†Y + Y†X͒:X†X = R,Y†Y = S͖ =2TrͱRS ͑16͒ X,YB͑H͒ B͑H͒ † 1/2 is achieved on any X satisfying the condition X X=R, say X=R , and Y =Yo satisfying the equation
† ͑ †͒1/2 † ͑ ͒ YoX = XSX = XYo. 17 Proof: To prove the lemma one can use either the polar decomposition or the method of Lagrange multipliers. We shall use the latter. Fixing an X satisfying X†X=R, we can write the Lagrange function as
L =Tr͑X†Y + Y†X − Y†YL͒, where L=L† B͑H͒ is the operator-valued Lagrange multiplier corresponding to the hermiticity condition S=Y†Y =S†. At the stationary point
␦L =Tr͑X† − LY†͒␦Y + ͑X − YL͒␦Y† =0, ͑ † † † † so Y =Yo must satisfy the equation YL=X the other equation, LY =X , corresponding to Y =Yo, ͒ −1 −1 is obtained by taking the Hermitian adjoint . Thus Yo =XL , where L should be determined from L−1X†XL−1=S. Multiplying this on the left by X and on the right by X† yields ͑XL−1X†͒2 =XSX†,or
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−1 † ͑ †͒1/2 † ͑ †͒1/2 † XL X = XSX . Thus, we indeed have that YoX = XSX =XYo, and therefore that
͑ † †͒ ͑͑ †͒1/2͒ ͑ ͒ Tr YoX + XYo =2Tr XSX . 18 This extremal value is precisely the maximal value due to convexity of the function being maxi- mized in Eq. ͑16͒. Note that, since ͑U†XSX†U͒1/2=U†͑XSX†͒1/2U for any unitary U, the value of the supremum in Eq. ͑16͒, which coincides with Eq. ͑18͒, does not depend on the choice of X satisfying X†X=R. Indeed, by virtue of the polar decomposition X=UR1/2,
2Tr͑XSX†͒1/2 =2Tr͑U†͑XSX†͒1/2U͒ =2Tr͑͑R1/2SR1/2͒1/2͒. ͑ † ͒ Rewriting this trace in the equivalent form 2 Tr X Yo with
† 1/2͑ 1/2 1/2͒1/2 −1/2 ϵ ͱ X Yo = R R SR R RS corresponding to X=R1/2, we obtain the extremal value in Eq. ͑16͒. ᮀ We shall also need the following simple, but useful, result: Lemma 2: Let S be a compact subset of a complex Banach space V, such that xS implies xS for all C with ͉͉=1. Let f :V→C be a continuous function which is homogeneous of order 1, i.e., f͑x͒=f͑x͒ for all C and all xV. Then
sup͉f͑x͉͒ = sup Re f͑x͒. ͑19͒ xS xS * ͉ ͑ *͉͒ ͉ ͑ ͉͒ ͑ *͒ ͉ ͑ *͉͒ iargf͑x*͒ ** Proof: Let x S be such that f x =supxS f x , with f x = f x e . Let x −i arg f͑x*͒ * ªe x . By the homogeneity of f,
* f͑x**͒ =e−i arg f͑x ͒f͑x*͒ = ͉f͑x*͉͒.
But then ͉f͑x**͉͒= f͑x**͒=Re f͑x**͒. Since Re ഛ͉͉ for all C, the lemma is proved. ᮀ
III. OPERATIONAL FIDELITIES AND DISTANCES
A. Classical kernel fidelity The fidelity distinguishing different quantum operations without the restriction on the Hilbert space dimensionality was suggested by Belavkin in Ref. 21 on the basis of a noncommutative generalization of the maximal Hellinger distance between two positive kernels. Namely, given a ͑ B ͒ locally compact space X and a measure space Y , Y , , where is a positive measure, let us A C͑ ͒ B ϵC ͑ ͒ denote by the algebra X of bounded continuous functions on X, and by T T Y the space of absolutely -integrable complex functions on Y.Apositive kernel P is then given in terms of ͑ ͉ ͒ ϫ → + ͑ ͉ ͒B ͐ ͑ ͉ ͒ ͑ ͒ a function p · · :Y X R , such that Px ªp · x T for all x X, while Pª Yp y · d y A. Given two positive kernels P and Q, the squared pointwise Hellinger distance
1 1 d2 ͑P ,Q ͒ ͵ ͑ͱp͑y͉x͒ − ͱq͑y͉x͒͒2d͑y͒ = ͵ͫ ͑p͑y͉x͒ + q͑y͉x͒͒ − ͱp͑y͉x͒q͑y͉x͒ͬd͑y͒ H x x ª 2 2 ͑20͒ is well defined and finite for each xX, so that we can define
1 2 ͑ ͒ ͵ ͑ͱ ͑ ͉ ͒ ͱ ͑ ͉ ͒͒2 ͑ ͒ϵʈ 2 ͑ ͒ʈ ͑ ͒ dH P,Q sup p y x − q y x d y dH Px,Qx , 21 ª 2 xX 2 ͑ ͒ the last expression indicating the fact that dH P,Q is given by the supremum of the squared ͑ ͒ 2 ͑ ͒ pointwise Hellinger distance 20 over all x X. Note that the squared Hellinger distance dH P,Q between two positive distributions P=p͑·͒ and Q=q͑·͒ is the minimal mean quadratic distance
Downloaded 25 Aug 2008 to 152.3.194.16. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp 062106-9 Fidelity for quantum channels J. Math. Phys. 46, 062106 ͑2005͒
1 2 ͑ ͒ ͭ͵ ͉͑ ͒ ͑ ͉͒2 ͑ ͒ ͉͑ ͉͒2 ͑ ͒ ͉͑ ͉͒2 ͑ ͒ͮ dH P,Q = inf y − y d y : · = p · , · = q · 2 ,C͑Y͒
͑ 1 ͑ ͒͒ ͵ ͱ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ = 1, 2 P + Q − sup p y Re y d y , 22 :͉͑·͉͒2=q͑·͒ ͑ ͒ ͐ ͑ ͒ ͑ ͒ ͑ ͒ C͑ ͒ C ͑ ͒ where f , P = f y p y d y denotes the integral pairing of f Y with P T Y . The relative fidelity ͱ 1 ͱ ͑1, PQ͒ f͑P,Q͒ = ͱ sup ͵ p͑y͒ Re ͑y͒d͑y͒ = ͱ ͑23͒ ͑1,P͒͑1,Q͒ :͉͑·͉͒2=q͑·͒ ͑1,P͒͑1,Q͒ of the distributions P and Q is obviously related to the distance ͑22͒ by
2 ͑ ͒ ͱ͑ ͒͑ ͒ ͑ ͒ ͑ 1 ͑ ͒͒ ͑ ͒ dH P,Q + 1,P 1,Q f P,Q = 1, 2 P + Q . 24 ͑ ͉ ͒ ͑ ͉ ͒ ͑ ͒ If Px ªp · x and Qx ªq · x are conditional distributions with constant integrals 1,Px and ͑ ͒ 1,Qx , e.g., normalized to unity, this relation also remains valid for the minimal fidelity ͑ ͒ ͑ ͒ f P,Q = inf f Px,Qx , xX which can alternatively be defined by the minimax formula
͑ ͱ ͑ ͉ ͒͒ 1, Px Re · x f͑P,Q͒ = inf sup ͱ , ͑25͒ xX ͉͑ ͉ ͉͒2 ͑ ͒ ͑ ͒͑ ͒ : · x =Qx · 1,Px 1,Qx where the supremum is achieved on ͑·͉x͒ϵؠ͑·͉x͒ satisfying ͑y͉x͒=ͱq͑y͉x͒. In particular, if P ͑ ͒ ͑ ͒ and Q are probability kernels, 1,Px =1= 1,Qx for all x X, then
2 ͑ ͒ ͵ ͱ ͑ ͉ ͒ ͑ ͉ ͒ ͑ ͒ϵ ͑ ͒ dH P,Q =1−inf p y x q y x d y 1−f P,Q , xX where
͑ ͒ ͵ ͱ ͑ ͉ ͒ ͑ ͉ ͒ ͑ ͒ϵ ͑ ͱ ͒͑͒ f P,Q = inf p y x q y x d y inf 1, PxQx 26 xX xX is the minimax fidelity of the classical channels described by these kernels.
B. Quantum operational fidelity Generalizing Eq. ͑21͒, one can define the squared Hellinger distance between quantum opera- tions ⌽ and ⌿ with the respective operational densities ⌽ ,⌿ B͑H͒, H=g h,as
2 ͑⌽ ⌿͒ 1 ͕ʈ ͑⌫ ⌼͒†͑⌫ ⌼͒ʈ ⌫†⌫ ⌽ ⌼†⌼ ⌿ ͖ ͑ ͒ dH , = 2 inf Trh − − : = , = . 27 ⌫,⌼B͑H͒
† † The operators ⌫,⌼B͑H͒, such that ⌫ ⌫=⌽ and ⌼ ⌼=⌿, are naturally thought of as the purifications of ⌽ and ⌿, respectively. This means that we can fix an orthonormal basis ͕͉j͖͘ of H, say the product basis ͉j͘=͉i͘ ͉k͘ϵ͉i,k͘, where ͕͉i͖͘ and ͕͉k͖͘ are some fixed orthonormal bases of g and h, respectively, and represent any such ⌫ and ⌼ as strongly convergent sums
⌫ ͉ ͗͘ ͉⌫ ϵ ͉ ͑͘ ͉ ⌼ ͉ ͗͘ ͉⌼ ϵ ͉ ͑͘ ͉ ͑ ͒ = ͚ j j ͚ j Fj , = ͚ j j ͚ j Vj , 28 j j j j ͑ ͉ → where the generalized bra-vectors Fj define the bounded operators Fj ,Vj :g h through
Downloaded 25 Aug 2008 to 152.3.194.16. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp 062106-10 Belavkin, D’Ariano, and Raginsky J. Math. Phys. 46, 062106 ͑2005͒
͗ ͉ ͉ ͘ ͑ ͉͉͑ ͘ ͉ ͒͘ ͗ ʈ⌫͉ ͘ ͗ ͉ ͉ ͘ ͑ ͉͉͑ ͘ ͉ ͒͘ ͗ ʈ⌼͉ ͘ k Fj i = Fj i k = j i,k , k Vj i = Vj i k = j i,k . aF+bG. UsingۋF is linear: ͑aF+bG͉ۋAs seen directly from this definition, the mapping ͑F͉ Eq. ͑28͒, we may write
⌽ ͉ ͒͑ ͉ϵ⌫†⌫ ⌿ ͉ ͒͑ ͉ϵ⌼†⌼ ͑ ͒ = ͚ Fj Fj , = ͚ Vj Vj , 29 j j where the sums converge in the strong operator topology. This determines the Kraus 17 ⌽͑ ͒ ͚ † ⌿͑ ͒ ͚ † ⌽ ⌿ B→A decompositions B = jFj BFj, B = jVj BVj of the maps , : . Analogously, upon defining the mappings F,V:g→h H by
͉ ͘ ͉ ͘ F ª ͚ Fj j , V ª ͚ Vj j , j j
22 † † we can write the maps ⌽,⌿ in the Stinespring form as ⌽͑B͒=F ͑B 1H͒F and ⌿͑B͒=V ͑B 1H͒V. ʈ † ʈ ͑ † ͒ Taking into account the fact that A A =sup%S͑g͒% A A and defining the positive function ͑ ͒ B͑H͒ ϫ B ͑ ͒ → c · ; · : T g R,
c͑A;͒ 1 Tr͑A͑ ͒A†͒, ª 2 1h we can rewrite the fidelity distance ͑27͒ in the following minimax form:
2 ͑⌽ ⌿͒ ͑⌫ ⌼ ͒ ⌫†⌫ ⌽ ⌼†⌼ ⌿ ͑ ͒ dH , = inf sup c − ; : = , = . 30 ⌫,⌼B͑H͕͒%S͑g͒ ͖ On the other hand, generalizing Eq. ͑20͒ to quantum operations, we can define the squared pointwise distance
2 ͑⌽ ⌿͒͑ ͒ ͕ ͑⌫ ⌼ ͒ ⌫†⌫ ⌽ ⌼†⌼ ⌿ ͖͑͒ dH , % inf c − ; : = , = 31 ª ⌫,⌼B͑H͒ between ⌽ and ⌿ on the set S͑g͒ of all normal states on A=B͑g͒. Just as with the probability 2 ͑⌽ ⌿͒ kernels in the commutative setting described in the preceding section, dH , coincides with the 2 ͑⌽ ⌿͒͑ ͒ S͑ ͒ ⌽ ⌿ ͑ supremum of dH , % over all normal states % g whenever and are proportional to͒ quantum channels: Theorem 1: Let ⌽,⌿:B→A be quantum operations with the respective operational densities ⌽ ,⌿ B͑H͒. Suppose that for all %S͑g͒ the pairings
͑⌽ ͒ϵ ͓⌽͑ ͔͒ ͑⌿ ͒ϵ ͓⌿͑ ͔͒ ͑ ͒ , 1h % 1h , , 1h % 1h 32 are constant. Then
2 ͑⌽ ⌿͒ 2 ͑⌽ ⌿͒͑ ͒ ͑ ͒ dH , = sup dH , % . 33 %S͑g͒ Furthermore, then we have that
2 ͑⌽ ⌿͒ ͱʈ⌽ʈʈ⌿ʈ ͑⌽ ⌿͒ 1 ͑ʈ⌽ʈ ʈ⌿ʈ͒ ͑ ͒ dH , + f , = 2 + , 34 where
1/2 Re Tr͓⌽ ⌼͑ 1 ͔͒ f͑⌽,⌿͒ = inf sup h ͑35͒ S͑ ͒ † ͱ ͓⌽͑ ͔͒ͱ ͓⌿͑ ͔͒ % g ⌼B͑H͒:⌼ ⌼=⌿ % 1h % 1h is the minimax fidelity between ⌽ and ⌿. Proof: Fix an arbitrary %S͑g͒. From Eq. ͑32͒ it follows that
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ʈ⌽ʈ ͓⌽͑ ͔͒ ͓⌽͑ ͔͒ = sup % 1h = % 1h , %S͑g͒
† † and the same goes for ⌿. Therefore, given any pair ⌫,⌼B͑H͒ such that ⌫ ⌫=⌽ and ⌼ ⌼ =⌿, we can write
͑⌫ ⌼ ͒ 1 ͑͑⌫ ⌼͒†͑⌫ ⌼͒͑ ͒͒ 1 ͑͑⌽ ⌿ ͒͑ ͒ ͑⌫†⌼ ⌫⌼†͒͑ ͒͒ c − ; = 2 Tr − − 1h = 2 Tr + 1h − + 1h 1 ͑ʈ⌽ʈ ʈ⌿ʈ ͓͑⌫†⌼ ⌫⌼†͒͑ ͔͒͒ = 2 + −Tr + 1h , whence it follows that
2 ͑⌽ ⌿͒͑ ͒ ͑⌫ ⌼ ͒ 1 ʈ⌽ʈ ʈ⌿ʈ ͓͑⌫†⌼ ⌫⌼†͒͑ ͔͒ dH , % = inf c − ; = 2 + − sup Tr + 1h . ⌫ ⌫†⌫ ⌽ ͩ ⌫ ⌫†⌫ ⌽ ͪ : = : = † † ⌼:⌼ ⌼=⌿ ⌼:⌼ ⌼=⌿ Taking the supremum of both sides over all %S͑g͒, we obtain
2 ͑⌽ ⌿͒͑ ͒ 1 ʈ⌽ʈ ʈ⌿ʈ ͓͑⌫†⌼ ⌫⌼†͒͑ ͔͒ ͑ ͒ sup dH , % = 2 + − inf sup Tr + 1h . 36 %S͑g͒ ͩ %S͑g͒ ⌫ ⌫†⌫ ⌽ ͪ : = † ⌼:⌼ ⌼=⌿ On the other hand,
2 ͑⌽ ⌿͒ ͑⌫ ⌼ ͒ 1 ͑ʈ⌽ʈ ʈ⌿ʈ ͓͑⌫†⌼ ⌫⌼†͒͑ ͔͒͒ dH , = inf sup c − ; = 2 inf sup + −Tr + 1h , † S͑ ͒ † S͑ ͒ ⌫:⌫ ⌫=⌽ % g ⌫:⌫ ⌫=⌽ % g † † ⌼:⌼ ⌼=⌿ ⌼:⌼ ⌼=⌿ which yields
2 ͑⌽ ⌿͒ 1 ʈ⌽ʈ ʈ⌿ʈ ͓͑⌫†⌼ ⌫⌼†͒͑ ͔͒ ͑ ͒ dH , = 2 + − sup inf Tr + 1h . 37 ͩ ⌫ ⌫†⌫ ⌽ %S͑g͒ ͪ : = † ⌼:⌼ ⌼=⌿ Note that the right-hand sides of Eqs. ͑36͒ and ͑37͒ differ only in the order of the extrema. Thus, establishing the validity of Eq. ͑33͒ amounts to justifying the interchange of the extrema. According to Lemma 1, the supremum over ⌫ and ⌼ in Eq. ͑36͒ can be evaluated by fixing 1/2 † ⌫=⌽ first and then varying only over all ⌼B͑H͒ such that ⌼ ⌼=⌿. By the polar decom- 1/2 position, any such ⌼ has the form U⌿ for some partial isometry U. Thus we have
͓͑⌫†⌼ ⌫⌼†͒͑ ͔͒ ͓⌽1/2⌼͑ ͔͒ sup Tr + 1h = 2 sup Re Tr 1h † † ⌫:⌫ ⌫=⌽ ⌼:⌼ ⌼=⌿ † ⌼:⌼ ⌼=⌿ ͓⌽1/2 ⌿1/2͑ ͔͒ ͑ ͒ = 2 sup Re Tr U 1h , 38 U where the supremum in Eq. ͑38͒ is taken over all partial isometries U such that
1/2 † 1/2 ⌿ U U⌿ = ⌿. Since the expression being minimized is linear in U and the isometries are the extreme points of ͑H͒ ͕ B͑H͒ ʈ ʈഛ ͖ 23 the unit ball B1 ª X : X 1 of all bounded operators, we may instead take the supremum over the entire unit ball:
͓⌽1/2⌼͑ ͔͒ ͓⌽1/2 ⌿1/2͑ ͔͒ ͑ ͒ sup Re Tr 1h = sup Re Tr X 1h . 39 † ͑H͒ ⌼:⌼ ⌼=⌿ X B1 Since the expression being maximized in the right-hand side of Eq. ͑39͒ is affine in both X and , ͑H͒ S͑ ͒ B͑H͒ B ͑ ͒ and since B1 and g are closed convex subsets of and T g , respectively, it follows from standard minimax arguments24 that we can indeed interchange the extrema to obtain ͑⌽ ⌿͒ ͑⌽ ⌿͒ f− , = f+ , , where
Downloaded 25 Aug 2008 to 152.3.194.16. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp 062106-12 Belavkin, D’Ariano, and Raginsky J. Math. Phys. 46, 062106 ͑2005͒
͑⌽ ⌿͒ ͓⌽1/2 ⌿1/2͑ ͔͒ f− , inf sup Re Tr X 1h ª S͑ ͒ ͑H͒ % g X B1
͑⌽ ⌿͒ ͓⌽1/2 ⌿1/2͑ ͔͒ f+ , sup inf Re Tr X 1h , ª ͑H͒ S͑ ͒ X B1 % g which proves the claim of Eq. ͑33͒. The rest is straightforward. ᮀ As seen immediately from Theorem 1, when ⌽ and ⌿ are quantum channels, then 2 ͑⌽ ⌿͒ ͑⌽ ⌿͒ dH , + f , =1, with the minimax fidelity given by
͑⌽ ⌿͒ ͓⌽1/2⌼͑ ͔͒ ͑ ͒ f , = inf sup Re Tr 1h . 40 S͑ ͒ † % g ⌼:⌼ ⌼=⌿
IV. EVALUATING THE FIDELITY DISTANCES A. Fidelities for quantum states and quantum effects Consider two normal states %, on B=B͑h͒ as quantum channels from B into the Abelian algebra A=B͑g͒ with gӍC. In this case, the operational densities % , of %, coincide with the A Ӎ →B corresponding density operators , :% = and = . The predual maps %T , T : T C T can ͑͒ ͑͒ then be thought of as the state creation operations, %T = and T = for C. In order to compute the minimax fidelity f͑%,͒, we have to consider all ,B that give the decompositions =† and =†. Note that we can always write these decompositions as purifications
͉ ͗͘ ͉ ͉ ͗͘ ͉ = ͚ j j , = ͚ j j , j j ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͕͉ ͖͘ where j ª j , j ª j with respect to a fixed orthonormal basis j of h. We then have the minimum quadratic distance
2 ͑ ͒ 1 ͓͑ ͒†͑ ͔͒ ϵ 1 ͓͑ ͒†͑ ͔͒ dH %, = inf sup − − inf Tr − − , 2 S͑ ͒ 2 B:†= g B:†= B:†= B:†= where the last equality is due to the fact that dim g=1. Expanding the product under the trace, we can write
2 ͑ ͒ 1 ͑ ͒ ͕ ͑†͒ † † ͖ ͑ ͒ dH %, = 2 Tr + − sup Re Tr : = , = 41 ͓ ,B ͔
=1 − sup Re Tr͑†͒͑42͒ B:†= B:†=
ϵ1−f͑%,͒. ͑43͒ According to Lemma 1, the supremum in Eq. ͑42͒ is attained at any B satisfying the condition † 1/2 † † 1/2 † : =, say = , and =ؠ satisfying the equation ؠ =͑ ͒ =ؠ
f͑%,͒ = sup Re Tr͑†͒ = sup ͕ReTr͑1/2͒:† = ͖ =Trͱ. B͑ ͒ B:†= h B:†= Observe that the standard Uhlmann fidelity between the density operators and ,F͑,͒ in Eq. ͑ ͒ ͑ ͒ ʈ1/21/2ʈ ͱ 1 , can be written as F , = T =Tr . Thus the minimax fidelity between two normal
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B →B ͑ ͒ states % and on , or, equivalently, between the state creation operations %T , T :C T h , agrees with the Uhlmann fidelity between the respective density operators and of % and . Next we turn to the other extreme case, namely that of the state annihilation operations ⌽,⌿ ⌽ ͑͒ ͑⌽ ͒ ⌿ ͑͒ ͑⌿ ͒ with the preduals T = , , T = , , corresponding to dim h=1. They are completely ⌽ ⌿ B͑ ͒ ഛ⌽ ⌿ ഛ specified by the effects, i.e., the positive operators , g satisfying 0 , 1g, which ͑ ͒ ⌫ ͗ ͉⌫ ⌼ ͗ ͉⌼ can be purified as in 29 , where j = j , j = j are the bra-vectors corresponding to an ortho- normal basis ͕͉j͖͘ in g. The squared pointwise minimax distance between the state annihilation operations ⌽,⌿, or, equivalently between the effects ⌽ ,⌿, on the set S͑g͒ of normal states % =T on B͑g͒ is given by the minimum
2 ͑⌽ ⌿͒͑ ͒ 1 ͕ ͓͑⌫ ⌼͒†͑⌫ ⌼͔͒ ⌫†⌫ ⌽ ⌼⌼ ⌿ ͖ dH , % = 2 inf Tr − − : = , = ⌫,⌼B͑g͒ of the quadratic distance between their purifications ⌫,⌼B͑g͒. The solution of this problem is likewise given by Lemma 1 with R=⌽ and S=⌿. Thus the optimum
2 ͑⌽ ⌿͒͑ ͒ 1 ͓͑⌽ ⌿ ͔͒ ͱ⌽ ͑⌿ ͒ dH , % = 2 Tr + −Tr † 1/2 is attained at any ⌫B satisfying the condition ⌫ ⌫=⌽, say ⌫=⌽ , and the corresponding ⌼ † ͱ † † ⌼ؠ satisfying the equation ⌼ؠ⌫ = ⌫⌿⌫ =⌫⌼ؠ . The maximum of this distance over all= states,
2 ͑⌽ ⌿͒ 2 ͑⌽ ⌿͒͑ ͒ dH , = sup dH , % %S͑g͒ ϵ ͑ 1 ͓͑⌽ ⌿ ͔͒ ͱ⌽ ͑⌿ ͒͒ sup 2 Tr + −Tr %S͑g͒ † † † = sup inf ͕Tr͓͑⌫ − ⌼͒ ͑⌫ − ⌼͔͒:⌫ ⌫ = ⌽,⌼ ⌼ = ⌿͖, %S͑g͒ ⌫,⌼B͑g͒ is given by the minimax quadratic distance
2 ͑⌽ ⌿͒ 1 ͕ʈ⌫ ⌼ʈ2 ⌫†⌫ ⌽ ⌼†⌼ ⌿ ͖ dH , = 2 inf − : = , = , ⌫,⌼B͑g͒ interchange of the extrema following from standard minimax arguments,24 and the fact that all ⌫, † † ⌼ satisfying, respectively, ⌫ ⌫=⌽ and ⌼ ⌼=⌿ are contained in the unit ball of B͑g͒.
B. Semiclassical fidelity It is straightforward to extend the formalism of Sec. III A involving the commutative Hell- inger distance between two positive kernels to the case of mappings from a set X into positive ͒ ͑ۋ ͑ ͒B ͑ ͒ ۋ trace-class operators on the Hilbert space h, i.e., :x X x T h and :x X x B ͑ ͒ ͑ ͒ ͑ ͒ജ T h with x , x 0 for all x X. We thus have the pointwise Hellinger distance 2 ͑͑ ͒ ͑ ͒͒ ͑ 1 ͓͑ ͒ ͑ ͔͒͒ ͱ͑ ͑ ͒͒͑ ͑ ͒͒ ͑͑ ͒ ͑ ͒͒ dH x , x = 1, 2 x + x − 1, x 1, x f x , x ͑ ͒ ͑ ͒ B B͑ ͒ B B ͑ ͒ in terms of the trace pairing B, =Tr B˜ of B = h and T = T h , where ͱ ͱ ͑1, ͑x͒͑x͒͒ Tr ͑x͒͑x͒ f͑͑x͒,͑x͒͒ = ͱ = ͱ . ͑1,͑x͒͒͑1,͑x͒͒ Tr ͑x͒Tr ͑x͒ The semi-classical operational distance between =͑·͒ and =͑·͒ can then be defined as
2 ͑ ͒ 2 ͑͑ ͒ ͑ ͒͒ϵʈ 2 ͑͑ ͒ ͑ ͒͒ʈ ͑ ͒ dH , = sup dH x , x dH · , · . 44 xX When Tr ͑x͒=1=Tr͑x͒ for all xX, i.e., when and are classical-to-quantum, c-q ͑or ͒ ͑ ͒ 2 ͑ ͒ ͑ ͒ semiclassical , channels, Eq. 44 can be written as dH , =1−f , , where
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f͑,͒ = inf Tr͓͑x͒1/2͑x͒͑x͒1/2͔1/2 = inf Trͱ͑x͒͑x͒ϵinf F͑͑x͒,͑x͒͒ xX xX xX is the minimax fidelity of relative to .
C. Semiquantum fidelity Next we consider the opposite of semiclassical operations—namely, the semiquantum opera- tions which correspond to quantum measurements as quantum-to-classical ͑q-c͒ channels. Such operations are given as
⌽͑ ͒ ͵ ͑ ͒⌽ ͑ ͒ ͑ ͒ϵ͑ ⌽ ͒ b = b y y d y b, Y B C͑ ͒ → ͑ B ͒ on the algebra = Y of continuous bounded functions b:Y C, where Y , Y , is a measure space, by specifying the positive operator-valued Bochner -integrable functions ⌽ :Y →A =B͑g͒.If ⌽͑ ͒ ͑ ⌽ ͒ 1 = 1, = 1g, ͒ ͑ ⌽ ͑͒͑ ͒CۋA the predual maps T · T Y , ⌽ ͑͒͑ ͒ ͑⌽ ͑ ͒ ͒ϵ ͓⌽ ͑ ͔͒ T y ª y , % y , S͑ ͒ ͑ B ͒ define for each input quantum state % g a classical probability density on Y , Y , , that is, they describe quantum measurements by the positive operator-valued measures ͑POVMs͒ M͑dy͒=⌽͑y͒d͑y͒. In order to avoid technicalities in defining the semi-quantum fidelity distance between two q-c channels ⌽,⌿:B→A, we shall assume that ⌽͑y͒,⌿͑y͒ are weakly continuous bounded func- 2 ͑⌽ ⌿͒ tions on Y. Then the squared distance dH , can be written as
2 ͑⌽ ⌿͒ Ͱ͵ ͑⌫͑ ͒ ⌼͑ ͒͒†͑⌫͑ ͒ ⌼͑ ͒͒ ͑ ͒Ͱ ͑ ͒ dH , = inf y − y y − y d y , 45 † † ⌫,⌼:⌫ ⌫=⌽,⌼ ⌼=⌿ † † where the decompositions ⌫ ⌫=⌽ and ⌼ ⌼=⌿ are understood in the pointwise sense as
† † ⌽͑y͒ = ⌫͑y͒ ⌫͑y͒, ⌿͑y͒ = ⌼͑y͒ ⌼͑y͒, ∀ y Y.
͑ ͒ ⌫A C ͑ ͒ ⌫†⌫ ⌽ The infimum in Eq. 45 is achieved at any T Y satisfying the condition = , say ⌫͑ ͒ ⌽ ͑ ͒1/2 ⌼ ⌼ y = y , and the corresponding = o satisfying the equation ⌼ ͑ ͒⌫͑ ͒† ͓⌫͑ ͒⌿ ͑ ͒⌫͑ ͒†͔1/2 ⌫͑ ͒⌼ ͑ ͒† o y y = y y y = y o y . The maximum of this minimal distance over all states,
1 2 ͑⌽ ⌿͒ ͵ͩ ͓͑⌽ ͑ ͒ ⌿ ͑ ͔͒͒ ͱ⌽ ͑ ͒͑⌿ ͑ ͒͒ͪ ͑ ͒ dH , = sup Tr y + y −Tr y y d y , %S͑g͒ 2 2 ͑⌽ ⌿͒ ͑⌽ ⌿͒ ⌽͑ ͒ ⌿͑ ͒ is equal to dH , =1−f , in the measurement operation case 1 =1g = 1 , where
͑⌽ ⌿͒ ͵ ͱ⌽ ͑ ͒͑⌿ ͑ ͒͒ ͑ ͒ ͑ ͒ f , = inf Tr y y d y . 46 %S͑g͒
D. Operational fidelity formula Now we can easily evaluate the minimax formula ͑30͒ for the fidelity of two general quantum operations ⌽,⌿:B→A, B=B͑h͒, A=B͑g͒. The solution of this problem is also given by Lemma ⌽ ͑ ͒⌿ ͑ ͒ S͑ ͒ 1 with R= and S= 1h 1h . For a given % g , the supremum in
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2 ͑⌽ ⌿͒͑ ͒ 1 ͑ ͓͑⌽ ⌿ ͒͑ ͔͒ ͕ ͓⌫†⌼͑ ͔͒ ⌫†⌫ ⌽ ⌼†⌼ ⌿ ͖͒ dH , % = 2 Tr + 1h − 2 sup Re Tr 1h : = , = ⌫,⌼B͑H͒
ͱ⌽ ͓͑ ͒⌿ ͑ ͔͒ ⌫B͑H͒ is equal to Tr 1h 1h , and is achieved at any satisfying the condition ⌫†⌫ ⌽ ⌫ ⌽1/2 ⌼ ⌼ = , say = , and the corresponding = o satisfying the equation
⌼ ͑ ͒⌫† ͓⌫͑ ͒⌿ ͑ ͒⌫†͔1/2 ⌫͑ ͒⌼† o 1h = 1h 1h = 1h o. ⌽ ⌿ ⌽ ⌿ When , are quantum channels, or, equivalently, when the preduals T , T are trace preserv- ing, Theorem 1 says that the maximum of this distance over all states,
2 ͑⌽ ⌿͒ ͑ 1 ͑⌽ ⌿ ͒͑ ͒ ͱ⌽ ͓͑ ͒⌿ ͑ ͔͒͒ ͑ ͒ dH , = sup Tr 2 + 1h − 1h 1h , 47 %S͑g͒ 2 ͑⌽ ⌿͒ ͑⌽ ⌿͒ can be written as dH , =1−f , , where
͑⌽ ⌿͒ ͱ⌽ ͓͑ ͒⌿ ͑ ͔͒ ͑ ͒ f , = inf Tr 1h 1h 48 %S͑g͒ is the minimax fidelity between ⌽ and ⌿.
E. Operational fidelity in terms of Kraus and Stinespring decompositions Consider, as before, two quantum channels ⌽,⌿:B→A, where B=B͑h͒ and A=B͑g͒. Given the minimax fidelity
͑⌽ ⌿͒ ͓⌫†⌼͑ ͔͒ ͉ ͓⌫†⌼͑ ͔͉͒ f , = inf sup Re Tr 1h = inf sup Tr 1h S͑ ͒ † S͑ ͒ † % g ⌫:⌫ ⌫=⌽ % g ⌫:⌫ ⌫=⌽ † † ⌼:⌼ ⌼=⌿ ⌼:⌼ ⌼=⌿ between ⌽ and ⌿, where the second equality follows from Lemma 2, the supremum over all ⌫ † † and ⌼ satisfying, respectively, ⌫ ⌫=⌽ and ⌼ ⌼=⌿ can be replaced with the supremum over all ⌽ ⌿ ͕ ͖ ͕ ͖ Kraus decompositions of and , i.e., over all collections Fj , Vj of bounded operators g →h, determined from ⌽ ,⌿ via Eqs. ͑29͒ and ͑28͒:
͑⌽ ⌿͒ ͑ † ͒ ͑ ͒ f , = inf sup ͚ͯ % Fj Vj ͯ. 49 S͑ ͒ ͕ ͖ ͕ ͖ % g Fj , Vj j Just as in the proof of Theorem 1, we may restrict ourselves only to those ⌫,⌼ that can be written 1/2 1/2 1/2 1/2 as ⌫=U⌽ ,⌼=V⌿ for some isometries U,V. Thus, if we write ⌽ and ⌿ in the form of Eq. ͑28͒ as
⌽1/2 ͉ ͑͘ ˆ ͉ ⌿1/2 ͉ ͑͘ ˆ ͉ = ͚ j Fj , = ͚ j Vj , j j then it follows that, given isometries U, we can write
⌫ ⌽1/2 ͉ ͘ ˆ ϵ ͉ ͑͘ ˆ ͑ ͉͒ = U = ͚ j ͚ͩ UjᐉFᐉͯ ͚ j Fj U , j ᐉ j 1/2 and similarly for ⌼=V⌿ . Thus
͑⌽ ⌿͒ ͓ ˆ ͑ ͒† ˆ ͑ ͔͒ ͓ ˆ ͑ ͒† ˆ ͔ f , = inf sup͚ͯ % Fj U Vj V ͯ = inf sup͚ͯ % Fj U Vj ͯ. %S͑g͒ U,V j %S͑g͒ U j Turning now to the infimum over all normal states % on AϵB͑g͒, we may equivalently consider :representation of A on a Hilbert space K-ء all pairs ͕,K͖, where is a normal
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͑⌽ ⌿͒ ͉͓͗ ˆ ͑ ͒† ˆ ͔͉͘ f , ª inf sup͚ͯ Fj U Vj ͯ. ͕,K͖;K,ʈʈ=1 U j representations of the full operator algebra B͑g͒ are unitarily equivalent to an-ء Since all normal ۋ amplification B B 1k for some Hilbert space k, we can write
͑⌽ ⌿͒ ͚ ͉͗ ˆ ͑ ͒† ˆ ͉͘ ͑ ͒ f , ª inf supͯ Fj U Vj 1k ͯ. 50 gk;ʈʈ=1 U j Introducing the vectors ͉,⌽͘,͉,⌿͘g k H, defined by
͉ ⌽͘ ͚ ͑ ˆ ͒ ͉ ͘ ͉ ⌿͘ ͚ ͑ ˆ ͒ ͉ ͘ , ª Fj 1k j , , ª Vj 1k j , j j we obtain yet another form of the minimax fidelity:
͑⌽ ⌿͒ ͉͗ ⌽͉ ͉ ⌿͉͘ ͑ ͒ f , = inf sup , 1gk U , . 51 gk U For a fixed g k, taking the supremum over U is tantamount to taking the supremum of ͉͉͉͗͘ over all pairs of unit vectors ,g k H such that
͉͉͗͘ ͑˜ˆ ͉͉͒͗͑͘˜ˆ ͒† ϵ ⌽ ͉͉͑͗͒͘ TrH = ͚ Fj 1k Fj 1k T id , j
͉͉͗͘ ͑˜ˆ ͉͉͒͗͑͘˜ˆ ͒† ϵ ⌿ ͉͉͑͗͒͘ TrH = ͚ Vj 1k Vj 1k T id , j which, in conjunction with the standard results on the Uhlmann fidelity ͑1͒ between density operators,6,7 finally yields
͑⌽ ⌿͒ ͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ ͑⌽ ͑͒ ⌿ ͑͒͒ f , = inf F T id , T id = inf F T id , T id . gk:ʈʈ=1 %S͑gk͒ Note that we may always take k isomorphic to g:
͑⌽ ⌿͒ ͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ ͑ ͒ f , = inf F T id , T id . 52 gg,ʈʈ=1 ͕ ͖ ͕ ͖ ⌽ ⌿ Given some Kraus decompositions Fj , Vj of and , respectively, we may define the operators
͉ ͘ ͉ ͘ F ª ͚ Fj j , V ª ͚ Vj j j j
† from g into h H and express ⌽ and ⌿ in the Stinespring form ⌽͑B͒=F ͑B 1H͒F, ⌿͑B͒ † =V ͑B 1H͒V ͑cf. Sec. III B͒. Then we may rewrite Eq. ͑49͒ as
f͑⌽,⌿͒ = inf sup͉Tr͑FV†͉͒, %S͑g͒ F,V where the supremum is over all F,V:g→h H giving the Stinespring decompositions of ⌽ and ⌿, respectively. We may, as before, fix F and V, say, by considering the “canonical” Kraus ͕ ˆ ͖ ͕ ˆ ͖ ͑H͒ decompositions Fj , Vj , and instead take the supremum over all unitaries U U :
͑⌽ ⌿͒ ͉ ͓͑ ͒ †͔͉ ͉ ͓ ͑ †͔͉͒ f , = inf sup Tr 1h U F V = inf sup Tr U Trh F V , %S͑g͒ U %S͑g͒ U which yields another useful formula
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͑⌽ ⌿͒ ʈ ͑ †͒ʈ ͑ ͒ f , = inf Trh F V T 53 %S͑g͒ for the minimax fidelity between the channels ⌽,⌿. It is, in fact, not hard to show that the right-hand side of Eq. ͑53͒ does not depend on the particular choice of the Stinespring operators F,V, as long as we agree to dilate the input Hilbert space h by the “canonical” auxiliary Hilbert space H=g h. We note that the constructions of this section are valid more generally for channels given in terms of the continual Kraus decompositions
⌽͑B͒ = ͵ F͑z͒†BF͑z͒d͑z͒, ⌿͑B͒ = ͵ V͑z͒†BV͑z͒d͑z͒, Z Z provided that the measures and are equivalent, i.e., absolutely continuous with respect to each other. Then Eq. ͑49͒ is a special instance of the more general expression
f͑⌽,⌿͒ = inf sup ͯ%ͩ͵ ͱd/dF͑z͒†V͑z͒d͑z͒ͪͯ, %S͑g͒ ͕F͑z͖͒,͕V͑z͖͒ Z where d/d is the Radon-Nikodym derivative of with respect to , for the case when both and are counting measures, d=d=1, on a finite or countably infinite set.
V. PROPERTIES OF THE OPERATIONAL FIDELITY In this section we establish several key properties of the minimax fidelity between quantum operations. These properties follow almost immediately from the corresponding properties en- joyed by the fidelity ͑1͒ on density operators: ͑F.1͒ F is symmetric, F͑,͒=F͑,͒, bounded between 0 and 1, and F͑,͒=1 if and only if =. ͑F.2͒ F is jointly concave over all pairs of density operators. ͑F.3͒ F is unitarily invariant, i.e., F͑,͒=F͑UU† ,UU†͒ for any unitary U. ͑ ͒ ͑⌽ ͑͒ ⌽ ͑͒͒ജ ͑ ͒ F.4 F is monotone with respect to quantum channels: F T , T F , for every quantum channel ⌽. ͑ ͒ ͑ ͒ ͱ ͑ ͒ F.5 The Bures distance dB ·, · ª 1−F ·, · is topologically equivalent to the trace-norm half-distance D͑·, ·͒:
−1/2 ͑ ͒ ഛ ͑ ͒ ഛ ͱ ͑ ͒ 2 D , dB , D , ͓cf. Eq. ͑3͔͒. Property ͑F.2͒, in fact, follows from strong concavity of F,2 i.e.,
ജ ͱ ͑ ͒͑͒ F͚ͩ pi i,͚ qi iͪ ͚ piqiF i, i 54 i i i ഛ ഛ ͚ ͚ for all 0 pi ,qi 1 such that ipi =1= iqi. Using Eq. ͑52͒, we can immediately obtain for the minimax fidelity f͑·, ·͒ on pairs of quantum channels the following analogs of properties ͑F.1͒–͑F.4͒ of the fidelity F͑·, ·͒ on pairs of density operators: ͑f.1͒ f is symmetric, bounded between 0 and 1, and f͑⌽,⌿͒=1 if and only if ⌽=⌿. ͑f.2͒ f is jointly concave over all pairs of channels. ͑f.3͒ f is invariant under both left and right composition with unitarily implemented channels, i.e.,
⍜ ؠ ⌽ ⍜ ؠ ⌿͒ ͑⌽ ⌿͒͑ f U , U = f , and
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⌽ ؠ ⍜ ⌿ ؠ ⍜ ͒ ͑⌽ ⌿͒͑ f V, V = f , for any two channels ⌽,⌿:B͑h͒→B͑g͒ and any two unitaries UU͑g͒, VU͑h͒, where ⍜ ͑ ͒ † ⍜ U B ªU BU, and V is defined analogously. ͑f.4͒ f is monotone with respect to both left and right composition with quantum channels, i.e., ⌽ؠ⌽ ⌿ؠ⌽ ͒ജ ͑⌽ ⌿͒ ͑⌽ ؠ⌽ ⌽ ؠ⌿͒ജ ͑⌽ ⌿͒ ⌽ ⌿ B→A͑ f 1 , 1 f , and f 2 , 2 f , for any two channels , : , all ⌽ B ⌽ A channels 1 into , and all channels 2 on . Just as in the case of the fidelity between density operators, the minimax fidelity f possesses the strong concavity property
⌽ ⌿ ജ ͱ ͑⌽ ⌿ ͒ ͑ ͒ f͚ͩ pi i,͚ qi iͪ ͚ piqif i, i . 55 i i i On the other hand, deriving for the minimax fidelity f an analog of property ͑F.5͒ of the Uhlmann fidelity F requires a bit more work. To this end, let us consider two channels ⌽,⌿:B→A,B =B͑h͒,A=B͑g͒. Suppose first that g is infinite dimensional and separable. Then gӍᐉ2, and we can rewrite Eq. ͑55͒ as
͑⌽ ⌿͒ ͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ f , = inf F T id , T id . gᐉ2;ʈʈ=1 ᐉ2 ᐉ2 The space contains, as a dense subset, the pre-Hilbert space 0 of all infinite sequences of complex numbers with all but finitely many components equal to zero. Using this fact and the continuity property ͑F.5͒ of the fidelity F, we obtain
͑⌽ ⌿͒ ͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ f , = inf F T id , T id . ᐉ2 ʈʈ g 0; =1 Using this expression in conjunction with Eq. ͑3͒, we get the bounds
͑⌽ ⌿͒ ജ ͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ ͑ ͒ f , 1 − sup D T id , T id , 56 ᐉ2 ʈʈ g 0: =1
2͑⌽ ⌿͒ ഛ 2͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ ͑ ͒ f , 1 − sup D T id − T id . 57 ᐉ2 ʈʈ g 0: =1 Now, for any completely bounded map ⌳:B͑h͒→B͑g͒, the image of the set ͕͉͉͗͘:g ᐉ2 ʈʈ ͖ ⌳ B ͑ ᐉ2͒→B ͑ ᐉ2͒ 0 , =1 under the predual map T id: T g 0 T h 0 is contained in the trace- ͕͉͉͗͘ ᐉ2 ʈʈ ͖ norm closure of the linear span of : h 0 , =1 , which is dual to the tensor product B͑h͒ K͑ᐉ2͒, where K͑ᐉ2͒ is the space of compact operators on ᐉ2. Thus, by duality we have
͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ 1 ʈ͑⌽ ⌿͒ ʈϵD͑⌽ ⌿͒ sup D T id , T id = 2 − idK͑ᐉ2͒ , , ᐉ2 ʈʈ g 0: =1 D͑⌽ ⌿͒ 1 ʈ⌽ ⌿ʈ where , denotes the CB-norm half-distance 2 − CB, and the last equality follows from ʈ⌳ʈ ʈ⌳ ʈ ⌳ the formula CB= idK͑ᐉ2͒ for any completely bounded map . On the other hand, when dim g=mϽϱ, we can use the fact20 that, for any completely bounded map ⌳ into B͑g͒,
ʈ⌳ʈ = ʈ⌳ idM ʈ = ʈ⌳ idM ʈ , CB m T m T M ϫ where m denotes the algebra of m m complex matrices, it follows that
͑⌽ ͉͉͑͗͒͘ ⌿ ͉͉͑͗͒͒͘ D͑⌽ ⌿͒ sup D T 1 , T 1 = , . gg:ʈʈ=1 In either case, we immediately derive the inequality
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1−D͑⌽,⌿͒ ഛ f͑⌽,⌿͒ ഛ ͱ1−D2͑⌽,⌿͒, ͑58͒ ͑ ͒ ͱ ͑ ͒ which, when expressed in terms of the Hellinger distance dH ·, · ª 1−f ·, · as
−1/2D͑⌽ ⌿͒ ഛ ͑⌽ ⌿͒ ഛ ͱD͑⌽ ⌿͒ ͑ ͒ 2 , dH , , , 59 yields the desired property ͑ ͒ ͑ ͒ ͱ ͑ ͒ f.5 the Hellinger distance dH ·, · ª 1−f ·, · is topologically equivalent to the CB-norm distance ͓cf. Eq. ͑59͔͒. This completes our survey of the basic properties of the minimax fidelity f.
VI. SOME EXAMPLES AND APPLICATIONS The expressions for the minimax fidelity, derived in Sec. IV for different kinds of quantum operations encountered in quantum information theory, share the common feature of being set up as variational problems, namely, as minimizations of a concave functional over a convex set. This feature of the minimax fidelity renders the problem of computing it amenable to robust numerical methods ͑see Ref. 3 for detailed discussion of numerical optimization methods for the calculation of fidelitylike measures in quantum information theory͒. However, there are instances in which the minimax fidelity between two quantum channels can be written down in a more explicit form. In this section we sketch some examples of such instances. Before we proceed, we would like to remind the reader of the assumption we made in Sec. II B, namely that all the channels we deal with are completely majorized by the trace in the sense of Ref. 14. This assumption, while allowing us to circumvent certain technicalities involving unbounded operators, is somewhat restrictive, as one can easily find examples of quantum chan- nels between infinite-dimensional algebras ͑e.g., unitarily or isometrically implemented channels; see Ref. 21 for details͒ that do not satisfy this condition of complete majorization. However, owing to the CB-continuity of the minimax fidelity ͑cf. Sec. V͒, we may always regard such channels as CB-limits of sequences of channels with finite-dimensional output algebras. Thus, ⌽ B→A B B͑ ͒ A B͑ ͒ ϱ ͕ ͖ given a channel : , = h , = g with dim g= , we consider a sequence Pn of → ͕⌽ ͖ finite-dimensional projections such that Pn 1g strongly, and the corresponding sequence n of ⌽ ͑ ͒ ⌽͑ ͒ ⌽ ͑ ͒→⌽͑ ͒ →ϱ quantum operations n B ª Pn B Pn, so that n B B uniformly as n for each B B ⌽ B A ʈ⌽ ⌽ ʈ , and each n is a channel from into Pn Pn, with limn→ϱ − n CB=0. With this in mind, in the examples below we shall not worry about the issue of bounded versus unbounded operational densities.
A. Unitary maps ⍜ ⍜ → ⍜ ͑ ͒ In the case of channels U , V implemented by the unitaries U,V:h h, i.e., U B † ⍜ ͑ ͒ † ͑⌽ ⌿͒ ͑ ͒ =U BU and V B =V BV, the minimax fidelity f , is easily evaluated using Eq. 49 :
͑⍜ ⍜ ͒ ͉ ͑ ͉͒ f U, V = inf % W , %S͑g͒ † ͑ ͒ where we have defined WªU V. Let Sp W denote the spectrum of W, which is a closed compact subset of the unit circle T in the complex plane, and let EW͑dz͒ denote the corresponding spectral measure of W. Then we can write
͑⍜ ⍜ ͒ ͯ͵ W,%͑ ͒ͯ f U, V = inf zM dz , %S͑g͒ Sp͑W͒ where MW,%͑dz͒ is the probability measure %͓EW͑dz͔͒ϵ͑EW͑dz͒,͒. Thus
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͑⍜ ⍜ ͒ ͑ ͑ ͒͒ ͑ ͒ f U, V = dist 0,coSp W , 60 ͑ ͒ ͑ ͒ ͑ ͒ ͕͉ Ј͉ Ј ͖ where co Sp W denotes the closed convex hull of Sp W , and dist z,S ªinf z−z :z S for ʚ ͑⍜ ⍜ ͒ ʚ any z C and S C. Clearly, f U , V =1 if and only if co Sp W T, i.e., if and only if W ͉͉ ⍜ ⍜ = 1h with =1, which is equivalent to U = V. When dim hϽϱ,Sp͑W͒ is a finite subset of T, so that co Sp͑W͒ is a polygon in the complex ͑ ͒ ͑⍜ ⍜ ͒ plane, and Eq. 60 shows that f U , V is simply the distance d from this polygon to the origin. 12 D͑⍜ ⍜ ͒ ͱ 2 On the other hand, recalling the formula U , V = 1−d , we see that the upper bound in Eq. ͑58͒ is saturated by the unitarily implemented channels.
B. Random unitary channels Continuing with the setup from the preceding example, let us consider channels of the form
⌽͑B͒ = ͚ p ⍜ ͑B͒, ⌿͑B͒ = ͚ q ⍜ ͑B͒, ͑61͒ i Ui i Ui i i where ⍜ are unitarily implemented channels and pϵ͕p ͖, qϵ͕q ͖ are probability distributions. It Ui i i then follows immediately from the strong concavity property ͑55͒ of the minimax fidelity that
͑⌽ ⌿͒ ജ ͱ ϵ ͑ ͒ ͑ ͒ f , ͚ piqi F p,q . 62 i Ͻϱ ͑ ͒ When dim h , the inequality in 62 becomes equality when the unitaries Ui are orthogonal † ␦ in the Hilbert-Schmidt sense, Tr Ui Uk =dim h· ik. On the other hand, when h is infinite dimen- sional, this orthogonality condition does not make sense unless we consider channels given in terms of continual Kraus decompositions, so that the sums in Eq. ͑62͒ are replaced with integrals with respect to some positive measure , and agree to understand orthogonality in the sense of operator-valued Schwartz distributions. As an example, consider the following. Let h=F, the boson Fock space, let a and a† be the field annihilation and creation operators, ͑ ͒ ͑ † ¯ ͒ and let D z ªexp za −za , z C, be the unitary displacement operators obeying the Weyl rela- ͑ ͒ ͑ Ј͒ iImzzЈ ͑ Ј͒ 2͑ ͒ ͑ ͒ ͑ ͒ tion D z D z =e D z+z . Given a function f L C,dz , where dzªd Re z d Im z ,we define its Weyl-Fourier transform as D͑f͒ −1/2͐ f͑z͒D͑z͒dz. Since f is square integrable, D͑f͒ ª C is a Hilbert-Schmidt operator, and it can be easily shown that
† 2 ͓ ͑ ͒ ͑ ͔͒ ͵ ͑ ͒ ͑ ͒ ϵ͗ ͘ 2 ∀ ͑ ͒ Tr D f D g = f z g z dz f,g L ͑C͒, f,g L C C ͓ ͑ ͒† ͑ Ј͔͒ ␦͑2͒͑ Ј͒ Ј ␦͑2͒͑͒ ␦͑ ͒␦͑ ͒ so that Tr D z D z = z−z ,z,z C, where ª Re Im is the Dirac ␦-function in the complex plane. ͑͒ With this in mind, consider the family of channels ⌫ :B͑F͒→B͑F͒,R+, with the predu- als given by
͑͒ 1 ⌫ ͑͒ ͵ D͑z͒D͑z͒† exp͑− ͉z͉2/͒dz T ª C ͑in quantum optics these channels model the so-called Gaussian displacement noise25͒. Then the minimax fidelity between ⌫͑͒ and ⌫͑͒ is given by
͑͒1/2 f͑⌫͑͒,⌫͑͒͒ = . ͑63͒ 1 ͑ + ͒ 2 Owing to the inequality between the geometric and the arithmetic means, the right-hand side of Eq. ͑63͒ is always bounded between 0 and 1, and the maximum value of 1 is attained if and only
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if =, i.e., ⌫͑͒ =⌫͑͒. This, of course, agrees with the properties of the minimax fidelity ͑cf. Sec. V͒.
C. Master equation ͕⌽͑t͒ B͑ ͒→B͑ ͖͒ Consider a strongly continuous semigroup of channels : h h tR+, with the ⌽͑t͒ 26 preduals T satisfying the Lindblad master equation ͑ ͒ d⌽ t ͑͒ 1 T = XX† − ͑X†X + X†X͒͑64͒ dt 2 for some XB͑h͒. Introducing the dilating Hilbert space H=h h with the basis ͕͉0͘,͉1͘,...͖,we can, for an infinitesimal time t=, write the predual of the channel ⌽͑͒ in the Stinespring form ⌽͑͒͑͒ † ͑ ͒ T =TrHA A, 65
where the map A :h→h H is given by
1 † ͱ 2 A ͑ − X X͒ ͉0͘ + X ͉1͘ + O͑ ͒, ͑66͒ ª 1h 2 O͑2͒ indicating terms with norm bounded from above by M2 for some constant M ജ0. Note that ͉ ͘ ͑0͒ A0 = 0 , so that T =id. We can then evaluate the partial trace ͓ †͔ ͑ 1 ͗ † ͘ ͉͒ ͗͘ ͉ ͱ͗ ͘ ͉ ͗͘ ͉ ͑2͒ ͑ ͒ Trh A A0 = 1− 2 X X 0 0 + X 1 0 + O , 67 ͗ ͘ ͑ ͒ B͑ ͒ ͑2͒ where B ªTr B for B h . Then, again up to an additive term of operator norm O , ͓ †͔† ͓ †͔Ϸ͓͑ 1 ͗ † ͘ ͒2 ͉͗ ͘ ͉2͔͉ ͗͘ ͉ ͑ ͒ Trh A A0 Trh A A0 1− 2 X X + X 0 0 , 68 which allows us to compute, up to O͑2͒, the minimax fidelity between the channel T͑͒ after an infinitesimal time and the identity map. Using Eq. ͑53͒, we obtain
͑ ͑⑀͒ ͒ ʈ ͓ †͔ʈ Ϸ ͱ ͑ ͒ f T ,id = inf Trh A A0 T 1− C, 69 %S͑g͒ where
† 2 C = inf ͑͗X X͘ − ͉͗X͉͘ ͒. ͑70͒ %S͑g͒
D. Impossibility of quantum bit commitment The statement of topological equivalence of the noncommutative Hellinger distance and the CB-norm distance between a pair of quantum channels, i.e., Eq. ͑59͒, is essentially the “continuity argument” at the heart of a proof of “impossibility of quantum bit commitment ͑QBC͒.”27 Quan- tum bit commitment is a cryptographic objective in which one party, Alice, commits a bit to another party, Bob, in such a way that the corresponding protocol is concealing ͑i.e., Bob is not able to retrieve the bit before the opening͒ and binding ͑i.e., Alice cannot change the bit after the commitment͒. The impossibility proof asserts that if the protocol is perfectly concealing, then it is necessarily not binding, and invokes a continuity argument for “asymptotically” concealing pro- tocols, stating that Alice’s probability of successful cheating approaches unity, while Bob’s cheat- 1 ͑ ͒ ͑ ing probability becomes close to the value 2 pure guessing . The reader should be aware that the impossibility proof in Ref. 27 is valid for a restricted class of protocols, i.e., those that are nonaborting and have a single commitment step. For wider classes of protocols, it is still a matter of debate whether a secure QBC protocol exists.28͒ In this example we derive the continuity argument from the expression of Alice’s and Bob’s respective cheating probabilities as a conse- quence of the topological equivalence between the Hellinger distance and the CB-norm distance in Eq. ͑59͒.
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From the point of view of Bob, Alice’s action of committing the bit is equivalent to a channel ⌽ B͑ ͒ Ͻϱ ͑b͒ A͑b͒ on an algebra h , dim h , for each value of the committed bit b=0,1, where A ͑b͒ ͑b͒† ͑b͒ ϵ͕ ͖k ͚k ⌽ ͑ ͒ Aj j=1 is a collection of operators satisfying the Kraus condition j=1Aj Aj =1, and A b denotes the channel induced by this Kraus decomposition. At the opening, Alice informs Bob about which element of the Kraus decomposition A͑b͒ she actually used in the commitment. However, prior to unveiling the label j, Alice can perform an EPR attack with the purpose of ͑b͒͑ ͒ϵ͕ ͑b͒͑ ͖͒ changing the Kraus decomposition to another equivalent decomposition A V Aj V , where ͑b͒ ͑b͒ k A ͑V͒ ͚ᐉA V ᐉ for some VU͑ ͒. The EPR attack is achieved by Alice via the unitary j ª l j C transformation V on an ancillary k-dimensional space H. The conditional probability that Alice can cheat successfully by convincing Bob that she has committed, say, b=1, while having suc- cessfully committed b=0 instead, is given by
͑0͒† ͑1͒ 2 ͉͉͗A ͑V͒A 1H͉͉͘ PA͑V,͒ = ͚ j j , ͑71͒ c ʈ͑ ͑1͒ ͒ʈ2 j Aj 1H ͑ ͒ ʈ͑ 0 ͒ʈ2 where Aj 1H is the probability that the jth Kraus element is unveiled. Which V should Alice use? Without any knowledge of ͉͘, the best she can do is to adopt a conservative strategy of choosing the V that will maximize her cheating probability in the worst-case scenario, namely ͉͘ A͑ ͒ for the anonymous state chosen by Bob to minimize Pc V, . This is the minimax choice of V, corresponding to the cheating probability
¯ A A͑ ͒ ͑ ͒ Pc sup inf Pc V, . 72 ª k H ʈʈ VU͑C ͒ h ; =1 On the other hand, for equiprobable bit values b͕0,1͖ Bob’s optimal probability of cheating is given by the probability of error in discriminating between the corresponding output states, more precisely
1 1 1 ¯ B ʈ ʈ ͓ D͑⌽ ͑ ͒ ⌽ ͑ ͔͒͒ ͑ ͒ Pc = 2 + 4 sup A͑0͒ − A͑1͒ T = 2 1+ A 0 , A 1 , 73 hH;ʈʈ=1
⌽ ͉͉͑͗͒͘ where we have defined A ª A id . Using Jensen’s inequality, we can bound Alice’s A͑ ͒ cheating probability Pc V, from below as
2 A͑ ͒ ജ ͉͉͗ ͑0͒͑ ͒† ͑1͒ ͉͉͘ ͑ ͒ Pc V, ͚ͯ Aj V Aj 1H ͯ . 74 j Note that the value of the max-min in Eq. ͑72͒ will not change if we perform the maximization over the closed convex hull of U͑Ck͒, i.e., the set K͑Ck͒ of all linear contractions on Ck, and the minimization over the closed convex hull of the pure states on h H, i.e., the set S͑h H͒ of states on B͑h H͒, thus completing the domain of the max-min to the product K͑Ck͒ϫS͑h H͒ of compact convex sets. Now, the functional
͑ ͒ ͚ ͕͓ ͑0͒͑ ͒† ͑1͒ ͔͖ ͑ ͒ F V, ª Re Tr Aj V Aj 1H 75 j 24 is affine in both VK͑Ck͒ and S͑h H͒, so that we can use standard minimax arguments to justify the interchange of extrema in Eq. ͑72͒, and then apply Lemma 2 to obtain
sup inf ͉F͑V,͉͉͉͗͒͘ = sup inf ͉F͑V,͉͒ ͑76͒ k H k S͑ H͒ VU͑C ͒ h VK͑C ͒ % h
= inf sup ͉F͑V,͉͒ ͑77͒ S͑ H͒ k % h VK͑C ͒
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= inf sup ͉F͑V,͉͉͉͗͒͘. ͑78͒ hH;ʈʈ=1 VU͑k͒ Now, since a monotone function does not affect the saddle point, we can use Eqs. ͑50͒, ͑72͒, ͑74͒, and ͑78͒ to obtain
¯ A ജ 2͑⌽ ͑ ͒ ⌽ ͑ ͒͒ Pc f A 0 , A 1 . Using Eq. ͑59͒ and then Eq. ͑73͒, we finally obtain the chain of estimates
¯ A ജ 2͑⌽ ͑ ͒ ⌽ ͑ ͒͒ ജ ͓ D͑⌽ ͑ ͒ ⌽ ͑ ͔͒͒2 ജ ͓ ͑¯ B ͔͒2 Pc f A 0 , A 1 1− A 0 , A 1 1−2 Pc − 1/2 , ¯ B → 1 where it follows that, for “asymptotically” concealing protocols, i.e., those for which Pc 2 , Alice’s probability of cheating will approach unity, and the protocol will not be binding.
ACKNOWLEDGMENTS This work has been sponsored by the Multiple Universities Research Initiative ͑MURI͒ pro- gram administered by the U.S. Army Research Office under Grant No. DAAD19-00-1-0177. V.P.B. acknowledges support from EC under the program ATESIT ͑Contract No. IST-2000- 29681͒. G.M.D. also acknowledges support by EC and Ministero Italiano dell’Università e della Ricerca ͑MIUR͒ through the cosponsored ATESIT project IST-2000-29681 and Cofinanziamento 2002. M.R. acknowledges the kind hospitality of the Quantum Information Theory Group at Università di Pavia, and support from MIUR under Cofinanziamento 2002 and from the European Science Foundation. The authors would like to thank the referee for several suggestions, which resulted in improved presentation.
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