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Contravariant Densities, Complete Distances and Relative Fidelities for Quantum Channels

Introducing contravariant trace-densities for quantum states, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operator-valued contravariant kernels. The CB-norm distance between two quantum operations with type one input algebras is explicitly expressed in terms of these densities, and this formula is also extended to a generalized CB-distances between quantum operations with type two inputs. A larger C-distance is given as the natural norm-distance for the channel densities, and another, Helinger type distance, related to minimax mean square optimization problem for purification of quantum channels, is also introduced and evaluated in terms of their contravariant trace-densities. It is proved that the Helinger type complete fidelity distance between two channels is equivalent to the CB distance at least for type one inputs, and this equivalence is also extended to type two for the generalized CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings as generalized entanglements of quantum states opposite to the inputs and the output states.


Introduction
Quantum channels, which are usually described by trace preserving operations T : A * → B * , preadjoint to normal completely positive (CP) maps Ψ = T * of the output algebra B = B (H) into an input operator algebra A with respect to the Hilbert-Schmidt pairing ̺|Ψ (B) := τ ̺ † Ψ (B) = τ T (̺) † A ≡ T (̺) |B , can be completely characterized in terms of their positive densities Ψ τ with respect to the standard trace τ = tr on B, see for example [1]. Such densities, defined as non-commutative Radon-Nikodym derivatives [2] of Ψ with respect to τ , even in the case of states Ψ = ρ are actually different from the usually used densities ̺. The Radon-Nikodym derivative of a normal state ρ on the matrix algebra A = B (H) with respect to the reference trace τ = tr is canonically identified by a normal representation with not the covariant density matrix ̺ ∈ A * but with the contravariant density ρ τ as the complex conjugated ρ τ =̺ (or equivalently transposed, ρ τ =̺) matrix in the commutant of the standard representation of A. In general the contravariant state densities ρ τ are uniquely defined as affiliated elements of the opposite algebra A, the positive normalized elements of which describe normal states ρ on A majorized by τ as ρ (A) = A, ρ τ with respect to the bilinear pairing A, ρ τ = ρ|A of A and A * . They are more suitable for operational generalizations than the usual, covariant densities ρ * = ̺ which retain only partial (transpose) positivity when they are replaced by covariant channel densities Ψ * ∈ A ⊗ B * defining the maps Ψ by the partial trace Ψ (B) = tr B (I ⊗ B) Ψ * . The positive, contravariant densities Ψ τ describe CP maps Ψ by partial tracing corresponding to B, σ τ = tr Bσ τ , and they transform the contravariant input densities ρ τ into contravariant densities σ τ =ς of output states ς = T (̺) by partial tracing Ψ τ , ρ τ = tr A Ψ τ (ρ τ ⊗ I B ) .
In the case of infinite dimensional Hilbert space H, the channel densities Ψ τ , unlike the state densities, might be unbounded even in the simple cse B = B (H), and in general should be understood in a distribution sense. Such densities were defined in [2] as Hermitian-positive kernels, characterized as unbounded self-adjoint operators in Stinespring representation [3] of another, dominating CP map Φ. In the case of tracial Φ (B) = I A trB the channel densities Ψ τ are defined as Hermitian-positive kernels affiliated to the tensor product A ⊗ B of the input channel algebra A with the opposite (transposed) algebra B to the output B. The Schmidt decompositions of the positive densities Ψ τ are in one-to-one correspondence with the Kraus decompositions of the channels Ψ, and the spectral decompositions of the positive self-adjoint operators Ψ τ , corresponding to the orthogonal Kraus decompositions of Ψ, completely describe the properties of quantum channels in terms of their spectral measures on R + .
Here we develop a consistent metric space theory of contravariant quantum channel kernels describing quantum operations in terms of trace-pairings of the observables and the contravariant densities respectively to generalized traces, introduced in [1]. We discuss several distances for comparing two quantum channels Φ and Ψ, among them the complete boundedness (CB) distance and the complete Helinger (CH), or operational Bures distance, and define the complete relative fidelity of these channels. All these distances, conditioned upon the input state, are explicitly evaluated in terms of the contravariant densities Φ τ , Ψ τ of the channels. Thus, the CB distance is expressed as the maximum D cb (Φ, Ψ) = sup ρ D ρ cb (Φ, Ψ) of the standard entanglement trace distance over the input state densities ρ * ∈ A * as contravariant densities of normal states on A. The CH distance d c (Φ, Ψ) between two quantum channels Φ and Ψ is found τ for a given ρ. Here and below |A| means the "modulus" of an operator A, for which under the trace can be taken any of the expressions √ A † A and √ AA † (which are different in the case of not normal A).
CB distance has recently found an extensive use quantum information theory, see for example the review paper [ [4]]. While the CH distance, as will be shown here, is topologically equivalent to the CB distance between two channels, from the operational and computational point of view this new (Helinger) complete distance have certain advantages over the CB distance.
Note that in the finite dimensional case one can use for discriminating of the channels the standard entanglement relative fidelity with respect to the input tracial state ρ corresponding to ρ * = d −1 I. In fact such fidelity has been recently suggested by Raginsky [5] who used to define his fidelity relative to the maximal entangled state. However the corresponding fidelity distance is not equivalent to the CB distance, and there is no such measure in the infinite dimensional case. The normalized tracial states do not exist on type one algebras if d = ∞, and there is no maximal entangled state. We prove that our complete fidelity distance is equivalent to the CB distance, and give an operational interpretation of this fidelity in terms of a minimax problem for quantum encodings and decompositions, purifying the channels, in complete parallel to Uhlman's theorem [6].

Some facts and notation
Quantum state σ, identified as usual with a list of all expectations σ (A), is defined as a linear normalized functional A → σ (A) on the algebra B (H) of all bounded operators A in a separable Hilbert space H. The functional also satisfies the positivity condition σ A † A ≥ 0 for any A ∈ B (H) such that σ (I) = 1 for the identity operator I on H. As usual we shall consider only normal states σ (A) = A, σ , identifying them with density operators σ = |i σ ik k| defining the expectations σ (A) as sums, or the series Here σ is given in an orthogonal basis of real units i| ∈ H and |i = i| † as a Hermit-positive σ ≥ 0 matrix of σ ik = σ (|i k|) with the unit trace For the reason which will be explained later we prefer to use the tensor form (2.1) for the bilinear pairing A and σ rather than sesquilinear form TrσA = σ ik a ki which pairs A with the covariant formσ = [σ ik ] of the density matrix σ = σ ik in terms of the complex conjugated elementsσ ik ≡ σ ik = σ ki , coinciding with the transposed ones due to σ † = σ. Equivalently (2.1) can be written as A, σ = TrσA, whereσ denotes the transposed matrix σ ki . The matrix σ ik is called the contravariant density of the state σ with respect to the standard trace τ = Tr. As we shall see, contravariant density gives more adequate representation for quantum channels than the matrixσ = [ i|σ|k ] =σ, and its name can be explained in terms of the contravariant transformations.  on the predual space B ⊺ (H) given by an operator A ∈ B (H). Note that A ⊺ is not the transposed operator in H but is described in terms of the transposed operator A = A † as A ⊺ (σ) = Tr Aσ. The transposition A → A is related not to the operator but to the Hilbert space pairing φ, ψ = ϕ|ψ defining the vector transpose ψ →ψ as the reverse to |ψ → ψ | by identifying the complex conjugate elementsψ ∈ H with bra-vectors ψ|.
All of that can be easily generalized [1] to an arbitrary operator subalgebra B ⊆ B (H) with the standard, or a nonstandard normal faithful semifinite trace [7] ν when the standard one, τ = Tr, is not semifinite, e.g. is trivial on B in the sense that Tr A † A = ∞ for all operators A = 0 from the algebra B. Such a trace ν is defined as a nonnormalized state on B, finite on a weakly dense part of B, with the property ν A † A = ν AA † and separating B in the sense that ν A † A = 0 ⇒ A = 0. A quantum system will be called semifinite if it is described by the semifinite algebra, i.e. admits a semifinite, not necessarily standard, trace ν. (There exist also quantum infinite systems which are not semifinite.) The only difference is that the predual space B ⊺ of contravariant densities σ with respect to the trace ν on B may not be apart of the algebra B but a part of an opposite algebra B, or, if unbounded, affiliated to B. The opposite algebra can be defined on the same Hilbert space equipped with a complex conjugation χ →χ, i.e. isometric involution in H, as the subalgebra B = A ∈ B (H) : A ∈ B of complex conjugated operators A = A † . This B, equipped with the reference traceν A := ν (A), does not necessarily coincide with (B, ν) as in the simple case B = B (H). The "opposite" trace µ =ν, coinciding withν (A) = ν A on the opposite algebra B, defines the µ-pairing Below S τ ⊆ A * denotes the convex set of all positive normalized contravariant densities ̺ ≥ 0, τ (̺) = 1 for normal states on the von Neumann algebra A with respect to a normal faithful semifinite trace λ =τ . They can also be regarded as covariant densities, ρ * = ̺ ∈ A * of the normal states ρ (A) = τ (̺A) on the opposite algebra A with the reference trace τ .

Quantum Bures and trace distances
The difference between two states ρ, σ ∈ S (H) is usually measured by trace-norm distance given in terms of the modulus-difference of their trace densities ρ = ρ τ and σ = σ τ . It is also defined for non positive ρ and σ and thus doesn't have much informationoperational meaning. Another measure of this difference is relative entropy [8] which has clearly more information-operational meaning. However the quantum relative entropy is not symmetric and not unique, and is not a distance on S τ . The natural operational distance between two quantum states is quantum Bures distance which can be defined as the square root of the minimal Euclidean squared distance Here the infimum is taken over all all Schmidt decompositions of the trace-densities ρ and σ. Uhlman's theorem [6] states that this infimum is actually achieved at the value d (ρ, σ) is the relative fidelity of the states ρ and σ. Note that this can obviously be written as the trace of the matrix giving a much simpler formula Tr √ ρσ for the fidelity f (ρ, σ). It is also valid even if ρ and/or σ are not invertible as soon as the product ρσ = T ΛT −1 remains similar to a positive diagonal matrix Λ to make sense of √ ρσ = T Λ 1/2 T −1 . Thus, quantum Bures distance is a noncommutative generalization of the statistical Helinger distance which in the case of the commutative ρ and σ has the familiar form Tr √ ρ − √ σ 2 . Since Schmidt decompositions ψ † ψ = ψ † |j j|ψ correspond to purifications of the states σ, the infimum (3.2) has clear informationoperational meaning.
Although the squared fidelity distance d 2 is smaller [6] than D, they achieve the same maximal value d 2 = 2 = D on S (H) when f = 0, i.e. when the range of ρ is orthogonal to the range of σ. In fact, as it follows from the inequality D ≤ 2 1 − f 2 , see for example [9], D cannot be larger than 2d. Thus both distances are topologically equivalent, All of that can be easily generalized to a more general semifinite algebra B ⊂ B (H) with respect to a trace ν, inducing the (nonstandard) reference trace µ =ν on the opposite algebra B, and the pairing of B with the predual space B ⊺ affiliated to the opposite algebra B is understood in the sense of (2.2). The only difference is that the contravariant densities ρ = ρ µ , σ = σ µ , normalized with respect to trace µ =ν on B, may not be bounded, but they are still uniquely described by the positive selfadjoint operators in H affiliated to B. The trace and the Bures distance formulae remain the same with the reference trace µ =ν replacing the standard trace, and the fidelity formula with respect to the arbitrary trace µ is generalized to Here √ ρσ (and √ σρ) is still well-defined as the function T √ ΛT −1 of the operator ρσ = T ΛT −1 (and √ σρ = T ΛT −1 ) as being similar to a positive diagonal operator Λ with respect to a similarity transformation T . If ρ and σ are invertible, one can take either σ −1/2 U or ρ 1/2 V † as such T (respectively either ρ −1/2 V or σ 1/2 U † ), where U and V are unitary transformation diagonalizing respectively σ 1/2 ρσ 1/2 and ρ 1/2 σρ 1/2 .

Operational densities and quantum channels
In order to compare quantum channels we need to generalize these results to linear completely positive (CP) trace preserving mappings Φ ⊺ from the predual space A ⊺ of the "Alice" algebra A ⊆ B (g) on an input Hilbert space g into the predual space B ⊺ of the "Bob" algebra B ⊆ B (h) on the same or different output Hilbert space h. The notation H will be kept for the Hilbert product g ⊗ h with the total trace Tr = τ g ⊗ τ h inducing the trace λ ⊗μ on the entangled input-output system A ⊗ B as the product of the standard traces λ = τ g |A andμ = τ h |B if τ g = tr g and τ = tr remain on these subalgebras semifinite. Otherwise we should take non-standard reference traces τ on A and ν on B and define λ ⊗μ as the opposite trace to τ ⊗ µ on A⊗B.
As in the case of the states it is more convenient to describe quantum channels by the normal unital CP operations defined as the maps Φ : B → A on the output algebra B into the input algebra A, like the expectations σ mapping B into the trivial algebra A = C. These can always be defined as the Recall that a normal map Φ : Moreover, at least in the case of the simple algebra B = B (H), Φ is normal on B iff it is weak continuation of the CP map Φ|B 0 (h). This means that Φ like a normal state is completely determined on finite rank operators B = ik |i b ik k| by the operator entries Φ ik µ = Φ (|i k|) ∈ A of the Hermitian-positive matrix Φ µ = Φ ik µ , and thus has the form Φ (B) = B, Φ µ µ in terms of the (B, B ⊺ ) µ pairing 0 is a kernel on g ⊗ h given in an orthogonal basis of real units i| ∈ h and |i = i| † by the Hermit-positive A-valued matrix Φ ik µ having the unit partial trace The kernel Φ µ , called the density of a quantum operation Φ : B → A with respect to the traceμ (B) = tr h B on B, defines the density operators of output states σ = ρ • Φ by the partial tracing in terms of the (A, A ⊺ ) pairing with respect to the reference trace λ =τ on A.
Note that the usual trace-form of the pairing defines the covariant formσ = τ Φ ′ µ (ρ ⊗ I h ) of the output matrix σ by the partial transposition Φ ′ µ = Φ ki µ which may be not Hermitian-positive for the positive Φ µ . This indicates that covariant trace-pairing is not natural for describing quantum channelling and explains our preference to use the tensor form of the pairings corresponding to the contravariant form of the trace-densities. Otherwise there would not be one-to-one correspondence between complete positivity of operations and Hermitian positivity of their densities as kernels.
In general the density Φ µ is an unbounded Hermitian-positive operator, or even a generalized one, defined only as a kernel of a positive Hermitian form in g ⊗ h. Nevertheless we will still use the notations of the partial tracings and write µ (Φ µ ) = I h , Φ µ µ , defining the channel densities by the Hermitian positivity and the normalization conditions condition Φ (I h ) = I g . In particular, the density Φ µ of the operation Φ (B) = F † BF with respect to the standard trace µ = tr h is described as the kernel Φ µ = |F )(F |, where the generalized bra-vector (F | is well-defined on the finite linear combinations of the products ξ ⊗ η ∈ g ⊗ h by the partial transposition It cannot be defined as an element of the Hilbert space H = g ⊗ h for any isometry, a unitary operator F = U say, if the space g has infinite dimensionality dim g = ∞. Note that for the arbitrary output algebra B the quantum channel density as a positive self-adjoint, possibly unbounded operator (kernel) affiliated to A ⊗ B was first introduced in [2] with respect to operator-valued weights φ as CP maps into A densely defined in B, which correspond in our case to φ (B) = ν (B) I g . However, in order to avoid technicalities one can consider here only the channels with the bounded densities Φ µ with respect to the trace µ =ν.

An explicit formula for operational CB distance
As a measure of difference between two quantum channels Φ and Ψ : B → A one can adopt the usual boundedness norm distance D b (Φ, Ψ) := Φ − Ψ b (or B-distance for short), defined in terms of the density operators Φ µ and Ψ µ as where τ [∆ µ (ρ)] is dual action ∆ ⊺ (ρ) = ∆ µ , ρ λ of ∆ = Φ − Ψ which us equal to the partial tracing (4.2) of the operator in terms of the µ-density operator ∆ µ = Φ µ −Ψ µ . Here S τ ⊗µ (σ) denotes the convex set of normal state density operators ω ∈ A ⊗ B ⊺ with respect to the product trace τ ⊗ µ in A ⊗ B, having fixed partial trace τ (ω) = σ.
However it is more appropriate to use the larger CB-distance D cb (Φ, Ψ) := ∆ cb since the difference ∆ = Φ − Ψ of two CP maps is not just bounded, but it is also completely bounded, ∆ cb < ∞, where · cb ≥ · b is the so-called norm of complete boundedness [10] (or CB-norm for short). Instead of maximizing over normal input states ρ ∈ S λ one should maximize over normal input entanglements with any probe quantum system A, or at least with the standard one, described by the "matrix algebra" A = B (k) = A on the Hilbert space k = ℓ 2 of all squaresummable sequences indexed by N. The normal entanglements are usually given by the densitiesρ ∈ Sτ of the normal statesρ on the algebra A ⊗ A with respect to the standard traceτ = τ k ⊗ τ .
This maximization can be written as the supremum over all input states of the conditional CB semidistance as the maximal trace distance of the entangled output statesφ =ρ •Φ andσ =ρ •Ψ on A ⊗ B with respect to the input entangled statesρ having fixed margin ρ on A.
of trace class matrices λ ρ ik in the natural basis of ℓ 2 and A ⊆ B (g), withτ = τ k ⊗ µ. They are applied as∆ ⊺ (ρ) = ∆ ⊺ ρ ik on the convex set Sτ (ρ) of the predual space (A⊗A) ⊺ = A ⊺ ⊗A ⊺ of all A ⊺ -valued density matriceŝ ρ = ρ ik ,ρ ik ∈ A ⊺ with respect to the traceτ on A⊗A = A⊗A, having the partial trace τ k (ρ) := iρ ii = ρ. Taking into account that anyρ ∈ Sτ (ρ) can be written in blocks as ρ 1/2 Π ik ρ 1/2 with Π ik ∈ A defining a normal unital CP map Π Ā =ā ik Π ik on the matrix algebra A into A, we can write whereΠ = Π ⊗ Id is the right ampliation A ⊗ B → A ⊗ B of the normal unital CP map Π given by the A-valued density matrix Π ik with respect to τ k . Thus the supremum in (5.2) can be expressed in the form of the supremum over all normal unital CP maps Π as where we used the obvious inequality Π ⊺ (ω) ⊺ ≤ ω ⊺ := (τ ⊗ µ) |ω| valid for any linear trace preserving CP map Π ⊺ on A ⊺ into A ⊺ = B ⊺ (k) and ω ∈ A ⊗ B ⊺ . The inequality in (5.3) is actually the equality in the case of separable A as it can easily be seen for the simple algebras A = B (g) when the supremum is obviously achieved at Π = Id on the opposite input algebra A = A coinciding with B (k) in a representation g = k.
In order to obtain the same formula for an arbitrary semifinite algebra A, the supremum in (5.2) overρ ∈ Sτ (ρ) should also be extended to the nonstandard A. This can be seen as optimization of the operational distance via all quantum encodings [1] as CP mappings of not only standard algebra A = B (k) into A ⊺ but also less simple algebras, including A = A. Quantum encodings were defined in [1] by normal CP maps π : A → A on any semifinite "quantum alphabet" algebra A with fixed normalization π (I k ) = ρ. They describe normal couplings of the corresponding state ρ on A with the normal states on A, having the densities P = Π ⊺ (ρ) = π ⊺ (I g ) with respect to a trace τ on A, byρ where Πτ ∈ A ⊺ ⊗ A is the density operator of the unital CP map Π. The maximal distance over all such encodings is obviously achieved on the standard coupling π * Ā = ρ 1/2Ā ρ 1/2 as in the case of the simple algebra A = B (g), but the optimal "encoding alphabet" system algebra A = A does not coincide with A if A = A but be antiisomorphic (opposite) to A. The standard entanglement defines the optimal compound state where υ = ρ 1/2 and ω := |υ)(υ| ≡ ω (ρ) is the optimal density operatorρ * of this standard entangled state with respect toτ = τ ⊗ λ and the partial trace τ (ω) = ρ. Thus the formula (5.4) gives an expression for the CB-distance as the supremum of the conditional CB semidistance in terms of the standard input entangled states ω (ρ) ∈ A ⊗ A ⊺ . It is maximal trace-distance of the optimally entangled states on A ⊗ B described by the densitiesφ =Φ ⊺ (ω) andσ =Ψ ⊺ (ω) with the same partial trace P = λ (ω) =ρ.
One can also show that the CB-distance is majorized by the natural densityoperator distance (complete distance) D c (Φ, Ψ) ≥ D cb (Φ, Ψ) having the particularly simple form Ψ).) However D c is not equivalent to the CB-distance, that is closeness of Ψ n to Φ in the sense D cb (Φ, Ψ n ) ց 0 does not guarantee the closeness with respect to D c (Φ, Ψ n ) ≥ D cb (Φ, Ψ n ), and it is difficult to give an operational meaning of the optimality criterion defined by this distance. This is why an operational fidelity distance is even more desirable for quantum channels than for states.

Helinger distance and relative channel fidelity
One can define a Helinger like operational square-distance between two quantum operations Φ and Ψ as the Bures square-distance d (ϕ, σ) 2 for the output states ϕ = ρ • Φ, σ = ρ • Ψ in terms of their output µ-densities and similar for φ = Φ ⊺ (ρ) in the notation of the previous section, maximized over the input states ρ. In the case of the channels described by the normal unital CP maps Φ and Ψ this can be expressed as d (Φ, Ψ) 2 /2 = 1 − f (Φ, Ψ) in terms of the the minimal fidelity of the output states over all inputs ρ ∈ S λ with respect to λ =τ . As for any two output states φ, σ on B the following equivalence inequality obviously holds for this Helinger (H-) distance: However there is no such equivalence inequality between this fidelity distance and the CB distance (5.2), and it is difficult to give a definition of this fidelity without reference to the input states ρ.
Since the map Φ and Ψ are not just positive but CP, it is more appropriate to define the complete fidelity distance of the operations Φ and Ψ as the supremum for the input-output entangled statesφ =ρ •Φ andσ =ρ •Φ over the densitieŝ ρ ∈ (A ⊗ A) ⊺ describing quantum encodings of a fixed input state ρ, similar to (5.2). Since in the case of the unital Φ, Ψ this can be written as d ρ c (Φ, Ψ) 2 /2 = 1 − f ρ c (Φ, Ψ) in terms of the complete relative quantum channel fidelity conditioned by an input state ρ. Here as in (5.2) the minimization is given over quantum encodingsρ = I k ⊗ ρ 1/2 Π τ I k ⊗ ρ 1/2 ∈ Sτ (ρ) described by normal unital CP maps Π : A → A in terms of their densities Π τ with respect to τ = τ k for a fixed ρ ∈ S λ . Using the monotonicity [8,9] f Π , Ψ µ (ρ)) of the quantum state relative fidelity with respect to such Π, we obtain the inequality This inequality is obviously equality achieved at Π = Id in the case the simple input algebra A = B (g) on g = ℓ 2 , and this lower bound is also achieved for any separable A. Thus we arrive to the following formula for the complete relative fidelity f c of quantum channels in terms of their densities Φ µ and Ψ µ with respect to the trace µ and a given input state ρ. This defines the complete Helinger operational half-square semidistance for each ρ: where we used an equivalent representation (7.8) for the fidelity (6.3) which will be derived in the next Section. Since this is simply relative fidelity for two optimal input-output entangled states described by the densitiesφ = Φ µ (ρ) andσ = Ψ µ (ρ), the following equivalence inequality obviously holds for each ρ: . By allowing the arbitrary semifinite quantum alphabet algebras A we can extend this formula to the complete channel fidelity as the infimum µ over all input states ρ on any algebra A. It defines the complete operational Helinger (CH-) distance by d 2 c /2 = 1−f c as the supremum of (6.4) over all standard encodings ω such thatΦ ⊺ (ω) = Φ µ (ρ) andΨ (ω) = Ψ µ (ρ), corresponding to the optimal quantum alphabet algebra A = A. Applying the equivalence inequality (3.4) to these states and taking then the supremum over all ρ ∈ S λ we obtain the equivalence inequality also for the complete fidelity distance and the CB-distance (5.5). Thus the complete relative fidelity (6.2) refines the CB-norm inequalities D b ≤ D cb ≤ D c by providing the channel fidelity distance d c which satisfies the equivalence inequalities (6.6) due to the inequality D cb ≤ 2 1 − f 2 c .The complete channel fidelity f c has clear operational meaning as the minimal relative fidelity of the compound states achieved over all input-output entanglements via two quantum channels described by the densities Φ µ , Ψ µ [1]. In particular, as we show in the next section if Φ is given in Kraus form Φ (B) = F † j BF j and Ψ is pure Ψ (B) = V † BV , given by an isometry V .

Operational CH-distance as a minimax problem
Following analogy with quantum Bures distance as a variational problem (3.2) we should define the operational Bures distance as the square root of the natural quadratic distance between generalized Schmidt decompositions Γ, Υ ∈ B (f ⊗ g) of their density operators Φ µ and Ψ µ : without explicit reference to the input state ρ. In the case of the standard trace µ induced by tr h this can be written as the minimization of the Hilbert module square distance corresponding to the purified Schmidt decomposition Here (F | and (V | are the columns of (F j |, (V j | which are the components of in an orthonormal basis {|j } of H, say |j = |i ⊗ |k ≡ |i, k , defining the bounded operators F j , V j : g → f as acting on the right of the bra-vectors k| ∈ g: over all input densities ρ ∈ S λ with respect to λ =τ . Taking into account that A † A = sup ρ∈S λ τ ρA † A , and that the positive function given by the total trace τ ⊗ µ is convex with respect to Γ − Υ and concave with respect to ρ, we can rewrite the minimal distance (7.1) in the form where we exchanged the extrema since since S λ is convex and the function c is actually a positive quadratic form with respect to Γ − Υ, certainly achieving its infimum. Thus, (7.1) can be represented in the form where the first term achieves the value 2 for the unital Φ and Ψ, and Now we will prove that this supremum is in fact the conditional relative fidelity (6.3), and the complete relative fidelity (6.5) for quantum channels coincides with minimax for the real part of the Hilbert-Schmidt scalar product (τ ⊗ µ) X † Y of X = Γ (ρ ⊗ I g ) 1/2 and Y = Υ (ρ ⊗ I g ) 1/2 with respect to the total trace τ ⊗ µ. Since the operators X and Y define the Schmidt decompositions Φ µ (ρ) = X † X and Ψ µ (ρ) = Y † Y of the density operatorsΦ ⊺ [ω (ρ)] andΨ ⊺ [ω (ρ)] for the optimal entangled states on the algebra A ⊗ B = A ⊗ B, the complete fidelity (6.5) is simply the minimax relative fidelity for the input-output entangled states obtained by two different transformationsΦ ⊺ andΨ ⊺ of the input standard entangled states on algebra A ⊗ A = A ⊗ A. It is first maximized over all Schmidt decompositions corresponding to the Kraus decompositions (7.2) of Φ and Ψ, and then minimized with respect to all density operators ρ of the input states.
Lemma 1. Let R, S ∈ A ⊗ B be positive bounded operators such that they have finite trace τ ⊗ µ. Then where T = √ XSX † . This supremum is achieved at any X ∈ A ⊗ B satisfying the condition X † X = R, X = R 1/2 ≡ X o say, and at Y = Y o satisfying the equation Proof. First we observe, by applying the Schwarz inequality that this supremum is finite. In order to find it one can use Lagrangian multiplier method. Fixing X satisfying X † X = R, X o = R 1/2 say, we can write the Lagrangian function as where L = L † ∈ A ⊗ B is the Lagrangian multiplier corresponding to the Hermitian condition S = Y † Y = S † . At the stationary point Multiplying this from the left by X and from the right by X † this gives This extremal value is the maximal value because of convexity of the maximizing function in (7.6). Note that due to √ U † XSX † U = U † √ XSX † U for any unitary U , the supremum (7.6), coinciding with (7.7) does not depend on the choice of X satisfying X † X = R. Indeed, by virtue of polar decomposition X = U R 1/2 corresponding to X = R 1/2 , we obtain the optimal value in (7.6) in either form (τ ⊗ µ) (T ) = (τ ⊗ µ) √ RS or (τ ⊗ µ) (T ) = (τ ⊗ µ) √ R 1/2 SR 1/2 .
Note that choosing X = R 1/2 for R = Φ µ (ρ) and Y = R −1/2 √ RS for S = Ψ µ (ρ) gives the formula (6.3). This proves that the operational Bures distance d * where we used the identity (V | (ρ ⊗ I h ) |F ) = tr g ρF † V . This defines the expression used in (6.7), and if Φ is also pure, Φ (B) = F † BF , we obtain even simpler formula f ρ c (Φ, Ψ) = tr g ρF † V . Therefore, the CF-distance of such Φ and Ψ is evaluated in d 2 c /2 = 1 − f ρ c (Φ, Ψ) by the infimum which is minimal magnitude inf ρ tr g ρF † V of correlation between F and V in the case of pure Φ. Note that the CB distance cannot be so explicitly evaluated in the case of one pure channel, and it does not have such simple interpretation even when both channels are pure.