Fidelity and Fisher information on quantum channels

The fidelity function of quantum states have been widely used in quantum information science and frequently arises in the quantification of optimal performances for the estimation and distinguish of quantum states. A fidelity function on quantum channel is expected to have same wide applications in quantum information science. In this paper we propose a fidelity function on quantum channels and show that various distance measures on quantum channels can be obtained from this fidelity function, for example the Bures angle and the Bures distance can be extended to quantum channels via this fidelity function. We then show that the distances between quantum channels lead naturally to a new Fisher information which quantifies the ultimate precision limit in quantum metrology, the ultimate precision limit can thus be seen as a manifestation of the distances between quantum channels. We also show that the fidelity on quantum channels provides a unified framework for perfect quantum channel discrimination and quantum metrology, in particular we show that the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.

The fidelity function of quantum states have been widely used in quantum information science and frequently arises in the quantification of optimal performances for the estimation and distinguish of quantum states. A fidelity function on quantum channel is expected to have same wide applications in quantum information science. In this paper we propose a fidelity function on quantum channels and show that various distance measures on quantum channels can be obtained from this fidelity function, for example the Bures angle and the Bures distance can be extended to quantum channels via this fidelity function. We then show that the distances between quantum channels lead naturally to a new Fisher information which quantifies the ultimate precision limit in quantum metrology, the ultimate precision limit can thus be seen as a manifestation of the distances between quantum channels. We also show that the fidelity on quantum channels provides a unified framework for perfect quantum channel discrimination and quantum metrology, in particular we show that the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.

I. INTRODUCTION
Fidelity, as a measure of the distinguishability between quantum states [1][2][3], plays an important role in many areas of quantum information science, for example it is related to the precision limit in quantum metrology [4], serves as a measure of entanglement preservation through noisy quantum channels [5], and a measure of entanglement preservation in quantum memory [6]; it has also been used as a characterization method for quantum phase transitions [7], and a criterion for successful transmission in formulating quantum channel capacities [8].
Unlike the fidelity of quantum states which is defined directly on quantum states, most commonly used measures for the distinguishability of quantum channels are defined indirectly through the effects of the channels on the states. For example the diamond norm, which is defined as K 1 − K 2 = max ρ SA K 1 ⊗ I A (ρ SA ) − K 2 ⊗ I A (ρ SA ) 1 [9][10][11]( here X 1 = T r √ X † X, ρ SA denotes a state on system+ancilla, and I A denotes the identity operator on the ancillary system), is induced by the trace distance on quantum states ρ 1 − ρ 2 1 ; another measure on quantum channels which is defined as arccos F min (K 1 , K 2 ) = arccos min ρ SA F S [K 1 ⊗I A (ρ SA ), K 2 ⊗I A (ρ SA )] [12,13], is induced by the fidelity on quantum states F S (ρ 1 , ρ 2 ) = T r ρ 1 2 1 ρ 2 ρ 1 2 1 . These induced measures through quantum states lack a direct connection to the properties of quantum channels, which severely restrict the insights that can be gained from these measures. A direct measure on quantum channels is expected to provide more insights thus highly desired.
In this paper we provide a fidelity function defined directly on quantum channels, and show that this fidelity function on quantum channels, together with the classical fidelity on probability distribution and the fidelity on quantum states, form a hierarchy of fidelity functions in terms of optimization. This fidelity function on quantum channels also lead to various distance measures defined directly on quantum channels, in particular we show the Bures angle and the Bures distance can be extended to quantum channels. We then show the distance between quantum channels leads naturally to a new Fisher information on quantum channels which quantifies the ultimate precision limit in quantum metrology. We also show that this fidelity function provides a unified framework for perfect quantum channel discrimination and quantum metrology, in particular we show the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.

II. FIDELITY FUNCTION ON QUANTUM CHANNELS
We start by defining the fidelity function on unitary channels then extend it to noisy channels. For a m × m unitary matrix U , we denote e −iθj as the eigenvalues of U , where θ j ∈ (−π, π] for 1 ≤ j ≤ m and we call θ j the eigen-angles of U . We define(see also [14][15][16]) U max = max 1≤j≤m | θ j |, and U g as the minimum of e iγ U max over equivalent unitary operators with different global phases, i.e., U g = min γ∈R e iγ U max . We then define (1) Quantitatively C(U ) is equal to the maximal angle that U can rotate a state away from itself [16,17,21], i.e., cos[C(U )] = min |ψ | ψ|U |ψ |. For mixed states it can be written as cos[C(U )] = min ρ F S (ρ, U ρU † ). If θ max = θ 1 ≥ θ 2 ≥ · · · ≥ θ m = θ min are arranged in decreasing order, then C(U ) = θmax−θmin 2 when θ max − θ min ≤ π [16]. We then define Θ QC (U 1 , U 2 ) = C(U † 1 U 2 ), here U 1 and U 2 are unitary operators on the same Hilbert space(we can expand the space if they are not the same). It is easy to see that Θ QC (U 1 , U 2 ) thus corresponds to the maximal angle between the output states of U 1 and U 2 (however we note that the definition of Θ QC (U 1 , U 2 ) is independent of the states). We then denote F QC (U 1 , U 2 ) = cos[Θ QC (U 1 , U 2 )] as the fidelity between U 1 and U 2 . For unitary channels this is equivalent to the fidelity function proposed previously in [17]. We now generalize this to noisy quantum channels. A general quantum channel K, which maps from m 1 -to m 2dimensional Hilbert space, can be represented by Kraus operators, Equivalently it can also be written as where |0 E denotes some standard state of the environment, and U ES is a unitary operator acting on both system and environment, which we call as the unitary extension of K.
We define Θ QC (K 1 , K 2 ) = min {U ES1 ,U ES2 } Θ QC (U ES1 , U ES2 ) and F QC (K 1 , K 2 ) = cos Θ QC (K 1 , K 2 ), where U ESi are unitary extensions of K i , i ∈ {1, 2}. In Appendix A, we show that the optimization can be taken by fixing one unitary extension and just optimizing over the other unitary extension, i.e., In terms of F QC (K 1 , K 2 ) it can be written as This can be seen as the counterpart of Uhlmann's purification theorem on quantum states [22](however the proof does not use Uhlmann's purification theorem [18]). In Appendix B, we show that Θ QC (K 1 , K 2 ) is a metric and can be computed directly from the Kraus operators of K 1 and K 2 as [18] and F 2j denote the Kraus operators of K 1 and K 2 respectively, w ij denotes the ij-th entry of a q × q matrix W with W ≤ 1 where · is the operator norm which corresponds to the maximum singular value, here W arises from the non-uniqueness of the Kraus representations. Thus We emphasize that F QC is defined directly on quantum channels without referring to the states, such direct connection, in contrast to the induced measure, is crucial when applying the fidelity to channel discrimination and quantum metrology as we will show later. Furthermore the fidelity can be formulated as a semi-definite programming and computed efficiently as max W ≤1 Analogous to the Bures distance on quantum states B S (ρ 1 , ρ 2 ) = 2 − 2F S (ρ 1 , ρ 2 ), we can similarly define a Bures distance on quantum channels as B QC (K 1 , K 2 ) = 2 − 2F QC (K 1 , K 2 ). In Appendix A, we prove an intriguing and useful connection between B QC (K 1 , K 2 ) and the minimum distances between the Kraus operators of K 1 and K 2 as where {F 1i }, {F 2i } are the sets of all equivalent Kraus representations of K 1 and K 2 respectively. This connection is particular useful in studying the scalings of the distance between quantum channels as we will show later.
In which sense we call F QC (K 1 , K 2 ) a fidelity function? It turns out that F QC (K 1 , To see this, it is proved in the supplemental material of Ref. [18] that which coincides with Eq. (6). From this relationship it is also immediate clear that F QC (K 1 , K 2 ) is stable, i.e., F QC (K 1 ⊗ I, K 2 ⊗ I) = F QC (K 1 , K 2 ). This result gives an operational meaning to F QC (K 1 , , K 2 ). We emphasize that although we made connections between F QC (K 1 , K 2 ) and the minimum fidelity of the output states, F QC (K 1 , K 2 ) is defined directly on quantum channels and does not depend on the states. The definition and the operational meaning of F QC (K 1 , K 2 ) play distinct roles in applications, the operational meaning provides a physical picture while the direct definition brings insights which enable or ease the proofs and computations, which will be demonstrated in the applications. This is in analogy to how fidelity of quantum states is connected to the classical fidelity F S (ρ 1 , [3], here similarly the fidelity between quantum states has the operational meaning as the minimum classical fidelity, however the fidelity between quantum states is defined directly on quantum states which is independent of the measurements and such direct definition has provided numerous insights which would be hindered with just the classical fidelity. It is known that the trace distance and the fidelity between quantum states have the following relationships [19] 1 from which it is straightforward to get the relationships between the diamond norm and the fidelity of quantum channels. This can be obtained by substituting which gives Since F QC (K 1 , K 2 ) can be computed directly from the Kraus operators, this also provides a way to bound the diamond norm using the Kraus operators.
In [20] the Choi matrices of the quantum channels are used to compute the fidelity between the channels, which corresponds to the fidelity between the output states of two quantum channels when the input state is taken as the maximal entangled state. As the maximal entangled state is in general not the optimal input state, the fidelity thus defined does not have operational meaning as the minimum fidelity of the output states, thus can not be related to the ultimate precision limit in quantum metrology etc(instead related to the precision limit when the probe state is taken as the maximally entangled state).

III. A UNIFIED FRAMEWORK FOR QUANTUM METROLOGY AND PERFECT CHANNEL DISCRIMINATION
Next we demonstrate the applications in quantum information science, in particular we show how the fidelity provides a unified platform for the ultimate precision in quantum metrology and the minimum number of uses needed for perfect channel discrimination.
The task of quantum metrology, or quantum parameter estimation in general, is to estimate a parameter x encoded in some channel K x , this can be achieved by preparing a quantum state ρ SA and let it go through the extended channel K x ⊗ I A with the output state ρ x = K x ⊗ I A (ρ SA ). By performing POVM, {E y }, on ρ x one gets the measurement result y with probability p(y|x) = T r(E y ρ x ). According to the Cramér-Rao bound [24][25][26][27], the standard deviation for any unbiased estimator of x is bounded below by δx ≥ , where δx is the standard deviation of the estimation of x, J C [p(y|x)] is the classical Fisher information and n is the number of times that the procedure is repeated. The classical Fisher information can be further optimized over all POVMs, which gives where the optimized value J S (ρ x ) is usually called the quantum Fisher information [4,24,25,28], here for distinguish we will call it the quantum state Fisher information. We first recall established connections between the fidelity functions and the Fisher information. Given ρ x and its infinitesimal state ρ x+dx , for a given POVM {E y }, the classical fidelity between p(y|x) = T r(E y ρ x ) and p(y|x The classical Fisher information is related to the classical fidelity as 1 up to the second order of dx [4], this can also be written as If we optimize over {E y } the classical fidelity then leads to the fidelity between quantum states as [4] min and the classical Fisher information leads to the quantum state Fisher information J S (ρ x ) = max {Ey} J C [p(y|x)] and up to the second order of dx [4,28] The precision can be further improved by optimizing over the probe states, which leads to the ultimate local precision limit of estimating x from K x . Intuitively, this ultimate precision limit should be quantified by the distance between K x and its infinitesimal neighboring channel K x+dx , in a way analogous to how Bures distance of quantum states quantifies the precision limit of estimating x from the state ρ x [4]. However although much progress has been made on calculating the ultimate precision limit [29][30][31][32][33][34][35][36][37], such a clear physical picture has still not been established after more than two decades since Braunstein and Caves's seminal paper [4], this is mainly due to the lack of proper tools on quantum channels. Here we show that the fidelity between quantum channels can be used to establish such a physical picture, which also leads naturally to a new Fisher information on quantum channel.
Further optimizing over the probe states this leads naturally to a quantum channel Fisher information J QC (K x ) = max ρ SA J S (ρ x ) which is similarly related to the distance on quantum channels as The quantum channel Fisher information quantifies the ultimate precision limit upon the optimization over the measurements and probe states This connects the precision limit directly to the distance between quantum channels which provides a clear physical picture for the ultimate precision limit. The scaling of the ultimate precision limit can now be seen as a manifestation of the scaling of the distances between quantum channels as we now show. Two schemes on multiple uses of quantum channels are usually considered in quantum parameter estimation, the parallel scheme and the sequential scheme as shown in Fig.1. We will show that for both schemes, the scaling of the distances between two quantum channels are at most linear, which underlies the scaling for the Heisenberg limit.
For parallel scheme with N uses of a channel K as shown in Fig.2, the total dynamics can be described by K ⊗N ⊗I A . If we denote U ES as one unitary extension of K, then U ⊗N ES is a unitary extension of K ⊗N as shown in Fig.3. Given two channels K 1 and K 2 , we choose U ES1 and U ES2 as the unitary extension for K 1 and K 2 respectively which satisfies we then have For the sequential scheme, we consider the general case that controls can be inserted between sequential uses of the channels. Any measurements that are used in the control can be substituted by controlled unitaries with ancillary systems, the controls interspersed between the channels can thus be taken as unitaries, which is shown in Fig.4. Parallel scheme can be seen as a special case of the sequential scheme by choosing the controls as SWAP gates on the system and different ancillary systems [36]. We show that with N uses of the channel, the distance is still bounded above by N Θ QC (K 1 , K 2 ).
We present the proof for the case of N = 2, same line of argument works for general N . For N = 2, one unitary extension of U 2 K 1 U 1 K 1 is U 2 U E2S1 U 1 U E1S1 , similarly U 2 U E2S2 U 1 U E1S2 is a unitary extension of U 2 K 2 U 1 K 2 , here U Ej Si denote a unitary extension of K i , i = 1, 2, with E j as the environment. We can choose U Ej Si such that Θ QC (K 1 , K 2 ) = Θ QC (U Ej S1 , U Ej S2 ), here all operators are understood as defined on the whole space so the  multiplication makes sense, for example the control U 1 , which only acts on the system and ancillaries, is understood as U 1 ⊗ I E , an operator on the whole space including the environment. We then have i.e., with two uses of the channel, the distance is bounded above by 2Θ QC (K 1 , K 2 ). With the same line of argument it is easy to show that with N uses of the channel the distance is bounded above by N Θ QC (K 1 , K 2 ).
Substitute K 1 with K x and K 2 with K x+dx , we have Θ QC (N K x , N K x+dx ) ≤ N Θ QC (K x , K x+dx ) for both schemes. From Eq.(19) the ultimate precision limit is then bounded by the scaling 1/N is called the Heisenberg scaling, which, as we showed, is just a manifestation of the fact that the distance between quantum channels can grow at most linearly with the number of channels.
For N uses of the channels under the parallel scheme we can also obtain a tighter bound as here K W = q i=1 q j=1 w ij F † 1i F 2j as previously defined, and the inequality holds for any W with W ≤ 1 (see Appendix C). In the asymptotical limit, N (N − 1) I − K W 2 is the dominating term, in that case we would like to choose a W minimizing I − K W to get a tighter bound. This can be formulated as semi-definite programming with If we let K 1 = K x and K 2 = K x+dx , then Eq.(23) provides bounds on the scalings in quantum parameter estimation, which is consistent with the studies in quantum metrology [29,30,32,35,36] but here with a more general context (see also Ref. [18]).
Given two quantum channels K 1 and K 2 , they can be perfectly discriminated with one use of the channels if and only if there exists a ρ SA such that K 1 ⊗ I A (ρ SA ) and K 2 ⊗ I A (ρ SA ) are orthogonal, i.e., min ρ SA F S [K 1 ⊗ I A (ρ SA ), K 2 ⊗ I A (ρ SA )] = 0, which is the same as Θ QC (K 1 , K 2 ) = π 2 . When K 1 and K 2 can not be perfectly discriminated with one use of the channel, finite number of uses may able to achieve the task [42]. This is in contrast to the perfect discrimination of non-orthogonal states which always requires infinite number of copies. The minimum number of uses needed for perfect channel discrimination should satisfy Θ QC (N K 1 , N K 2 ) = π 2 . The perfect channel discrimination is thus determined by the distances between quantum channels, and the scalings of Θ QC (N K 1 , N K 2 ) obtained before can be used to determine the minimum N . For example, from Θ QC (N K 1 , N K 2 ) ≤ N Θ QC (K 1 , K 2 ) we can obtain a lower bound on N as where x is the smallest integer not less than x. This bound is tighter than existing bounds for noisy channels [40] and for unitary channels it reduces to the formula which is known to be tight [17]. For noisy channels under the parallel scheme we can also substitute Θ QC (K ⊗N 1 , K ⊗N 2 ) = π 2 into the inequality (23) to get a tighter bound. The lower bound on minimum N can also be obtained via a connection to quantum metrology. Given two channels K 1 and K 2 , let K x , x ∈ [a, b] as a path connecting K 1 and K 2 . With N uses of the channel under the parallel strategy . From the triangular inequality This connects the prefect channel discrimination to the ultimate precision limit. By choosing different paths various useful lower bounds on the minimum number of uses for perfect channel discrimination can be obtained. For example, given K 0 (ρ) = e iθσ1 ρe −iθσ1 and K 1 = 1+η 2 ρ + 1−η 2 σ 3 ρσ 3 , where σ 1 , σ 2 and σ 3 are Pauli matrices and assume θ = 0.3, η = 0.5. For the parallel strategy the lower bound given by Eq. (25) is N ≥ π 2Θ QC (K0,K1) = 3. If we choose a simple path K x = (1 − x)K 0 + xK 1 , x ∈ [0, 1], which is a line segment connecting K 0 to K 1 , then with the connection provided by Eq. (26) we obtain N ≥ 4. Other paths may be explored to further improve the bound. By using the inequality (23) with the W obtained from the semi-definite programming that minimizes I − K W , we get N ≥ 5. For any N we can also choose the W to minimize ) with the increasing of N , it turns out that the minimum N such that Θ QC (K ⊗N 0 , K ⊗N 1 ) = π 2 is actually 6. All computations here are done with the CVX package in Matlab [44].

IV. SUMMARY
A fidelity function defined directly on quantum channels is provided, which leads to various distance measures defined directly on quantum channels, as well as a new Fisher information on quantum channel. This forms another hierarchy for fidelity functions and Fisher information as shown in the table: where cos Θ i = F i and J i = lim dx→0 In this table the functions on quantum states equal to the optimized value over all measurements of the corresponding functions on probability distribution, and the functions on quantum channels equal to the optimized value over all probe states of the corresponding functions on quantum states. This framework connects quantitatively the ultimate precision limit and the distance between quantum channels, which provided a clear physical picture for the ultimate precision limit in quantum metrology. It also provide a unified framework for the continuous case in quantum parameter estimation and the discrete case in perfect quantum channel discrimination, with this framework the progress in one field can then be readily used to stimulate the progress of the other field. We expect these tools will find wide applications in many other fields of quantum information science. We show that the distance between two quantum channels Θ QC (K 1 , can be computed from the Kraus operators of K 1 and K 2 as Θ QC (K 1 , , here U ESi are unitary extensions of K i , i ∈ {1, 2} and λ min (K W + K † W ) denotes the minimum eigenvalue of K W + K † W with K W = ij w ij F † 1i F 2j , F 1i , F 2j denotes the Kraus operators of K 1 and K 2 , w ij denotes the ij-th entry of a q × q matrix W with W ≤ 1( · is the operator norm which equals to the maximum singular value), q is the number of the Kraus operators. Furthermore the minimization on both U ES1 and U ES2 can be reduced to the minimization of just one We start by a general unitary extension for any given channel K(ρ) = q j=1 F j ρF † j with q j=1 F † j F j = I, which maps from a m 1 -to m 2 -dimensional Hilbert space, F q * * · · · * 0 * * · · · * . . . . . . . . .
where W E ∈ U (p) only acts on the environment and can be chosen arbitrarily, here U (p) denotes the set of p × p unitary operators with p ≥ q as p − q zero Kraus operators can be added. Here only the first m 1 columns of U are fixed, the freedom of other columns can be represented as F 1 * * · · · * F 2 * * · · · * . . . . . . . . .
It is easy to see that W ≤ 1, conversely for any W with W ≤ 1 it can be imbedded as the first q × q block of a unitary matrix [45]. Thus by varying W E1 and W E2 we can take W to be any q × q matrix with is now reduced to the optimizing over V 1 , V 2 and W .
Next we optimize over W . Basically we need to find W such that arccos 1 2 λ min [K W + K † W ] is minimized, which is equivalent to find max |W |≤1 Note that the freedom of global phase from · max to · g (see main text for definitions) has been included in the freedom of W and since max |W |≤1 It is obvious that the freedom of W can be achieved by only varying W 1 or W 2 , thus the equality can be attained by just exploring the freedom of V 1 and W 1 , or V 2 and W 2 . We then have Next we show that this distance measure has a connection to the minimum distance between equivalent Kraus operators. Given two quantum channels, (zero Kraus operators can be appended if the number of the Kraus operators are not the same), by appending additional p − q zero Kraus operators, we have the Kraus operators for K 1 and K 2 as {F 11 , F 12 , · · · , F 1q , 0, · · · , 0} and {F 21 , F 22 , · · · , F 2q , 0, · · · , 0} respectively. Equivalent Kraus operators for K 1 and K 2 can be represented asF 1i = k u ik F 1k andF 2i = k v ik F 2k where u ik and v ik are entries of U, V ∈ U (p) respectively, here 1 ≤ i ≤ p. Then where K W = q ij w ij F † 1i F 2j and w ij is the ij-th entry of W , which is the first q × q block of U † V and can be any q × q matrix with W ≤ 1 by varying U and V , i.e., by varying the equivalent representations of K 1 and K 2 . Thus we then have Appendix B: ΘQC (K1, K2) defines a metric on quantum channels We show that Θ QC (K 1 , K 2 ) defines a metric on quantum channels. First we show that Θ QC (U 1 , U 2 ) = C(U † 1 U 2 ), where C is defined in the main text, is a metric on unitary channels. We start by listing some useful properties of C(U ): where V is any unitary operator. The first equality is obvious from the definition; the second inequality can be easily verified using the formula C(U ) = θmax−θmin 2 when θ max −θ min ≤ π, the equality is saturated when C(U 1 )+C(U 2 ) ≤ π 2 ; proof of the third inequality can be found in [47,48].
It is obvious that Θ QC (U, U ) = 0 and Θ QC (U 1 , where for the inequality we have used the property that C(U 1 U 2 ) ≤ C(U 1 ) + C(U 2 ). This shows that Θ QC (U 1 , U 2 ) is a metric on unitary operators.
For two general channels, Θ QC (K 1 , where U ES1 and U ES2 are unitary extensions for K 1 and K 2 respectively. It is easy to see that Θ QC (K 1 , K 2 ) = Θ QC (K 2 , K 1 ) ≥ 0 and the equality is saturated only when K 1 = K 2 . We show that Θ QC also satisfies the triangular inequality as the last inequality is valid for any U ES2 , specially we can choose the U ES2 which minimizes C(U † ES2 U ES3 ), thus Θ QC (K 1 , K 2 ) thus defines a metric on the space of quantum channels.