Fidelity and Metrics of quantum states

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1 Fidelity and Metrics of quantum states Zhi-Hao Ma, Fu-Lin Zhang, Jing-Ling Chen arxiv: v [quant-ph] ov 008 Abstract Fidelity plays an important role in quantum information theory. In this paper, two pairs of metrics of quantum states are introduced based on the Uhlmann-Jozsa fidelity and super-fidelity. They are proved to satisfy the axioms of metrics and return to the Sine metric and Bures metric for the qubit case. The CP expansive property and convex property of the metrics are studied. These metrics have deep connections with the known important metrics such as trace metric, Sine metric, Bures metric, and spectral metric, and are useful to quantify entanglement. I. Background, problem, and result Suppose one has two quantum states ρ and σ, then the Uhlmann-Jozsa fidelity [] between ρ and σ is given by F(ρ, σ) = [Tr ρ σρ ]. Uhlmann-Jozsa fidelity plays an important role in quantum information theory and quantum computation [], []. We know that for the case of qubit, Uhlmann-Jozsa fidelity has a simple form. From the Bloch sphere representation of quantum states, a qubit is described by a density matrix as([3]): ρ(u) = (I + σ u), where I is the unit matrix and σ = (σ, σ, σ 3 ) are the Pauli matrices. Assume ρ(u) and ρ(v) are two states of one qubit, then they can represented by two vectors u and v in the Bloch sphere. The Uhlmann-Jozsa fidelity for qubits has an elegant form([4] and [5]): F(ρ(u), ρ(v)) = [ + u v + u v ], where u v is the inner product of two vectors u and v, and u is the magnitude of u. We know that for quantum states, the Uhlmann- Jozsa fidelity has no simple form like the case of qubits. To use the simple form of fidelity, we note that in [6], the authors introduce a new fidelity, called super-fidelity, and proved that Uhlmann-Jozsa fidelity can be bounded above by the super-fidelity G(ρ, ρ ): F(ρ, ρ ) G(ρ, ρ ) := Trρ ρ +. ( Trρ )( Trρ ) Zhi-Hao Ma was in Department of Mathematics, Shanghai Jiaotong University, Shanghai, 0040, China( mazhihao@sjtu.edu.cn), Lin-Fu Zhang, and Jing-Ling Chen were in Theoretical Physics Division, Chern Institute of Mathematics, ankai University, Tianjin, 30007, China ( chenjl@nankai.edu.cn). When ρ and ρ are two qubits, super-fidelity G(ρ, ρ ) coincides with Uhlmann-Jozsa fidelity F(ρ, ρ ). The super-fidelity G(ρ, ρ ) has some good properties. Let ρ u = (I + ( ) λ.u) be the density matrix of a qunit( quantum state), where I is the unit matrix, λ = (λ, λ,...λ ) are the generators of SU(), and u is the ( )-dimensional Bloch vector. Then super-fidelity is rewritten as G(ρ u, ρ v ) = [ + ( ) u.v + ( ) ( u )( v )]. This shows that super-fidelity only depends on the magnitudes of u, v and the angle between them(that is, u.v). Fidelity by itself is not a metric. It is a measure of the closeness of two states. As a metric defined on quantum states, d(x, y) is a function satisfies the following four axioms: (M). d(x, y) 0 for all states x and y; (M). d(x, y) = 0 if and only if x = y; (M3). d(x, y) = d(y, x) for all states x and y; (M4). The triangle inequality: d(x, y) d(x, z) + d(y, z) for all states x, y and z. One may expect that a metric, which is a measure of distance, can be built up from fidelity. Indeed, the following three functions A(ρ, σ) := arccos F(ρ, σ), B(ρ, σ) := F(ρ, σ), C(ρ, σ) := F(ρ, σ), exhibit such metric properties. They are now commonly known in the literature as the Bures angle, the Bures metric, and the Sine metric ([7],[8], [9]), respectively. For super-fidelity, one can define the metric as ([6],[0]): D M (ρ, ρ ) = G(ρ, ρ ). It was proved in [6] and [0] that D M (ρ, ρ ) is a genuine distance. So we know that based on fidelity or super-fidelity, one can generally define a metric D(ρ, σ) as D(ρ, σ) := φ(f(ρ, σ)) (or D(ρ, σ) := φ(g(ρ, σ))), where φ(t) is a monotonically decreasing function of t, and D(ρ, σ) is required to satisfy the axioms M-M4. From this way, one can define many useful metrics. All of the above metrics belong to this type and play important roles in quantum information theory. The purpose of this paper is to introduce a new way to define metric of quantum states based on fidelity. In Sec. II, the new metric is defined. We study the qubit case in detail and naturally connect the new metric with the well-known trace metric, the Sine metric and Bures metric. In Sec. III, we show that the new metric defined is truly a metric, i.e.,

2 it satisfies the axioms M-M4. The upper bound for the metric is also presented. In Sec.IV, we discussed the properties of the metrics, such as CP expansive property and convex property. Conclusions are made in the last section. II. Metric induced by Uhlmann-Jozsa fidelity Let us define a metric of quantum states as follows: D T (ρ, σ) = max F(ρ, ) F(σ, ), () where the maximization is taken over all quantum states (mixed or pure). We call this metric D T (ρ, σ) as the T- metric, and denote the state that attained the maximal as the optimal state for the metric D T (ρ, σ). ote that the T-metric was first introduced in [] for pure sates of an abstract transition probability space, here T means transition probability. In this paper, we define it for arbitrary quantum states in the Hilbert space. The above definition of metric is interesting: we know that fidelity is a measure of the closeness of two states, usually, if we want to get a distance for state ρ, σ from fidelity, we use a composition of a monotonically decreasing function and the fidelity F(ρ, σ), e.g.,define metric as C(ρ, σ) := F(ρ, σ), this definition uses the fidelity of ρ and σ directly. Our definition, however, does not use the fidelity F(ρ, σ) directly, we need a bridge state, and the metric D T (ρ, σ) is the biggest difference between the two fidelity F(ρ, ) and F(σ, ), this definition may has some clear geometrical meaning, and we leave as a further topic. On the other hand, our metric may be not easy to calculate. But we know when is a pure state, the Uhlmann-Jozsa fidelity can be simplified as F(ρ, ) = Tr(ρ), hence one can define another version of metric as follows: D PT (ρ, σ) = max F(ρ, ) F(σ, ), () where the maximization is taken over all pure states. We call this metric as the PT-metric, and call the pure state that attained the maximal as the optimal pure state. Also, we can define metric from super-fidelity. Define a metric as follows: D G (ρ, σ) = max G(ρ, ) G(σ, ). (3) where the maximization is taken over all quantum states. We call this metric as the G-metric. And call the state that attained the maximal as the optimal state. Similarly, we can define a metric as follows: D PG (ρ, σ) = max G(ρ, ) G(σ, ) (4) where the maximization is taken over all pure states. We call the metric D PG (ρ, σ) as PG-metric. And call the state that attained the maximal as the optimal pure state. In this section, we consider the case of qubits (twodimensional quantum system). Then we have the following Theorems. Theorem. For the qubit case, D PT (ρ(u), ρ(v)) equals to the trace metric, namely, D PT (ρ(u), ρ(v)) = u v = D tr (ρ(u), ρ(v)). Proof. Let ρ = ρ(u), σ = ρ(v) and = ρ(w), since is a pure state, which means w =, then we get F(ρ, ) F(σ, ) = (u v) w u v. The optimal pure state is = ρ(w), where w is a vector that parallels to u v. Thus D PT (ρ(u), ρ(v)) = u v = D tr (ρ(u), ρ(v)). Theorem. For the qubit case, D T (ρ(u), ρ(v)) equals to the Sine metric, namely, D T (ρ, σ) = F(ρ, σ) = C(ρ, σ). Proof. Let ρ = ρ(u), σ = ρ(v) and = ρ(w), then one obtains F(ρ, ) F(σ, ) = (u v) w + w ( u v ) [ u v w + w u ] v u v + u v = F(ρ(u), ρ(v)). The optimal state is = ρ(w), where w is a vector that u v parallels to u v, and w =. Thus we F(ρ(u),ρ(v)) have D T (ρ, σ) = F(ρ, σ) = C(ρ, σ). This significant result seems surprising, since we know that F(ρ, σ) is the Sine metric introduced in ([7],[8], [9]), which plays an important role in quantum information processing ([7],[8], [9]), but here we can recover it for the qubit case through the definition (). One may wonder whether the Bures metric can be obtained by the similar definition. The answer is positive. By using the same approach developed in Theorem, for the qubit case one can prove that Bures metric B(ρ, σ) := F(ρ, σ) can be expressed in the following equivalent form B(ρ, σ) = max [ + F(ρ, ) F(σ, ) F(ρ, ) F(σ, ) ], (5) where the maximization is taken over all quantum states. Since for qubits, Uhlmann-Jozsa fidelity coincides with super-fidelity. We get that for qubits, metric D PG (ρ, σ) equal the trace metric and metric D G (ρ, σ) equal the sine metric. III. Metric character of D T and D PT We come to discuss the case of qunit( density matrix). In this case, if is a pure state, then the Uhlmann-

3 3 Jozsa fidelity may have a simple form: F(ρ, ) = Tr(ρ), so we first show the metric character of D PT (ρ, σ), where the optimal state is restricted to pure state, and then turn to show the metric character of D T (ρ, σ). We need the following concepts: For two quantum state ρ and σ, let λ i, (i =,, 3,..., d), be all eigenvalues of ρ σ, and λ i s are arranged as λ λ... λ d. Similarly, let λ i be all eigenvalues of σ ρ. Define E(ρ, σ) := max λ i and define E(σ, ρ) := maxλ i, so we know that λ = max λ i. ow we give an interpretation of E(ρ, σ). Proposition. Let ρ and σ be two quantum states. The following hold: E(ρ, σ) = maxtr(p(ρ σ), (6) P where the maximization is taken over all pure states P. Proof. We can diagonal ρ σ as Tr[(ρ σ)p] = Tr[(U(ρ σ)u + UPU + )] = Tr(DP ), where D is a diagonal matrix, and satisfy U(ρ σ)u + = D [since P is an arbitrary pure state, in the following we still denote Tr(DP ) as Tr(DP)]. Suppose P = ψ ψ, then we get Tr[(ρ σ)p] = Tr(DP) = ψ D ψ, assume ψ = α i i, then we get Tr[(ρ σ)p] = i α i λ i λ. This ends the proof. ote that generally E(ρ, σ) is not a metric, since E(ρ, σ) may not equal to E(σ, ρ), but we can symmetrize it as: D S (ρ, σ) := max[e(ρ, σ), E(σ, ρ)] = max λ i, (7) where λ i is the absolute value of λ i. From the knowledge of matrix analysis, D S (ρ, σ) equals to the spectral metric between ρ and σ, which was defined as the largest singular value of ρ σ, hence we know that D S (ρ, σ) is nothing but the spectral metric. For the qubit case, the D S (ρ, σ) or the spectral metric is also equal to the trace metric, i.e., D S (ρ(u), ρ(v)) = u v. ow we show the metric character of D PT (ρ, σ). Proposition. For states ρ, σ, we have D PT (ρ, σ) = D S (ρ, σ), i.e, the PT-metric we studied is in fact the same as the spectral metric. Proof. From definition, D PT (ρ, σ) = max F(ρ, ) F(σ, ), since z is an arbitrary pure state, we have F(ρ, ) = Tr(ρ). It yields F(ρ, ) F(σ, ) = Tr(ρ) Tr(σ), then we get max Tr(ρ) Tr(σ) = max(e(ρ, σ), E(σ, ρ)) = D S (ρ, σ). The Proposition is proved. ow we know that the PT-metric equals to the spectral metric, so it is a true metric. In the following we shall prove that the T-metric is also a true metric. Proposition 3. The T-metric D T (ρ, σ) as shown in Eq. () is truly a metric, i.e, it satisfies conditions M-M4. Proof. From definition, it is easy to prove conditions M and M3 hold. What we need to do is to prove conditions M and M4. If ρ = σ, then of course D T (ρ, σ) = 0. If D T (ρ, σ) = 0, we will prove ρ = σ. From the definition, we know that D T (ρ, σ) D PT (ρ, σ), so we get D PT (ρ, σ) = 0, since D PT (ρ, σ) is a true metric, we get ρ = σ. ow we come to prove M4, the triangle inequality D T (ρ, σ) D T (ρ, ) + D T (σ, ). D T (ρ, σ) = max F(ρ, ) F(σ, ), and suppose is the state that attains the maximal, so D T (ρ, σ) = F(ρ, ) F(σ, ). We assume that F(ρ, ) F(σ, ) = F(ρ, ) F(σ, ), then we get F(ρ, ) F(σ, ) = F(ρ, ) F(w, )+F(w, ) F(σ, ) F(ρ, ) F(w, ) + F(w, ) F(σ, ) D T (ρ, w)+d T (w, σ). Thus one finally has D T (ρ, σ) D T (ρ, w) + D T (σ, w). Proof finished. It is easy to prove that the PG-metric equal to the PTmetric, so we know that the PG-metric is nothing but the spectral metric, so it is a true metric, and we can prove that the G-metric is also a true metric, i.e, it satisfies conditions M-M4. The method is the same as that of Proposition 3. IV. Super-fidelity and related Metrics We have proved that T-metric is a true metric, but this definition is not operational, because it uses taking maximal for all states, so we need an operational form of the metric. We know that for qubits, it has a clear form as: D T (ρ, σ) = F(ρ, σ), how about higher dimension? For the qunit case, one does not have the relation D T (ρ, σ) = F(ρ, σ) as in Theorem. However, the numerical computation indicates the following upper bound holds: D T (ρ, σ) F(ρ, σ). (8) ow we will give the rigorous proof: as was shown in [7], such an inequality holds: F(ρ, ) F(σ, ) F(ρ, σ) (9) for arbitrary quantum states ρ, σ,. Taking maximal in the left hand of inequality (9), we get the inequality (8). How about G-metric? Like the proof of lemma, we can expect the following holds: For qunits ρ(u) and ρ(v), D G (ρ(u), ρ(v)) = G(ρ(u), ρ(v)). ( ) The proof is the following: Let ρ = ρ(u), σ = ρ(v) and = ρ(w), then one obtains G(ρ, ) G(σ, ) = (u v) w + w ( u v ) [ u v w + w u ] v = ( ) = G(ρ(u), ρ(v)). u v + u v (0) [ u.v u v ] The optimal state is = ρ(w), where w is a vector that parallels to u v, and w = u v. G(ρ(u),ρ(v))

4 4 The above proof seems to be correct, but is is not solid. In fact, we know that D G (ρ, σ) = max G(ρ, ) G(σ, ) range from 0 to, since G(ρ, ) and G(σ, ) are all range ( ) from 0 to, but G(ρ(u), ρ(v)) range from ( ) 0 to, which is a contraction. The reason is subtle. When the maximal attained, i.e., ( ) D G (ρ, σ) = G(ρ(u), ρ(v)), we need that the inequality (0) becomes quality, that means the optimal state = ρ(w) is attained, where w is a vector that parallels to u v, and w = u v, but in G(ρ(u),ρ(v)) fact we can not always get such optimal state. Because such an operator may not be a density operator! It is well known that every density matrix can be represented by the ( )-dimensional Bloch vector as: ρ(u) = (I + ( ) λ.u), but the converse is not true, i.e., not all operator of the form (I + ( ) λ.u) is a density matrix, where u is an arbitrary ( )- dimensional Bloch vector. ote that a density matrix must satisfy three conditions: (a). Trace unity, Tr(ρ(u)) =. (b) Hermitian, ρ(u) = ρ(u); and (c). positivity, i.e., all eigenvalues of ρ(u) are non-negative. Indeed, the operator (I + ( ) λ.u) automatically satisfies the conditions (a) and (b). However, not every vector u, u, allows ρ(u) satisfies the positive condition (c), we only know that when u, ρ(u) is always a density operator, see([4]). To get that inequality (0) becomes equality, we need that the optimal state = ρ(w) is a density matrix, where w is a vector that parallels to u v, and w = u v G(ρ(u),ρ(v)), but this is not always true in general. So we can only get the following relation holds: ( ) D G (ρ, σ) G(ρ, σ). () The following counterexample will show that strictly inequality will occur. Example. Let ψ = , φ = Define ρ = ψ ψ, σ = φ φ, then we know that ρ, σ are all pure states, it is easy to prove that in this case D G (ρ, σ) = D PT (ρ, σ) = D S (ρ, σ) =, while (3) 4 G(ρ, σ) = 3 8 >. For the operational form of D G (ρ, σ) for general states, we leave as a further topic. Suppose d(ρ, σ) is a distance on quantum states, we wish that this distance has some useful properties, such as the following: (P.) Unitary invariance: d(ρ, ρ ) = d(uρ U, Uρ U ), for any unitary operator U. (P.) Convex: for all 0 λ, states ρ, ρ, σ, σ, d(λρ + ( λ)ρ, σ) λd(ρ, σ) + ( λ)d(ρ, σ). A stronger version of P is the following: (P.) Joint Convex: for all 0 λ, states ρ, ρ, σ, σ, d(λρ + ( λ)ρ, λσ + ( λ)σ ) λd(ρ, σ ) + ( λ)d(ρ, σ ). In quantum information theory, a quantum operation or a quantum channel is representated by a completely positive trace preserving (CPT) map. We wish that a metric function is contractive under CPT map, this lead the following: (P3.) CP non-expansive property: suppose φ is a completely positive trace preserving (CPT) map, ρ, σ are two density matrices, then the following inequality holds: d(φ(ρ), φ(σ)) d(ρ, σ). We know that the super-fidelity G(ρ, σ) is a good fidelity measure. Recently, we find the deep connection between super-fidelity and concurrence([]). Also, super-fidelity has the following appealing properties([6]): i ) Bounds: 0 G(ρ, ρ ). ii ) Symmetry: G(ρ, ρ ) = G(ρ, ρ ). iii ) Unitary invariance: G(ρ, ρ ) = G(Uρ U, Uρ U ), for any unitary operator U. iv ) Concavity: super fidelity is concave, that is for states ρ, σ, σ and λ [0, ]. G(ρ, λσ + ( λ)σ ) λg(ρ, σ ) + ( λ)g(ρ, σ ). v ) Properties of the tensor product: Super fidelity is super multiplicative, that is, for states ρ, ρ, σ, σ G(ρ ρ, σ σ ) G(ρ, σ )G(ρ, σ ) We know that the Uhlmann-Jozsa fidelity F(ρ, σ) has the CP expansive property, that is the following: If ρ and σ are density matrices, Φ is a CPT map, then F(Φ(ρ), Φ(σ)) F(ρ, σ). We may guess that if super-fidelity has the CP expansive property, the following counterexample shows that this property does not holds. Example ([0]). Let A = , B = , Define Φ(γ) = AγA + +BγB +, where γ is an arbitrary density operator, then we defined a completely positive trace preserving map. Let ρ and σ be the density operators defined by ρ = , σ = ,

5 5 Then Φ(ρ) = 0 0 0, Φ(σ) = 0 0 0, One then easily obtains G(ρ, σ) > G(Φ(ρ), Φ(σ)), which shows that the CP expansive property property does not hold for super-fidelity. It is interesting to note that this example was first appear in [3] to show that the Hilbert- Schmidt metric is not contractive under quantum operation. Based on super-fidelity, one can define the metric as ([6],[0]): D M (ρ, ρ ) = G(ρ, ρ ). it was proved in [6] and [0] that D M (ρ, ρ ) is a genuine distance. The example shows that D M (ρ, ρ ) does not satisfying the CP non-expansive property. However, based on super-fidelity, we can also define another two metrics, the G-metric and PG-metric. Do the two metrics have the CP non-expansive property? Let us first consider the PG-metric. We know that PGmetric is in fact the spectral metric, and it was proved in [4] that the spectral metric is contractive under quantum operation. ow we will give a simple proof for this. Proposition 4.(contractive of the PG-metric): The PGmetric D PG (ρ, σ) has the CP non-expansive property. Proof. Let us denote M n as the set of all matrices, M n becomes a Hilbert space with inner product < X, Y >:= Tr(XY ), X, Y M n A linear map T induces its adjoint map as: < T(X), Y >=< X, T Y >. If φ is a positive map, then the adjoint map φ is also a positive map. The trace preserving property of φ means that φ is unital, that is, φ (I) = I. ow suppose γ is the optimal pure state for quantum states φ(ρ), φ(σ), so we get D PG (φ(ρ), φ(σ)) = G(φ(ρ), γ) G(φ(σ), γ). = (Trφ(ρ)γ) (Trφ(σ)γ). Let γ = φ (γ). Then we have Theorem is proved. D PG (φ(ρ), φ(σ)) = (Trφ(ρ)γ) (Trφ(σ)γ) = (Trρφ (γ)) (Trσφ (γ)) = (Trργ ) (Trσγ ) D PG (ρ, σ) Proposition 5. (joint convexity of the PG-metric): Let {p i } be probability distributions over an index set, let ρ i and σ i be density operators with the indices from the same index set. Then D PG ( i p i ρ i, i p i σ i ) i p i D PG (ρ i, σ i ) We know that D PG (ρ, σ) = D S (ρ, σ) = max(e(ρ, σ), E(σ, ρ)), so we only need to prove the following holds: E( i p i ρ i, i p i σ i ) i p i E(ρ i, σ i ) since E(ρ, σ) = max Tr(γ(ρ σ)), where the maximization γ in the right hand is taken over all pure states γ, then there exists a pure state γ such that E( i p i ρ i, i p i σ i ) = i The proof is complete. p i Tr(γ(ρ i σ i )) i p i E(ρ i, σ i ). Let us see how about the G-metric? Using example, we get D G (ρ, σ) = max G(ρ, ) G(σ, ) = max Tr(ρ σ) + Tr ( Trρ Trσ ) = max Tr(ρ σ) =, while D G(Φ(ρ), Φ(σ)) = max Tr(Φ(ρ) Φ(σ)) =, this example shows that D G (ρ, σ) D G (Φ(ρ), Φ(σ)). So we get that the CP non-expansive property Fails for the G-matric D G (ρ, σ). The numerical method show that the T-metric and the G-metric D T (ρ, σ) = max F(ρ, ) F(σ, ) D G (ρ, σ) = max G(ρ, ) G(σ, ) both satisfying the CP non-expansive property. And how about the convex property? We find that, neither of the metrics D T and D G are convex, however, numerical experiment shows that their square are convex, that is mean, the following convex property holds: D T((λρ + ( λ)ρ ), σ) λd T(ρ, σ) + ( λ)d T(ρ, σ) D G((λρ + ( λ)ρ ), σ) λd G(ρ, σ) + ( λ)d G(ρ, σ) V. Conclusions In this paper, we give a new method to get metrics from fidelity. This method is remarkable different from the usual method. We first define new metric D T from Uhlmann- Jozsa fidelity and metric D G from the super-fidelity. For qubits case, the two metrics coincides with Sine metric. Then properties of both metrics are discussed. We find that they are all contractive under quantum operations, and their square are convex, these show that the two metrics are all good candidates for geometrical entanglement measure. ow we give an application for the metrics D T and D G. ote that there are many entanglement measures, one of them is the geometrical measure. Its idea is based on the following: The set of all separable states is a convex set, denoted as S, the geometrical entanglement measure is defined as the minimal distance of the state ρ to any state of S: E(ρ) = min D(ρ, σ). σ S

6 6 It was showed that for the entanglement measure defined as above to be a good entanglement measure, the distance D(ρ, σ) need verify the following properties[6]: (a). D(ρ σ) 0 and D(ρ ρ) = 0. (b). D(ε(ρ) ε(σ)) D(ρ σ) for any CPT map ε. Our metrics D T and D G satisfying the above conditions, so we can introduce two new entanglement measures by or E(ρ) = min σ (D T (ρ, σ)) E(ρ) = min σ (D G (ρ, σ)) Where D T (ρ, σ)(res.d G (ρ, σ)) is the T-metric (res.gmetric), and the infimum is taken over all separable states. Acknowledgments This work is supported by the ew teacher Foundation of Ministry of Education of P.R.China (Grant o ). J.L.C is supported in part by SF of China (Grant o ), and Program for ew Century Excellent Talents in University, and the Projectsponsored by SRF for ROCS, SEM. References [] A. Uhlmann, The transition probability in the state space of a *-algebra Rep. Math. Phys., 9, 73 (976). [] R. Jozsa, Fidelity for Mixed Quantum States J. Mod. Opt., 4, 35(994); B. Schumacher, Quantum coding Phys. Rev. A, 5, 738 (995). X. B. Wang, C. H. Oh, and L. C. Kwek, Bures fidelity of displaced squeezed thermal states Phys. Rev. A,58, 486 (998); X. Wang, Z. Sun, and Z. D. Wang, Operator fidelity susceptibility: an indicator of quantum criticality arxiv: [3] Michael A. ielsen, Isaac Chuang. Quantum Computation and Quantum Information. [4] J. L. Chen, L. Fu, A. A. Ungar, X. G. Zhao, Alternative fidelity measure between two states of an -state quantum system Phys. Rev. A, 65, (00). [5] J. L. Chen, L. Fu, A. A. Ungar, X. G. Zhao, Geometric observation for Bures fidelity between two states of a qubit Phys. Rev. A, 65, 04303(00). [6] J.A.Miszczak, Z. Pucha la, P.Horodecki, A.Uhlmann, K. Życzkowski, Sub and super fidelity as bounds for quantum fidelity arxiv: [7] A.E. Rastegin, Sine distance for quantum states arxiv: quant-ph/060. [8] A.E. Rastegin, Relative error of state-dependent cloning Phys. Rev. A, 66, 04304(00). [9] A. Gilchrist,. K. Langford, and M. A. ielsen, Distance measures to compare real and ideal quantum processes Phys. Rev. A, 7, 0630(005). [0] Paulo E. M. F. Mendonca, R. d. J. apolitano, Marcelo A. Marchiolli, C. J. Foster, Yeong-Cherng Liang, An alternative fidelity measure for quantum states arxiv: v. [] J. G. F.Belinfante, Transition probability spaces J. Math. Phys, vol. 7, 85 9 (976). [] Zhihao Ma, Dong-Ling Deng, Fu-Lin Zhang, Jing-Ling Chen, Simple proof for the bounds of concurrence arxiv: [3] Masanao Ozawa,, Entanglement measures and the HilbertC- Schmidt distance Phys. Lett. A,68, 58 (000). [4] D.Garca, M. Wolf, D.Petz, M.Ruskai, Contractivity of positive and trace-preserving maps under L p norms J. Math. Phys, 47, [5] W.Bruzda, V.Cappellini, H.Sommers, K.Życzkowski, Random Quantum Operations arxiv: [6] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying Entanglement Phys.Rev.Lett, 78, 75(997).