Equal-fidelity surface for a qubit state via equal-distance extended Bloch vectors and derivatives
NNoname manuscript No. (will be inserted by the editor)
Equal-fidelity surface for a qubit state viaequal-distance extended Bloch vectors andderivatives
Sang Min Lee · Heonoh Kim · Han SebMoon
Received: date / Accepted: date
Abstract
We describe and characterize an equal-fidelity surface in Blochspace targeted for a qubit state by means of equal-distance concept. The dis-tance is generalized and defined as the Euclidean distance between extendedBloch vectors for arbitrary dimensional states. The distance is a genuine dis-tance according to the definition and is related to other distances betweenquantum states and super-fidelity.
Keywords
Fidelity · Distance between quantum states · Geometry ofquantum states
Fidelity is the concept most frequently used to compare two quantum statesin quantum information science, and related to various distances between thequantum states, even though it is not a metric. For examples, Bures distanceis directly convertible to the fidelity [1,2], and fidelity gives upper and lowerbounds of trace distance [3]. In the paper, we introduce a distance between twoquantum states that is defined as the Euclidean distance between two extendedBloch vectors in R N space. This distance was already proved to be a metric,namely, the “modified root infidelity,” in a previous work [4]. However, we givea simple geometrical definition of the distance. From the distance between twoqubit states ( N = dim ( H ) = 2), we can directly calculate the fidelity. However, Sang Min LeeDepartment of Physics, Pusan National University, Busan 609-735, KoreaE-mail: [email protected] KimDepartment of Physics, Pusan National University, Busan 609-735, KoreaHan Seb MoonDepartment of Physics, Pusan National University, Busan 609-735, Korea a r X i v : . [ qu a n t - ph ] N ov Sang Min Lee et al. for the cases of
N >
2, the derived quantity is an upper bound of the fidelity(super-fidelity) [4].The beginning of this research is a curiosity about geometry of equal-fidelity states in Bloch space targeted for a qubit state, because two well-known examples show completely different features. For a pure state and themaximally mixed state, equal-fidelity states are represented by an orthogonalplane to the Bloch vector of the target state and a sphere of which center ison the target state, respectively. We try to explain the two different aspectsinto a unified mechanism (equal-distance of extended Bloch vectors) througha mediate example, a non-maximally mixed state.The paper is structured as follows. Section 2 briefly introduces the fidelityand the generalized Bloch vector for N ≥
2. In Sec. 3, we first show twoaforementioned examples of equal-fidelity surfaces, then describe schematicallyhow to obtain an equal-fidelity surface for a general qubit state and describeits properties. In Sec. 4, we discuss the distance between the extended Blochvectors for N ≥ – Bounds: 0 ≤ F ( ρ , ρ ) ≤ – F = 1 iff ρ = ρ , – F = 0 iff supp ( ρ ) ⊥ supp ( ρ ). – Symmetry: F ( ρ , ρ ) = F ( ρ , ρ ). – Unitary invariance: F ( ρ , ρ ) = F ( U ρ U † , U ρ U † ).The fidelity is defined as F ( ρ , ρ ) ≡ (cid:18) T r (cid:20)(cid:113) √ ρ ρ √ ρ (cid:21)(cid:19) (1)= T r [ ρ ρ ] + 2 (cid:88) i 2, and satisfies | λ | ≤ / 2. Then the density matrix is expressed as ρ = I/ (cid:80) i =1 λ i ˆ σ i , where { ˆ σ i } are the Pauli operators.For an N -dimensional Hilbert space, the Bloch vector is generalized viagenerators { ˆ λ , · · · , ˆ λ N − } of SU ( N ). In this paper, the generalized Blochvector [4,8,9] is defined as λ ( N ) = ( λ , · · · , λ N − ) in R N − , where λ i = T r [ ρ ˆ λ i ] / 2, and the density matrix is expressed as ρ = I/N + (cid:80) N − i =1 λ i ˆ λ i . Thelength of the generalized Bloch vector is bounded as (cid:12)(cid:12) λ ( N ) (cid:12)(cid:12) ≤ (cid:113) N − N (equalityfor only pure states). The generators of SU ( N ) are defined as { ˆ λ i } N − i =1 = { ˆ u jk , ˆ v jk , ˆ w m } , (3)ˆ u jk = | j (cid:105)(cid:104) k | + | k (cid:105)(cid:104) j | , ˆ v jk = − i | j (cid:105)(cid:104) k | + i | k (cid:105)(cid:104) j | , ˆ w m = (cid:115) m ( m + 1) m (cid:88) j =1 | j (cid:105)(cid:104) j | − m | m + 1 (cid:105)(cid:104) m + 1 | , where 1 ≤ j ≤ k ≤ N and 1 ≤ m ≤ N − 1. They satisfyˆ λ i = ˆ λ † i , T r [ˆ λ i ] = 0 , T r [ˆ λ i ˆ λ j ] = 2 δ ij . (4)The operators ˆ u jk , ˆ v jk , and ˆ w m are generalized Pauli operators of ˆ σ x , ˆ σ y , andˆ σ z , respectively. In general, density matrices of which the Bloch vectors are located at the samedistance from a target vector λ t have different fidelities with the target state ρ t . For a simple example, ( A ) equal-fidelity states for a pure target state arerepresented by an orthogonal plane of the target vector in Bloch space, asshown in Fig. 1 (a). The reason is as follows. When one of the states to becompared is pure, the fidelity is written in a simple form as F ( ρ t , ρ ) = T r [ ρ t ρ ] = 12 + 2 λ t · λ (5)using Eq. (1) and Eq. (4). Therefore, a set of { λ } on an orthogonal plane for λ t has the same fidelity with the target state. On the other hand, ( B ) for themaximally mixed state ( I/ F ( I/ , ρ ) = F ( I/ , U ρU † ) (6) The maximum length of the Bloch vector (for pure states) can be modified by adoptinga constant α as ρ = I/ α (cid:80) i =1 λ i ˆ σ i . Although the usual notation is α = 1 / λ i = T r [ ρ ˆ σ i ] and | λ | ≤ 1, we set α = 1 in this paper for a consistent argument later. Sang Min Lee et al. Fig. 1 Representations of equal-fidelity states in Bloch space, when the target state is (a)a pure state: case A , | λ t | = 1 / 2, and (b) the maximally mixed state: case B , | λ t | = 0. is satisfied. A unitary operation corresponds to a rotation operation in Blochspace, so Eq. (6) means that a set of { λ } , which are equivalent under rotation(have the same length), has the same fidelity with the target state I/ . These two extreme examples have completelydifferent features (flat plane for | λ t | = 1 / | λ t | = 0).To investigate general cases, 0 ≤ | λ t | ≤ / 2, we adopt modified forms of thefidelity and Bloch vector. The fidelity between two qubits can be representedby their Bloch vectors [2,5] as F ( ρ , ρ ) = 12 + 2 λ · λ + 2 (cid:113) (1 / − | λ | (cid:113) (1 / − | λ | . (7)If we extend the Bloch vector into L = (cid:16) λ x , λ y , λ z , (cid:113) (1 / − λ x − λ y − λ z (cid:17) [2,4,10,11], then Eq. (7) is modified concisely as F ( ρ , ρ ) = 12 + 2 L · L . (8)The extended Bloch vector L is on a hyperhemisphere of S , i.e., | L | = 1 / ≤ L , namely, an “Uhlmann hemisphere.” Equation (8) shows that a setof equal-fidelity states is represented by a hyperplane in R , similar to the caseof a pure target state in Eq. (5) and Fig. 1 (a). However, { L } are restricted onthe hyperhemisphere of S , so the solution is given by the intersection betweenthe hyperplane and the hyperhemisphere.In other words, the fidelity can be described by the distance between thetwo extended Bloch vectors. Since the Euclidean distance of two vectors isrepresented as | L − L | = 1 / − L · L , the fidelity is rewritten as F ( ρ , ρ ) = 1 − | L − L | (9) When the target state is the maximally mixed state for N = 2 (qubit), the same fi-delity states are equivalent to the same purity states (the same Bloch vector lengths).However, for N > 2, having the same purity states is a sufficient condition for havingthe same fidelity states for the target I/N . For example, ρ a = diag (0 . , . , . ρ b = diag (0 . , . , . 48) have different purities but the same fidelity with the target ρ t = (1 / , / , / using Eq. (8). The above relation shows that fidelity between two qubit statesis represented by the Euclidean distance between extended Bloch vectors ofthe states .Figure 2 shows schematic diagrams illustrating how to represent equal-fidelity states in Bloch space from equal-distance extended Bloch vectors. First,we assume that the target state is on the + z axis as λ t = (0 , , λ ≥ xz planein Bloch space, which is represented by the blue disk in Fig. 2 (a). When weignore y axis because of λ y = 0, the disk is converted to a hemisphere in theextended Bloch space in Fig. 2 (b). The extended Bloch vector of the target L t is projected on the hemisphere from λ t along the L direction, as shown inFig. 2 (b). A set of vectors { L } ed , which are located at the same distance fromthe target L t , is represented by the intersection of the blue hemisphere anda red sphere in Fig. 2 (c) and (d). As shown in Fig. 2 (e), the equal-distanceextended Bloch vectors { L } ed for L t are represented by the green ellipse onthe xz plane of the Bloch space through the reverse projection. This argument This distance differs from the Bures distance. Fig. 2 Schematic diagram illustrating how to obtain an equal-fidelity surface in Blochspace. (a) xz plane and target state λ t = (0 , , λ ≥ 0) in Bloch space. (b) Plane and targetvector λ t are represented by the hemisphere and the vector L t in the extended Bloch space.(c, d) The set of equal-distance extended Bloch vectors { L } ed from the target vector L t is represented by the intersection of two (blue and red) spheres in R . (e) Set of vectors { L } ed is represented by a green ellipse on xz plane in Bloch space. (f) The entire set ofequal-fidelity states is represented by an ellipsoid in Bloch space using z axis rotationalsymmetry. Sang Min Lee et al. Fig. 3 When D ( L t , L ) ≥ / − λ , there are spurious solutions. Since the extended Blochvectors are restricted by L > 0, the projection in black ( z > F − λ ) represents spurioussolutions for L < can be applied to an arbitrary disk plane in the Bloch space that containsthe origin (0 , , 0) and the target (0 , , λ ): z axis rotation symmetry. Thus, thetotal set of equal-fidelity states generally has the form of an ellipsoid in Blochspace, as shown in Fig. 2 (f). This schematic explanation in Fig. 2 clearlyshows the reason for the different features of the two examples: ( A ) λ = 1 / B ) λ = 0. The projected solutions in Fig. 2 (e) for cases ( A ) and ( B ) area straight line and a circle, respectively.For the target vector λ t = (0 , , λ ), where 0 ≤ λ ≤ / 2, the explicitexpression of equal-fidelity states in Bloch space is x + y F (1 − F ) + ( z − (2 F − λ ) F (1 − F )(1 − λ ) = 1 , (10)where z ≤ F − λ . The vectors on the ellipsoid where z > F − λ are spurioussolutions for the cases of L < 0, as represented by the black projection inFig. 3.The oblate ellipsoid solution (major axis: xy plane, minor axis: z axis) inEq. (10) has two properties. The length of the semimajor axis is a function ofthe fidelity as (cid:112) F (1 − F ), and the ratio between the major and minor axes isfixed as √ − λ . As shown in Fig. 2 (c) and (e), the solution (green ellipse on xz plane) for equal-fidelity states is projected from a circle on a tilted plane(not shown in Fig. 2) in the extended Bloch space. Since the length of themajor axis is not affected by the tilt angle of the plane, it is determined bythe distance in R or the fidelity. The ratio of the major and minor axes is afunction of the tilt angle θ (angle between L t and the L axis), so it is givenby cosθ = √ − λ . In Fig. 4, we show examples ( λ =1/2, 2/5, and 1/6) ofequal-fidelity states on the xz plane of the Bloch space. They clearly showthat the major axis is fixed as the fidelity and the eccentricities of the ellipsesare fixed as λ , the length of the target Bloch vector.In general, we think that two states are very close if their fidelity is 0.99.However, as shown in Fig. 4 (a), the angle between two pure states for F = 0 . itle Suppressed Due to Excessive Length 7(a) (b) (c) Fig. 4 Equal-fidelity states on xz plane in Bloch space for (a) λ = 1 / 2, (b) λ = 2 / 5, and(c) λ = 1 / 6. Large dots are target states; small dots are minimum fidelity states. Dottedlines in (b) and (c) represent spurious solutions. is about 11 . ◦ . If we consider a simple experiment using a polarization qubitsystem, it corresponds to about a 5 ◦ operational error of a half-wave plate.The experimental errors of wave plates are typically less than 1 ◦ and could befurther reduced via motorized rotation mounts. Recently, we experimentallydemonstrated operational error-insensitive approximate universal-NOT gatesin a polarization qubit system [12]. In the experiment, we measured the error-insensitivity of the gate via the fidelity deviation when the target state is anideally flipped pure state | ψ ⊥ (cid:105) (comparable to λ = 1/2) rather than an idealoutput state ρ (cid:48) = (cid:80) i ˆ σ i | ψ (cid:105)(cid:104) ψ | ˆ σ i (comparable to λ = 1/6) of the approximateUNOT gate, because the fidelities between the erroneous outputs and ρ (cid:48) arevery close to unity. In other words, when the target is ρ (cid:48) ( λ = 1/6) in Fig. 4(c), the fidelity deviations of the erroneous outputs are very small, since mosterroneous outputs are located inside the surface of F = 0 . 99. Thus, the fidelitydeviation can be changed by the target state, even though the distribution ofstates in Bloch space remains. N The extended Bloch vector L for a qubit state ( N = 2) is defined as | L | = 1 / N ≥ L N , as follows: L ≡ ( λ , · · · , λ N − , L N ) , (11) L N ≡ (cid:114) N − N − | λ | = (cid:112) (1 − T r [ ρ ]) / , (12)so L is on a hyperhemisphere of S N − with a radius of (cid:113) N − N . Using theEuclidean distance between the extended Bloch vectors for an arbitrary di- Sang Min Lee et al. mension, we define a distance D L between two density matrices as, D L ( ρ , ρ ) = | L ( ρ ) − L ( ρ ) | . (13)We redefine F (cid:48) between two density matrices similar with Eq. (9) as, F (cid:48) ( ρ , ρ ) ≡ − D L ( ρ , ρ )= 1 N + 2 (cid:32) λ · λ + (cid:114) N − N − | λ | (cid:114) N − N − | λ | (cid:33) = T r [ ρ ρ ] + (cid:113) (1 − T r [ ρ ]) (1 − T r [ ρ ]) , (14) D L ( ρ , ρ ) = (cid:112) − F (cid:48) ( ρ , ρ ) . (15) F (cid:48) has the following properties. It is unitary invariant, since the overlap oftwo density matrices and the purity of a density matrix are unitary invariant.When at least one of the density matrices ρ i is a pure state, F (cid:48) is the same asthe fidelity between the two states, because the square root term in Eq. (14) iszero, and the fidelity is reduced to T r [ ρ ρ ] in that case. However, in general,it is an upper bound of the fidelity, namely, the “super-fidelity” [4].As the definition itself (the Euclidean distance between two vectors in R N ), D L satisfies the general properties of a distance: – Non-negativity: d ( x , x ) ≥ d = 0 iff x = x . – Symmetry: d ( x , x ) = d ( x , x ). – Triangle inequality: d ( x , x ) + d ( x , x ) ≥ d ( x , x ).In previous works [4,13], D L is represented by the “modified root infidelity,” C (cid:48) ( ρ , ρ ), and is proved to be a genuine distance in a different way.We should note that the distance D L in Eq. (15) differs from the Buresdistance, which is defined as D B = 2 − (cid:112) F ( ρ , ρ ) [1,2] even for N = 2.However, if we assume that the fidelity is close to unity, as for F = 1 − δ ( δ (cid:28) 1) and cases where F = F (cid:48) , then D L and the Taylor-approximated D B are the same as δ .Now, we consider the inner distance ˜ D L between two extended Bloch vec-tors. Since L i are limited to the hyperhemisphere of S N − , the inner distanceis defined using the length of the vectors and the angle θ between two vectors If we define the Bloch vector as ρ = IN + (cid:80) i λ i ˆ λ i , where λ i = T r [ ρ ˆ λ i ], then L N = (cid:113) N − N − | λ | , and F (cid:48) ( ρ , ρ ) is defined as 1 − D ( L , L ) / as ˜ D L ( ρ , ρ ) ≡ | L i | θ = (cid:114) N − N cos − (cid:18) NN − L · L (cid:19) = (cid:114) N − N cos − (cid:18) N F (cid:48) − N − (cid:19) , (16)= 12 cos − (4 L · L ) for N = 2= 12 cos − (2 F ( ρ , ρ ) − cos − (cid:112) F ( ρ , ρ ) , (17)using Eqs. (8) and (14). For N = 2, the inner distance between two extendedBloch vectors ˜ D L is the same as the Bures length [7,11,14], as shown inEq. (17), and ˜ D L becomes δ when F = 1 − δ ( δ (cid:28) In this paper, we obtained the general expression for the equal-fidelity sur-faces of a qubit state in Bloch space via the concept of equal distances ofthe extended Bloch vectors and explained the properties of the equal-fidelitysurfaces. We generalized the extended Bloch vectors and their distance for anarbitrary-dimensional Hilbert space. The distance is a genuine distance ac-cording to the definition itself. From the distance between the extended Blochvectors, we define F (cid:48) for the two density matrices. In general, F (cid:48) is an upperbound of the fidelity, although F (cid:48) is reduced to the fidelity in restricted cases,i.e., N = 2 or at least one of the states is pure. We also show that D L is relatedto the Bures distance and Bures length. The distance D L is not a new distancebetween quantum states, but we introduce a definition in a new and intuitiveway. We expect and hope that our research will facilitate further work on basicstudies of the fidelity and quantum distances in quantum information science. Acknowledgements This work was supported by the National Research Foundation ofKorea (NRF) grant funded by the Korea government (MSIP) (No. 2015R1A2A1A05001819and No. 2014R1A1A2054719) References 1. D. Bures, Transactions of the American Mathematical Society , pp. 199 (1969)2. M. H¨ubner, Physics Letters A , 239 (1992)3. C. Fuchs, J. van de Graaf, Information Theory, IEEE Transactions on , 1216 (1999)4. J.A. Miszczak, Z. Pucha(cid:32)la, P. Horodecki, A. Uhlmann, K. ˙Zyczkowski, Quantum Info.Comput. , 103 (2009)5. R. Jozsa, Journal of Modern Optics , 2315 (1994)0 Sang Min Lee et al.6. A. Uhlmann, Reports on Mathematical Physics , 273 (1976)7. M.A. Nielsen, I.L. 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