Fidelity of Single Qubit Maps Mark D. Bowdrey,1 Daniel K. L. Oi,1 Anthony J. Short,1 Konrad Banaszek,1 and Jonathan A. Jones1,2, ∗ 1Centre for Quantum Computation, Clarendon Laboratory, University of Oxford, Parks Road, OX1 3PU, United Kingdom 2 0 2Oxford Centre for Molecular Sciences, 0 2 Central Chemistry Laboratory, University of Oxford, n a South Parks Road, OX1 3QH, United Kingdom J 4 email: [email protected] 2 FAX: +44 1865 272387 1 v (Dated: February 1, 2008) 6 0 Abstract 1 1 0 Wedescribeasimplewayofcharacterizingtheaveragefidelitybetweenaunitary(oranti-unitary) 2 0 operator and a general operation on a single qubit, which only involves calculating the fidelities / h p for a few pure input states, and discuss possible applications to experimental techniques including - t n Nuclear Magnetic Resonance (NMR). a u q PACS numbers: 03.67.-a,82.56.-b : v i X r a 1 Inquantum informationtheory[1]itisoftenuseful tocomparetheeffectsoftwo processes applied to a quantum system. The basic building blocks of quantum information processing are transformations (maps) on two level quantum systems known as quantum bits or qubits. Ideally, we would like to be able to compare any two single qubit maps, but unfortunately this is not always straightforward. The comparison is, however, much simpler if one map is unitary or anti-unitary. A natural approach to compare two maps is to calculate the state fidelity of their output states given identical inputs. The Uhlmann state fidelity of two density operators (ρ ,ρ ) is given by [2] 1 2 2 F(ρ ,ρ ) = Tr √ρ ρ √ρ . (1) 1 2 1 2 1 (cid:18) (cid:18)q (cid:19)(cid:19) This may be interpreted as the maximal overlap of all purifications of ρ and ρ . Under a 1 2 unitary or anti-unitary transformation, a pure input state maps to a pure output state and in this case we can simplify the state fidelity (1) to [3] F( ψ ψ ,ρ) = Tr( ψ ψ ρ). (2) | ih | | ih | The state fidelity of a unitary (or anti-unitary) map U and a general linear, trace-preserving, transformation acting on an initially pure state ψ ψ is given by M | ih | F = Tr U ψ ψ U [ ψ ψ ] . (3) ψ ψ † | ih | | ih | M | ih | (cid:0) (cid:1) The average map fidelity can then be defined by integrating over all pure input states, 1 F¯ = F dΩ, (4) ψ ψ 4π | ih | Z (where the integral is over the surface of the Bloch sphere) and this definition is widely used [4, 5, 6, 7]. There is, however, a simplification: using the fact that ψ ψ can be written | ih | in terms of the Pauli spin matrices and the identity matrix [8], cosθ sinθe iφ ψ ψ = 1 1+ − | θ,φih θ,φ| 2  sinθeiφ cosθ  − = 1 (σ +sinθ cosφσ +sinθsinφσ +cosθσ ) (5) 2 0 x y z σ j = c (θ,φ) j 2 j=0,x,y,z X 2 we can now express equation (4) as, 1 π 2π σ σ F¯ = Tr U c (θ,φ) j U c (θ,φ) k sinθdφdθ j † k 4π 2 M 2 Zθ=0Zφ=0 " j # " k #! X X (6) 1 σ σ j k = c c sinθdφdθ Tr U U j k † 4π 2 M 2 Xjk (cid:18) ZθZφ (cid:19) (cid:16) h i(cid:17) where we have used the linearity of U and . When integrated over the Bloch sphere the M coefficients of the off-diagonal terms go to zero, while the diagonal terms survive [9], leaving 2δ δ +δ σ σ F¯ = j0 k0 jk Tr U jU k † 3 2 M 2 Xjk (cid:18) (cid:19) (cid:16) h i(cid:17) σ σ σ σ = Tr U 0U† 0 + 1 Tr U jU† j (7) 2 M 2 3 2 M 2 (cid:16) h i(cid:17) j=Xx,y,z (cid:16) h i(cid:17) σ σ = 1 + 1 Tr U jU j , † 2 3 2 M 2 j=Xx,y,z (cid:16) h i(cid:17) where we have used the unit trace of σ and the fact that is trace-preserving. 0 M ExpressingtheaveragefidelityinthisformmaynotseemhelpfulasthePaulispinmatrices do not represent proper states. However, in NMR experiments where the states are highly mixed, single qubit states can be represented by 1σ , 1σ , 1σ [10, 11, 12] and therefore {2 x 2 y 2 z} we can use equation (7) directly. One application of this approach is to characterise the behaviour of composite rotation sequences [13, 14], which are widely used in NMR to reduce the effects of systematic errors. In conventional NMR experiments [13] composite rotations are used to effect particular motions on the Bloch sphere (such as inversion, which takes a spin from +z to z), and it suffices to determine the point-to-point fidelity, but when − used in NMR implementations of quantum computation [15] the initial state is unknown. One approach used to date is Levitt’s quaternion fidelity [14, 15] but this has the major disadvantagethatitcanonlybeusedtoassesthetheoreticalbehaviourofarotationsequence and cannot be determined by experiment. The average fidelity approach outlined above provides a simple approach which can be used for both theoretical and experimental studies. For experimental and theoretical work with pure state techniques we require a more appropriate form and so we use the substitutions σ 1+σ 1 j j = = ρ ρ j 0 2 2 − 2 − (8) 1 1 σ j = − = ρ ρ , 0 j 2 − 2 − − 3 where ρ represents a pure state in the j-direction and ρ is the maximally mixed state. j 0 ± ± This gives the two equivalent expressions F¯ = 1 + 1 Tr Uρ U [ρ ] Tr Uρ U [ρ ] (9) 2 3 j †M j − j †M 0 j=x,y,z X(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) F¯ = 1 + 1 Tr Uρ U [ρ ] Tr Uρ U [ρ ] , (10) 2 3 −j †M −j − −j †M 0 j=x,y,z X(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) and taking the average of (9) and (10) yields, F¯ = 1 + 1 Tr Uρ U [ρ ] +Tr Uρ U [ρ ] Tr U(ρ +ρ )U [ρ ] 2 6 j †M j −j †M −j − j −j †M 0 j=x,y,z X(cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) = 1 + 1 Tr Uρ U [ρ ] +Tr Uρ U [ρ ] 2Tr Uρ U [ρ ] 2 6 j †M j −j †M −j − 0 †M 0 j=x,y,z X(cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) = 1 + 1 Tr Uρ U [ρ ] +Tr Uρ U [ρ ] 1 2 6 j †M j −j †M −j − j=x,y,z X(cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:1) = 1 Tr Uρ U [ρ ] . 6 j †M j j= x, y, z ±X± ±(cid:0) (cid:0) (cid:1)(cid:1) (11) Hence, the fidelity of the map with the unitary or anti-unitary map U can be calcu- M lated by simply averaging the fidelities of the six axial pure states on the Bloch sphere, ρ ,ρ ,ρ ,ρ ,ρ ,ρ . We note that the average map fidelity (F¯) can in fact be char- +x x +y y +z z { − − − } acterized by only four pure states, 1(1+ 1 (+σ +σ +σ )), 1(1+ 1 ( σ σ +σ )), 1(1+ {2 √3 x y z 2 √3 − x− y z 2 1 ( σ +σ σ )), 1(1+ 1 (+σ σ σ )) . Indeed, the fidelity can be characterized using √3 − x y− z 2 √3 x− y− z } any four pure states forming a regular tetrahedron, or any six forming a regular octahedron; however the pure states at the six cardinal points provide a particularly natural approach. An obvious application of this result is to compare a desired unitary operation with its actual implementation that (due to experimental imperfections) may be more closely represented by a superoperator. A practical advantage of characterizing the fidelity by just testing six states is that this approach provides a simple means to verify the map fidelity by experiment. Similarly, we can also use this result to calculate the fidelity of a unitary or superoperator approximation to an anti-unitary map [4] in a convenient and intuitive manner. We thank E. Galv˜ao and L. Hardy for helpful conversations. M.D.B. and A.J.S thank EPSRC (UK) for financial support. D.K.L.O. thanks CESG (UK) and QAIP (contract IST-1999-11234) for financial support. K.B. thanks EQUIP (contract IST-1999-11053) for 4 financial support. J.A.J. is a Royal Society University Research Fellow. This is in part a contribution from the Oxford Centre for Molecular Sciences, which is supported by the UK EPSRC, BBSRC, and MRC. Electronic address: [email protected]; to whom correspondence should be addressed ∗ at the Clarendon Laboratory [1] C. H. Bennett and D. P. DiVincenzo, Nature 404, 247 (2000). [2] A. Uhlmann, Rep. Math. Phys. 9, 273 (1976). [3] D. Bruss, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello and J. A. Smolin, Phys. Rev. A 57, 2368 (1998). [4] L. Hardy and D. D. 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