Fidelity of Single Qubit Maps Mark D. Bowdrey,1, ∗ Daniel K. L. Oi,1, † Anthony J. Short,1, ‡ and Jonathan A. Jones1, 2, § 1 Centre for Quantum Computation, Clarendon Laboratory, University of Oxford, Parks Road, OX1 3PU, United Kingdom 2 Oxford Centre for Molecular Sciences, New Chemistry Laboratory, University of Oxford, South Parks Road, OX1 3QT, United Kingdom (Dated: March 15, 2001)
We give a simple way of characterising the average fidelity between a unitary and a general operation on a single qubit which only involves calculating the fidelities for a few pure input states.
arXiv:quant-ph/0103090v1 15 Mar 2001
PACS numbers: 03.67.-a
Consider transforming a quantum system by a desired operator, U . In practice, we may not be able to implement U exactly but may actually apply a map S instead. In general, the map S is a superoperator. To gauge how closely S approximates U (or vice versa), we require a measure of how close the output states are to each other, i.e. given identical input states, ρin , how does U [ρin ] compare with S[ρin ]. For the most general case where both output states are given by density operators, we can define their fidelity as [1], 2 q √ √ , ρ1 ρ2 ρ1 F (ρ1 , ρ2 ) = Tr F (| ψi hψ | , ρ) = Tr (| ψi hψ | ρ) ,
(2)
in the case where at least one of the states is pure [2]. This situation arises naturally when U is unitary and the input state, ρin , is pure, as in this case U [ρin ] will also be pure. In the case where our system is a single qubit, the allowed operations are restricted to affine contractions of the Bloch Ball. We will consider the case where U is
1 F¯ = 4π
π
θ=0
Z
2π
φ=0
Tr U
where F| ψihψ | = Tr U | ψi hψ | U † S [| ψi hψ |] .
(1)
which simplifies to
Z
a desired unitary transformation (rotation) of the Bloch ball, and S is a superoperator. It is assumed that S is a linear, trace-preserving map on the space of single qubit density operators [3]. We will define F¯ as the average fidelity over all pure input states, Z 1 F¯ = F| ψihψ | dΩ, (3) 4π
X
cj (θ, φ)
j
(4)
The state | ψi hψ | can be expressed in the basis of the Pauli spin matrices cos θ sin θe−iφ | ψi hψ | = 21 1 + sin θeiφ − cos θ σx σy σz σ0 + sin θ cos φ + sin θ sin φ + cos θ = 2 2 2 2 X σj cj (θ, φ) . = 2 j=0,x,y,z
(5)
Hence Eq. (3) can be expressed as
σj † U S 2
"
X
ck (θ, φ)
k
h σ i X 1 Z Z σj k cj ck sin θ dθ dφ Tr U U † S = 4π θ φ 2 2
#
σk sin θ dφ dθ 2
(6)
jk
where we have used the linearity of U and S. The integrals of the coefficients, cjk , are easily evaluated by symmetry: the cross terms vanish [4] leaving the simpler
expression h σ i X 2δj0 δk0 + δjk σj k Tr U U † S F¯ = 3 2 2 jk h σ i h σ i σ X σj 0 j 0 = Tr U U † S + 13 Tr U U † S 2 2 2 2 j=x,y,z h σ i X σj j † 1 1 Tr U U S =2+3 2 2 j=x,y,z
(7)
2 where we have used the fact that S[σ0 /2] is a density matrix and thus has unit trace. In order to express the average fidelity in terms of states, and to give a more intuitive picture of the above expression, we use the substitutions, σ0 + σj σ0 σj = − = ρj − ρ0 2 2 2 σ0 σ0 − σj = − = ρ0 − ρ−j , 2 2
1 2
+
1 Tr 6 j=x,y,z
=
1 2
+
1 Tr 6 j=x,y,z
=
1 2
+
1 Tr 6 j=x,y,z
=
1 Tr 6 j=±x,±y,±z
X X X
X
F¯ =
1 2
1 2
+
+
X
1 Tr 3 j=x,y,z
X
1 Tr 3 j=x,y,z
U ρj U † S[ρj ] − U ρj U † S[ρ0 ]
(9)
U ρ−j U † S[ρ−j ] − U ρ−j U † S[ρ0 ]
(10)
and taking their average yields,
U ρj U † S[ρj ] + U ρ−j U † S[ρ−j ] − U (ρj + ρ−j )U † S[ρ0 ] U ρj U † S[ρj ] + U ρ−j U † S[ρ−j ] − 2 U ρ0 U † S[ρ0 ] U ρj U † S[ρj ] + Tr U ρ−j U † S[ρ−j ] − 1
(11)
U ρj U † S[ρj ] .
Hence, the fidelity of the superoperator S with the unitary operator U can be calculated by simply averaging the fidelities of the six axial pure states on the Bloch sphere, {ρ+x , ρ−x , ρ+y , ρ−y , ρ+z , ρ−z }. In the case where S is unital (S[ρ0 ] = ρ0 ) it can be seen from Eq. 9 that this reduces to an average over only three states, {ρ+x , ρ+y , ρ+z }; similarly Eq. 10 shows that the average can be taken over {ρ−x , ρ−y , ρ−z }. We thank E. Galv˜ao and L. Hardy for helpful conversations. M.D.B. and A.J.S. thank EPSRC (UK) for a research fellowship. D.K.L.O thanks CESG (UK) for financial support. J.A.J. is a Royal Society University Research Fellow. This work is in part a contribution from the Oxford Centre for Molecular Sciences, which is supported by the UK EPSRC, BBSRC, and MRC.
∗
F¯ =
(8)
where ρ±j represents a pure state in the ±j-direction and ρ0 is the maximally mixed state. This gives the two
F¯ =
equivalent expressions,
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† ‡ §
[1] [2]
[3] [4]
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[email protected]; to whom correspondence should be addressed at the Clarendon Laboratory A. Uhlmann, Rep. Math. Phys. 9, 273 (1976). D. Bruss, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello and J. A. Smolin, Phys. Rev. A. 57, 2368 (1998). M. B. Ruskai, S. Szarek, and E. Werner, LANL e-print quant-ph/0101003. M. D. Bowdrey and J. A. Jones, LANL e-print quantph/0103060.
