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Fidelity of quantum operations
Line Hjortshøj Pedersen,1, ∗ Niels Martin Møller,2 and Klaus Mølmer1
1Lundbeck Foundation Theoretical Center for Quantum System Research,
Department of Physics and Astronomy, University of Aarhus,
Ny Munkegade, Bld. 1520, DK-8000 A˚rhus C, Denmark
2Department of Mathematical Sciences, University of Aarhus,
Ny Munkegade, Bld. 1530, DK-8000 A˚rhus C, Denmark
We present a derivation and numerous applications of a compact explicit formula for the average
fidelity of a quantum operation on a finite dimensional quantum system. The formula can be applied
to averages over particularly relevant subspaces; it is easily generalized to multi-component systems,
and as a special result, we show that when the same completely positive trace-preserving map is
applied to a large number of qubits with one-bit fidelity F close to unity, the average fidelity of the
operation on the full K-bit register scales as F 3K/2.
PACS numbers: 03.67.-a, 89.70.+c, 42.50.Lc
In quantum information theory one requires a measure
for the fidelity of quantum operations such as quantum
computing gates, quantum state storage and retrieval,
transmission and teleportation. The fidelity character-
izes the agreement between the actual outcome of the
operation and the desired state, and it will in general
depend on the initial state on which the operation is ap-
plied. Since the operation may generally be applied to
any arbitrary state, it is natural to average the fidelity
over all pure initial states, chosen uniformly in the sys-
tem Hilbert space. Derivations have been given in the
literature for the average fidelity for qubit [1] and qu-
dit operations [2], as well as for their evaluation as a
sum over a properly chosen discrete set of states [3], [4].
The latter approach is particularly relevant in connection
with Quantum Process Tomography [5], which provides a
procedure to deduce the quantum operation acting on a
system from experimental observations. In this Letter we
will present and analyze a simple expression for this aver-
age fidelity for the case where the quantum operation is
known, e.g., from a solution of the Schro¨dinger or Master
Equation of the system, and we focus on the calculation
of the fidelity with which the known actual operation re-
sembles the desired operation on the quantum system.
We shall first present a relation for uniformly averaged
matrix elements of a general linear operator, and subse-
quently we shall derive corollaries yielding the fidelity of
quantum operations in a number of different cases.
Theorem. For any linear operatorM on an n-dimensional
complex Hilbert space, the uniform average of |〈ψ|M |ψ〉|2
over state vectors |ψ〉 on the unit sphere S2n−1 in Cn is
given by
∫
S2n−1
|〈ψ|M |ψ〉|2dV (L)
=
1
n(n+ 1)
[
Tr(MM †) + |Tr(M)|2
]
(R)
(1)
∗Electronic address: lhp@phys.au.dk
where dV is the normalized measure on the sphere.
Proof. A recent detailed proof of this theorem is given
in [6], and it is also readily verified by use of a recent
result for the averages of general polynomials of state
vector amplitudes over the unit sphere [7]. Since both of
these proofs draw on elaborate mathematical results, we
shall present an alternative more straightforward deriva-
tion. We first show that (1) is valid if M is Hermitian.
A Hermitian matrix M can be diagonalized such that
UMU−1 = Λ, where Λ is a diagonal matrix with elements
λ1, . . . , λn ∈ R and U ∈ U(n). The left-hand and right-
hand sides of (1), denoted by L and R obey the conju-
gation invariance: L(UMU−1) = L(M), R(UMU−1) =
R(M) for any U ∈ U(n). This follows for the left-hand
side, L, by a change of variables ψ 7→ Uψ, and for R, it
is simply a property of the trace. It thus suffices to show
that L = R for a real, diagonal matrix Λ.
L(Λ) is a homogeneous polynomial of degree 2 in the
real variables λ1, . . . , λn, and unitary invariance implies
that it is invariant under the exchange of any two λi and
λj , which is only possible if L is on the form:
L = aTr(Λ2) + b (TrΛ)2
where a and b are constants that may depend on n. We
find that a = b = 1n(n+1) by picking two particularly
simple matrices and observing that
a · n+ b · n2 = L(I) = 1
and
a+ b = L






1
0
. . .





 =
∫
S2n−1
|c0|4dV =
2
n(n+ 1) ,
where |ψ〉 = ∑ cj |j〉 and the last equality follows by
evaluation of the integral. We have thus shown that
L(Λ) = R(Λ) and the theorem applies for any Hermi-
tian matrix.
We also have L(A) = R(A) where A denotes an anti-
Hermitian matrix (A† = −A), cf.:
L(A) = L(1i iA) = L(iA) = R(iA) = R(1i iA) = R(A)

2The general matrix M = M+M†2 + M−M
†
2 = S+A is de-
composed as a sum of a Hermitian and an anti-hermitian
matrix, respectively, and the explicit results,
L(S +A) =
∫
S2n−1
|〈ψ|S|ψ〉+ 〈ψ|A|ψ〉|2dV
=
∫
S2n−1
|〈ψ|S|ψ〉|2 + |〈ψ|A|ψ〉|2+
〈ψ|S|ψ〉 ·
(
〈ψ|A†|ψ〉+ 〈ψ|A|ψ〉
)
︸ ︷︷ ︸
0
dV
= L(S) + L(A)
and
R(S +A) = 1n(n+ 1){Tr((S +A)(S −A))+
(TrS +TrA)(TrS − TrA)}
= R(S) +R(A)
prove the theorem 
As noted above, the theorem has been known for a short
time, but its first application has been as an ingredient
in noise estimation and quantum process tomography [8].
We shall here apply the theorem to compute the average
fidelity explicitly in various cases, where the actual oper-
ation is known and fully characterized.
Average fidelity of a unitary transformation. We first
calculate the average fidelity of a process where our, pos-
sibly controllable, physical interactions yield a unitary
evolution U whereas we intend to perform the unitary
operation U0 on the system. For transmission, storage
and retrieval, U0 is the identity, and for quantum com-
puting U0 is the desired quantum gate. When different
initial states |ψ〉 are considered, the squared overlap be-
tween the actual outcome U |ψ〉 and the desired final state
U0|ψ〉, averages to
F =
∫
S2n−1
|〈ψ|M |ψ〉|2dV
=
1
n(n+ 1)
[
Tr(MM †) + |Tr(M)|2
]
,
(2)
with M = U †0U . Since M is unitary, Tr(MM †) equals
n, and if M multiplies all states by a single phase factor,
|Tr(M)|2 equals n2 and the fidelity is unity. A general
unitary matrix has eigenvalues of the form exp(iφj), and
|Tr(M)| will be smaller than n if these phases differ. The
”worst case” fidelity, i.e., the lowest fidelity obtained for
any state, is readily identified from the set of values φj
[9]. The design of composite pulses or smooth control
theory pulses to implement gates with good robustness
against variations in atomic parameters is significantly
facilitated with our simple fidelity expression.
Apart from the mean fidelity and the minimum, it
might be interesting to know the probability distribution
for F . This is a difficult problem, as witnessed by the
probability for the integrand in Eq.(1) for a Hermitian
matrix M , studied originally by von Neumann [10].
Subspace averaged fidelity of a unitary transformation.
In a number of quantum information scenarios, auxiliary
quantum levels are used to mediate the desired opera-
tions. Qubits may for example be encoded in ground
state atomic levels which are coupled by Raman tran-
sitions via optically excited states and two-qubit inter-
actions may involve state-selective excitation to states
with large dipole moments and strong long-range inter-
action [11]. In these protocols, the auxiliary levels of
the quantum system are unpopulated before the process,
and ideally, they are also unpopulated after the process.
But, if the fidelity is evaluated as above by identifying
M = U †0U on the complete system Hilbert space, this
may give far too pessimistic results. Consider for exam-
ple an atomic qubit which is initially and finally with cer-
tainty in the ground hyperfine states, and no population
has leaked to the excited atomic states during the Raman
process. If the average in (1) is made over the full system
Hilbert space, phases acquired by amplitudes on the ex-
cited states cause a reduction in the fidelity, even if the
final state is the correct one for all input qubit states. We
should only average the fidelity over the relevant input
states. Since the final state, ideally, is also in the qubit
subspace, we thus consider the matrix Mrel = PU †0UP ,
where P is the projection operator on the relevant, quan-
tum information carrying subspace. In matrix notation,
we thus address the square nrel × nrel submatrix of the
full matrix M , and use our theorem for this matrix.
F =
∫
S2nrel−1
|〈ψ|Mrel|ψ〉|2dV
=
1
nrel(nrel + 1)
[
Tr(MrelM †rel) + |Tr(Mrel)|2
]
,
(3)
Note that if population leaks to the auxiliary levels,Mrel
is not unitary, and hence both terms of the expression
have nontrivial values. If a measurement assures that
the final state does not populate the complement to the
relevant subspace, we get the average conditional fidelity
Fc = F/Q, where Q, given by the same expression (3)
with Mrel = PU †PUP , is the average acceptance prob-
ability of such a measurement.
Average fidelity for a general quantum operation. In
a number of quantum information scenarios, ancillary
quantum systems are used to mediate the desired op-
erations: quantum memory protocols in a very explicit
manner involve an extra quantum system, quantum tele-
portation requires an extra entangled pair of systems,
and in quantum computing with trapped ions, a mo-
tional degree of freedom is used to couple the particles.
The ancillary systems are ideally disentangled from the
qubits before and after the process, but in general they
act as an environment and cause decoherence of the quan-
tum system of interest. This forces us to generalize the
formalism even further and take into account the gen-
eral theory of quantum operations, according to which
density matrices are transformed by completely positive
maps. According to the Kraus representation theorem,

3any completely positive trace-preserving map G admits
the representation
G(ρ) =
∑
k
GkρG†k (4)
where
∑
k G
†
kGk = In is the n× n identity matrix [5]. If
there is only one Gk, this must be unitary and we return
to the case discussed above. The more general expression
allows interactions with an ancilla system followed by a
partial trace over that system, and it entails the effects
of dissipative coupling to the surroundings and of mea-
surements on the system, in which case the Gk may re-
present the associated projection operators and possible
conditioned feed-back operations applied to the system.
It is a powerful feature of (1) that it is easily extended to
allow computation of the average fidelity for protocols,
which apply or which are subject to these more general
operations.
If the pure input state ρ = |ψ〉〈ψ| is mapped to the
output state G(ρ) the mean fidelity with which our oper-
ation yields a unitary transformation U0 is
F =
∫
S2n−1
〈ψ|U †0G(|ψ〉〈ψ|)U0|ψ〉dV
=
∑
k
∫
S2n−1
|〈ψ|Mk|ψ〉|2dV
=
1
n(n+ 1)
{
Tr
(∑
k
M †kMk
)
+
∑
k
|Tr(Mk)|2
}
,
(5)
where Mk = U †0Gk, and {Gk} are the Kraus operators
for the map G.
The Kraus representation is not unique as the map G
might be represented by an alternative set of operators
{G′k}, where G′k =
∑
j VkjGj and V is a unitary trans-
formation among the Kraus operators [20]. We notice,
however, that the first term in (5) is the identity which
is unchanged, and since
∑
k
|Tr(Mk)|2 =
∑
k
Tr(M †k)Tr(Mk)
=
∑
ij
Tr(M ′†j )Tr(M ′i)
∑
k
V †jkVki
=
∑
j
Tr(M ′†j )Tr(M ′j) =
∑
k
|Tr(M ′k)|2,
the fidelity is invariant under the transformation.
Eq. (5) enables the calculation of the average fidelity
of any quantum operation, as soon as it has been put
on the Kraus form. If a system for example undergoes
dissipation, we can take this into account together with
other imperfections described in the previous sections,
and from the general solution of the associated Lindblad
master equation we can determine the average fidelity of
our quantum state operation. By the procedure outlined
in [5], the general propagator for the master equation can
be systematically put on the Kraus operator form, and
we also note the recent work [12], that explicitly provides
a Kraus form solution of the Lindblad master equation.
Let us demonstrate the formula (5) on the depolarizing
channel, where a qubit state is unchanged with probabil-
ity p and transformed into the random state I2/2 with
probability 1 − p. This may occur as consequence of a
continuous perturbation with a rate γ (p = exp(−γt)), or
it may represent the outcome of a teleportation process
[1], [2] where the channel is not occupying an ideal maxi-
mally entangled state two-qubit state |Ψ〉, but the mixed
state, ρEnt = p|Ψ〉〈Ψ| + 1−p4 I4. The transformation of
our qubit is represented by the action of four Kraus op-
erators, M0 =
√
3p+1
2 I2, Mχ =
√
1−p
2 σχ, where σχ with
χ = x, y, z are the Pauli matrices. The mean fidelity for
preservation of the state under the depolarizing channel,
or for the teleportation of a qubit by the mixed state,
follows readily from (5), and
Fdepolarized =
1
6
(2 + (3p+ 1)) = 1 + p
2
,
as expected.
Another example is atomic decay with a decay rate
Γ between an upper and a lower qubit level, where
the Kraus map is given by two operators, M0 =(
1 0
0 exp(−Γt/2)
)
, M1 =
(
0
√
1−exp(−Γt)
0 0
)
. In an analysis
using Monte-Carlo Wave Functions [13] these operators
represent the no-jump and the jump evolution, respec-
tively, weighted by the corresponding time dependent
probability factors. The fidelity (5) is readily evaluated
with the result:
Fdecay =
1
6
(3 + exp(−Γt) + 2 exp(−Γt/2)).
It is straightforward to apply our equations (2), (3)
and (5), and this is a great advantage, in particular for
higher dimensional systems, where the explicit integra-
tion over the space of initial states becomes cumbersome,
and where even the algebraic expressions in [2], [3] be-
come involved.
Average fidelity for general operations on composite
quantum systems. A particular example of higher dimen-
sional systems is composite systems, as, e.g., a K-qubit
or K-qudit quantum register, where our formalism for
not too large K can be used to determine the fidelity of
more complex operations, e.g., the achievements of error
correcting codes [14].
Let us consider here a simple example, where each
qubit or qudit is subject to the same operation, which
may be either unitary or a Kraus form operation. This
scenario is relevant for the teleportation of a full quan-
tum register by sequential or parallel teleportation of ev-
ery individual bit, or simply for the storage of a quan-
tum register which suffers from independent decoherence
mechanisms on every bit. If the Kraus operators for a
single qudit are denoted Mk, the Kraus operators for K
qudits are tensor product combinations of the Mk’s of
the form M~k =
⊗K
i Mki , with ~k = (k1, k2, . . . , kK). It

4follows from (5) that for the average over the nK dimen-
sional Hilbert space
FnK =
1
nK + 1 +
1
nK(nK + 1)
∑
~k
|Tr(M~k)|
2
=
1
nK + 1 +
1
nK(nK + 1)
(∑
k
|Tr(Mk)|2
)K
where we used that Tr
(
⊗K
i Mki
)
=
∏K
i Tr(Mki). Inter-
estingly, using (5), we can express the sum
∑
k |Tr(Mk)|2
in terms of the single qudit fidelity Fn, and hence we
obtain the relation between the K-qudit and the single-
qudit transformation fidelities:
FnK =
1
nK + 1
(
1 + ((n+ 1)Fn − 1)K
)
. (6)
We note that if Fn = 1, also FnK = 1, and due to (5),
Fn ≥ 1/(n+ 1) for any map, which consistently implies
that FnK ≥ 1/(nK + 1). Contrary to what one might
naively expect, the fidelity for K independent qudits is
not just FKn , but it is given by the more complicated
expression (6). This is due to the fact that the K’th
power of the one qudit fidelity formula only describes how
fast qubit product states decay on average, it does not
average over initial entangled states of the qudits in the
register. Eq. (6) implies that for Fn = 1−ǫ, where ǫ≪ 1,
FnK ∼ 1 − n
K
nK+1
n+1
n · Kǫ which is smaller than FKn ∼
1 −Kǫ. For the special case of many qubits, n = 2 and
K ≫ 1, we have the result FnK ∼ F 3K/2n . The result (6)
and the limiting values apply for any operation applied
to the individual qudits, it only assumes that they are all
subject to the same individual dynamics. For few qudit
systems one may well generalize the result to the case
of different operations on the individual components, if
for example part of the qubits in a quantum register are
transferred to an interaction region by teleportation.
In summary, we have presented a derivation and shown
various applications of a simple and compact expression
for the average fidelity of general quantum operations.
The problem has been addressed previously, and alter-
native expressions, involving discrete sums over a large
number of initial states, have been discussed in the lit-
erature with emphasis on quantum process tomography,
whereas, to our knowledge, the expression involving only
simple combinations of the trace of the relevant Kraus
operators has not been used in direct fidelity calcula-
tions. Our simple expression is a good starting point for
further analysis, e.g., of the achievements of error cor-
recting codes [14], decoherence free subspaces [15] and
protection of quantum information by dynamical decou-
pling [16]. Our expression can also be handled and gen-
eralized analytically as illustrated by our study of a K-
qudit register, which provides insight into the scaling of
errors which may have applications in quantum error cor-
rection, the capacity of quantum channels, and the way
that, e.g., communication with quantum repeaters [17]
and entanglement distillation should optimally be carried
out. We apply an average over pure states, but we note
that it is possible and also very interesting to define fideli-
ties for mixed initial states and to perform averages over
mixed states [18]. Since mixed states can be synthesized
as the partial trace of pure states on a larger system,
we imagine that the accomplishments of operations on
mixed states can also be addressed with our formalism.
The ability to restrict averages to subspaces may enable
generalization of our formalism to deal with non-uniform
averages, assuming nontrivial prior probability distribu-
tions on the Hilbert space. Also, in infinite-dimensional
Hilbert spaces we may use our formulas, omitting the
n(n + 1) prefactor, to define relative fidelity measures
for any trace class operators. By means of the regular-
ized traces used for quantum field theory [19], it is even
possible to extend such measures canonically to certain
non-trace class operators.
The authors gratefully acknowledge discussions with
Uffe V. Poulsen and Bent Ørsted. The project is sup-
ported by the European Integrated Project SCALA and
the ARO-DTO grant no. 47949 PHQC.
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