Fidelity of quantum states explained
In quantum mechanics, notably in quantum information theory, fidelity quantifies the "closeness" between two density matrices. It expresses the probability that one state will pass a test to identify as the other. It is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.
Definition
The fidelity between two quantum states
and
, expressed as density matrices, is commonly defined as:[1] [2] F(\rho,\sigma)=\left(\operatorname{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)2.
The square roots in this expression are well-defined because both
and
are positive semidefinite matrices, and the
square root of a positive semidefinite matrix is defined via the
spectral theorem. The Euclidean inner product from the classical definition is replaced by the
Hilbert–Schmidt inner product.
As will be discussed in the following sections, this expression can be simplified in various cases of interest. In particular, for pure states,
\rho=|\psi\rho\rangle\langle\psi\rho|
and
\sigma=|\psi\sigma\rangle\langle\psi\sigma|
, it equals:
This tells us that the fidelity between pure states has a straightforward interpretation in terms of probability of finding the state
when measuring
in a basis containing
.
Some authors use an alternative definition
and call this quantity fidelity. The definition of
however is more common.
[3] [4] [5] To avoid confusion,
could be called "square root fidelity". In any case it is advisable to clarify the adopted definition whenever the fidelity is employed.
Motivation from classical counterpart
with values
(
categorical random variables) and probabilities
and
, the fidelity of
and
is defined to be the quantity
F(X,Y)=\left(\sumi\sqrt{pi
.
The fidelity deals with the marginal distribution of the random variables. It says nothing about the joint distribution of those variables. In other words, the fidelity
is the square of the
inner product of
(\sqrt{p1},\ldots,\sqrt{pn})
and
(\sqrt{q1},\ldots,\sqrt{qn})
viewed as vectors in
Euclidean space. Notice that
if and only if
. In general,
. The
measure
is known as the
Bhattacharyya coefficient.
Given a classical measure of the distinguishability of two probability distributions, one can motivate a measure of distinguishability of two quantum states as follows: if an experimenter is attempting to determine whether a quantum state is either of two possibilities
or
, the most general possible measurement they can make on the state is a
POVM, which is described by a set of
Hermitian positive semidefinite operators
. When measuring a state
with this POVM,
-th outcome is found with probability
pi=\operatorname{tr}(\rhoFi)
, and likewise with probability
qi=\operatorname{tr}(\sigmaFi)
for
. The ability to distinguish between
and
is then equivalent to their ability to distinguish between the classical probability distributions
and
. A natural question is then to ask what is the POVM the makes the two distributions as distinguishable as possible, which in this context means to minimize the Bhattacharyya coefficient over the possible choices of POVM. Formally, we are thus led to define the fidelity between quantum states as:
} F(X,Y) = \min_ \left(\sum _i \sqrt\right)^.
It was shown by Fuchs and Caves[6] that the minimization in this expression can be computed explicitly, with solution the projective POVM corresponding to measuring in the eigenbasis of
\sigma-1/2|\sqrt\sigma\sqrt\rho|\sigma-1/2
, and results in the common explicit expression for the fidelity as
Equivalent expressions
Equivalent expression via trace norm
An equivalent expression for the fidelity between arbitrary states via the trace norm is:
F(\rho,\sigma)=\lVert\sqrt{\rho}\sqrt{\sigma}
=\left(\operatorname{tr}|\sqrt\rho\sqrt\sigma|\right)2,
where the absolute value of an operator is here defined as
|A|\equiv\sqrt{A\daggerA}
.
Equivalent expression via characteristic polynomials
Since the trace of a matrix is equal to the sum of its eigenvalues
F(\rho,\sigma)=\sumj\sqrt{λj},
where the
are the eigenvalues of
\sqrt{\rho}\sigma\sqrt{\rho}
, which is positive semidefinite by construction and so the square roots of the eigenvalues are well defined. Because the characteristic polynomial of a product of two matrices is independent of the order, the
spectrum of a matrix product is invariant under cyclic permutation, and so these eigenvalues can instead be calculated from
.
[7] Reversing the trace property leads to
F(\rho,\sigma)=\left(\operatorname{tr}\sqrt{\rho\sigma}\right)2
.
Expressions for pure states
If (at least) one of the two states is pure, for example
\rho=|\psi\rho\rangle\langle\psi\rho|
, the fidelity simplifies to
This follows observing that if
is pure then
, and thus
If both states are pure,
\rho=|\psi\rho\rangle\langle\psi\rho|
and
\sigma=|\psi\sigma\rangle\langle\psi\sigma|
, then we get the even simpler expression:
Properties
Some of the important properties of the quantum state fidelity are:
F(\rho,\sigma)=F(\sigma,\rho)
.
and
,
, and
.
- Consistency with fidelity between probability distributions. If
and
commute, the definition simplifies to
where
are the eigenvalues of
, respectively. To see this, remember that if
then they can be diagonalized in the same basis:
so that
\operatorname{tr}\sqrt{\rho\sigma}=
\operatorname{tr}\left(\sumk\sqrt{pkqk}|k\rangle\langlek|\right)=
\sumk\sqrt{pkqk}.
- Explicit expression for qubits.
If
and
are both
qubit states, the fidelity can be computed as
[8] F(\rho,\sigma)=\operatorname{tr}(\rho\sigma)+2\sqrt{\det(\rho)\det(\sigma)}.
Qubit state means that
and
are represented by two-dimensional matrices. This result follows noticing that
M=\sqrt{\rho}\sigma\sqrt{\rho}
is a
positive semidefinite operator, hence
\operatorname{tr}\sqrt{M}=\sqrt{λ1}+\sqrt{λ2}
, where
and
are the (nonnegative) eigenvalues of
. If
(or
) is pure, this result is simplified further to
F(\rho,\sigma)=\operatorname{tr}(\rho\sigma)
since
for pure states.
Unitary invariance
Direct calculation shows that the fidelity is preserved by unitary evolution, i.e.
F(\rho,\sigma)=F(U\rho U*,U\sigmaU*)
.
Relationship with the fidelity between the corresponding probability distributions
Let
be an arbitrary
positive operator-valued measure (POVM); that is, a set of positive semidefinite operators
satisfying
. Then, for any pair of states
and
, we have
where in the last step we denoted with
pk\equiv\operatorname{tr}(Ek\rho)
and
qk\equiv\operatorname{tr}(Ek\sigma)
the probability distributions obtained by measuring
with the POVM
.
This shows that the square root of the fidelity between two quantum states is upper bounded by the Bhattacharyya coefficient between the corresponding probability distributions in any possible POVM. Indeed, it is more generally true that where
F(\boldsymbolp,\boldsymbolq)\equiv\left(\sumk\sqrt{pk
, and the minimum is taken over all possible POVMs. More specifically, one can prove that the minimum is achieved by the projective POVM corresponding to measuring in the eigenbasis of the operator
\sigma-1/2|\sqrt\sigma\sqrt\rho|\sigma-1/2
.
[9] Proof of inequality
As was previously shown, the square root of the fidelity can be written as
\sqrt{F(\rho,\sigma)}=\operatorname{tr}|\sqrt\rho\sqrt\sigma|,
which is equivalent to the existence of a unitary operator
such that
Remembering that
holds true for any POVM, we can then write
where in the last step we used Cauchy-Schwarz inequality as in
|\operatorname{tr}(A\daggerB)|2\le\operatorname{tr}(A\daggerA)\operatorname{tr}(B\daggerB)
.
Behavior under quantum operations
is applied to the states:
[10] for any trace-preserving
completely positive map
.
Relationship to trace distance
We can define the trace distance between two matrices A and B in terms of the trace norm by
When A and B are both density operators, this is a quantum generalization of the statistical distance. This is relevant because the trace distance provides upper and lower bounds on the fidelity as quantified by the Fuchs–van de Graaf inequalities,[11]
1-\sqrt{F(\rho,\sigma)}\leD(\rho,\sigma)\le\sqrt{1-F(\rho,\sigma)}.
Often the trace distance is easier to calculate or bound than the fidelity, so these relationships are quite useful. In the case that at least one of the states is a pure state Ψ, the lower bound can be tightened.
1-F(\psi,\rho)\leD(\psi,\rho).
Uhlmann's theorem
We saw that for two pure states, their fidelity coincides with the overlap. Uhlmann's theorem[12] generalizes this statement to mixed states, in terms of their purifications:
Theorem Let ρ and σ be density matrices acting on Cn. Let ρ be the unique positive square root of ρ and
be a purification of ρ (therefore
is an orthonormal basis), then the following equality holds:
F(\rho,\sigma)=
|\langle\psi\rho|\psi\sigma\rangle|2
where
is a purification of σ. Therefore, in general, the fidelity is the maximum overlap between purifications.
Sketch of proof
A simple proof can be sketched as follows. Let
denote the vector
|\Omega\rangle=
|ei\rangle ⊗ |ei\rangle
and σ be the unique positive square root of σ. We see that, due to the unitary freedom in square root factorizations and choosing orthonormal bases, an arbitrary purification of σ is of the form
|\psi\sigma\rangle=(\sigma{1/{2}}V1 ⊗ V2)|\Omega\rangle
where Vi's are unitary operators. Now we directly calculate
|\langle\psi\rho|\psi\sigma\rangle|2
=|\langle\Omega|(\rho{1/{2}} ⊗ I)(\sigma{1/{2}}V1 ⊗ V2)|\Omega\rangle|2
=|\operatorname{tr}(\rho{1/{2}}\sigma{1/{2}}V1
)|2.
But in general, for any square matrix A and unitary U, it is true that |tr(AU)| ≤ tr((A*A)). Furthermore, equality is achieved if U* is the unitary operator in the polar decomposition of A. From this follows directly Uhlmann's theorem.
Proof with explicit decompositions
We will here provide an alternative, explicit way to prove Uhlmann's theorem.
Let
and
be purifications of
and
, respectively. To start, let us show that
|\langle\psi\rho|\psi\sigma\rangle|\le\operatorname{tr}|\sqrt\rho\sqrt\sigma|
.
The general form of the purifications of the states is:were
are the
eigenvectors of
, and
are arbitrary orthonormal bases. The overlap between the purifications is
where the unitary matrix
is defined as
The conclusion is now reached via using the inequality
|\operatorname{tr}(AU)|\le\operatorname{tr}(\sqrt{A\daggerA})\equiv\operatorname{tr}|A|
:
Note that this inequality is the
triangle inequality applied to the singular values of the matrix. Indeed, for a generic matrix
A\equiv\sumjsj(A)|aj\rangle\langlebj|
and unitary
U=\sumj|bj\rangle\langlewj|
, we have
where
are the (always real and non-negative)
singular values of
, as in the
singular value decomposition. The inequality is saturated and becomes an equality when
, that is, when
U=\sumk|bk\rangle\langleak|,
and thus
AU=\sqrt{AA\dagger}\equiv|A|
. The above shows that
|\langle\psi\rho|\psi\sigma\rangle|=
\operatorname{tr}|\sqrt\rho\sqrt\sigma|
when the purifications
and
are such that
\sqrt\rho\sqrt\sigmaU=|\sqrt\rho\sqrt\sigma|
. Because this choice is possible regardless of the states, we can finally conclude that
Consequences
Some immediate consequences of Uhlmann's theorem are
- Fidelity is symmetric in its arguments, i.e. F (ρ,σ) = F (σ,ρ). Note that this is not obvious from the original definition.
- F (ρ,σ) lies in [0,1], by the Cauchy–Schwarz inequality.
- F (ρ,σ) = 1 if and only if ρ = σ, since Ψρ = Ψσ implies ρ = σ.
So we can see that fidelity behaves almost like a metric. This can be formalized and made useful by defining
\cos2\theta\rho\sigma=F(\rho,\sigma)
As the angle between the states
and
. It follows from the above properties that
is non-negative, symmetric in its inputs, and is equal to zero if and only if
. Furthermore, it can be proved that it obeys the triangle inequality, so this angle is a metric on the state space: the
Fubini–Study metric.
[13] References
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