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

\rho

and

\sigma

, 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

\rho

and

\sqrt\rho\sigma\sqrt\rho

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:F(\rho, \sigma)=|\langle\psi_\rho|\psi_\sigma\rangle|^2.This tells us that the fidelity between pure states has a straightforward interpretation in terms of probability of finding the state

|\psi\rho\rangle

when measuring

|\psi\sigma\rangle

in a basis containing

|\psi\rho\rangle

.

Some authors use an alternative definition

F':=\sqrt{F}

and call this quantity fidelity. The definition of

F

however is more common.[3] [4] [5] To avoid confusion,

F'

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

X,Y

with values

(1,...,n)

(categorical random variables) and probabilities

p=(p1,p2,\ldots,pn)

and

q=(q1,q2,\ldots,qn)

, the fidelity of

X

and

Y

is defined to be the quantity

F(X,Y)=\left(\sumi\sqrt{pi

2
q
i}\right)
.

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

F(X,Y)

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

F(X,Y)=1

if and only if

p=q

. In general,

0\leqF(X,Y)\leq1

. The measure

\sumi\sqrt{piqi}

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

\rho

or

\sigma

, 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

\{Fi\}

. When measuring a state

\rho

with this POVM,

i

-th outcome is found with probability

pi=\operatorname{tr}(\rhoFi)

, and likewise with probability

qi=\operatorname{tr}(\sigmaFi)

for

\sigma

. The ability to distinguish between

\rho

and

\sigma

is then equivalent to their ability to distinguish between the classical probability distributions

p

and

q

. 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(\rho,\sigma)=

min
\{Fi\
} 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 asF(\rho, \sigma) = \left(\operatorname \sqrt\right)^2.

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}

2
\rVert
\operatorname{tr}

=\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

λj

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

\rho\sigma

.[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 toF(\rho,\sigma)=\operatorname(\sigma\rho)=\langle \psi_\rho|\sigma|\psi_\rho\rangle.This follows observing that if

\rho

is pure then

\sqrt\rho=\rho

, and thusF(\rho, \sigma) = \left(\operatorname \sqrt \right)^2= \langle \psi_\rho | \sigma | \psi_\rho \rangle \left(\operatorname \sqrt \right)^2= \langle \psi_\rho | \sigma | \psi_\rho \rangle.

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:F(\rho, \sigma)=|\langle\psi_\rho|\psi_\sigma\rangle|^2.

Properties

Some of the important properties of the quantum state fidelity are:

F(\rho,\sigma)=F(\sigma,\rho)

.

\rho

and

\sigma

,

0\leF(\rho,\sigma)\le1

, and

F(\rho,\rho)=1

.

\rho

and

\sigma

commute, the definition simplifies to F(\rho,\sigma) =\left[\operatorname{tr}\sqrt{\rho\sigma}\right]^2 =\left(\sum_k \sqrt \right)^2 =F(\boldsymbol p, \boldsymbol q),where

pk,qk

are the eigenvalues of

\rho,\sigma

, respectively. To see this, remember that if

[\rho,\sigma]=0

then they can be diagonalized in the same basis: \rho = \sum_i p_i | i \rangle \langle i |\text\sigma = \sum_i q_i | i \rangle \langle i |,so that

\operatorname{tr}\sqrt{\rho\sigma}= \operatorname{tr}\left(\sumk\sqrt{pkqk}|k\rangle\langlek|\right)= \sumk\sqrt{pkqk}.

If

\rho

and

\sigma

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

\rho

and

\sigma

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

λ1

and

λ2

are the (nonnegative) eigenvalues of

M

. If

\rho

(or

\sigma

) is pure, this result is simplified further to

F(\rho,\sigma)=\operatorname{tr}(\rho\sigma)

since

Det(\rho)=0

for pure states.

Unitary invariance

Direct calculation shows that the fidelity is preserved by unitary evolution, i.e.

F(\rho,\sigma)=F(U\rhoU*,U\sigmaU*)

U

.

Relationship with the fidelity between the corresponding probability distributions

Let

\{Ek\}k

be an arbitrary positive operator-valued measure (POVM); that is, a set of positive semidefinite operators

Ek

satisfying

\sumkEk=I

. Then, for any pair of states

\rho

and

\sigma

, we have \sqrt \le \sum_k \sqrt\sqrt \equiv \sum_k \sqrt,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

\rho,\sigma

with the POVM

\{Ek\}k

.

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 F(\rho,\sigma)=\min_ F(\boldsymbol p,\boldsymbol q), where

F(\boldsymbolp,\boldsymbolq)\equiv\left(\sumk\sqrt{pk

2
q
k}\right)
, 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

U

such that

\sqrt=\operatorname(\sqrt\rho\sqrt\sigma U).Remembering that

\sumkEk=I

holds true for any POVM, we can then write\sqrt=\operatorname(\sqrt\rho\sqrt\sigma U)=\sum_k\operatorname(\sqrt\rho E_k \sqrt\sigma U)=\sum_k\operatorname(\sqrt\rho \sqrt \sqrt\sqrt\sigma U) \le\sum_k\sqrt,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

lE

is applied to the states:[10] F(\mathcal E(\rho),\mathcal E(\sigma)) \geF(\rho,\sigma), for any trace-preserving completely positive map

lE

.

Relationship to trace distance

We can define the trace distance between two matrices A and B in terms of the trace norm by

D(A,B)=

1
2

\|A-B\|\rm.

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

| \psi _ \rangle = \sum_^n (\rho^ | e_i \rangle) \otimes | e_i \rangle \in \mathbb^n \otimes \mathbb^n

be a purification of ρ (therefore

style\{|ei\rangle\}

is an orthonormal basis), then the following equality holds:

F(\rho,\sigma)=

max
|\psi\sigma\rangle

|\langle\psi\rho|\psi\sigma\rangle|2

where

|\psi\sigma\rangle

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

style|\Omega\rangle

denote the vector

|\Omega\rangle=

n
\sum
i=1

|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}}V1V2)|\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}}V1V2)|\Omega\rangle|2 =|\operatorname{tr}(\rho{1/{2}}\sigma{1/{2}}V1

T
V
2

)|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

|\psi\rho\rangle

and

|\psi\sigma\rangle

be purifications of

\rho

and

\sigma

, 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:\begin |\psi_\rho\rangle &=\sum_k\sqrt|\lambda_k\rangle\otimes|u_k\rangle, \\ |\psi_\sigma\rangle &=\sum_k\sqrt|\mu_k\rangle\otimes|v_k\rangle,\endwere

|λk\rangle,|\muk\rangle

are the eigenvectors of

\rho,\sigma

, and

\{uk\}k,\{vk\}k

are arbitrary orthonormal bases. The overlap between the purifications is\langle\psi_\rho|\psi_\sigma\rangle =\sum_\sqrt \langle\lambda_j|\mu_k\rangle\,\langle u_j|v_k\rangle =\operatorname\left(\sqrt\rho\sqrt\sigma U\right),where the unitary matrix

U

is defined asU=\left(\sum_k |\mu_k\rangle\!\langle u_k| \right)\,\left(\sum_j |v_j\rangle\!\langle \lambda_j|\right).The conclusion is now reached via using the inequality

|\operatorname{tr}(AU)|\le\operatorname{tr}(\sqrt{A\daggerA})\equiv\operatorname{tr}|A|

: |\langle\psi_\rho|\psi_\sigma\rangle|=|\operatorname(\sqrt\rho\sqrt\sigma U)| \le\operatorname|\sqrt\rho\sqrt\sigma|.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\begin|\operatorname(AU)| &=\left|\operatorname\left(\sum_j s_j(A)|a_j\rangle\!\langle b_j| \,\,\sum_k |b_k\rangle\!\langle w_k| \right)\right| \\ &= \left|\sum_j s_j(A)\langle w_j|a_j\rangle\right|\\ &\le\sum_j s_j(A) \,|\langle w_j|a_j\rangle| \\&\le\sum_j s_j(A) \\&= \operatorname|A|,\endwhere

sj(A)\ge0

are the (always real and non-negative) singular values of

A

, as in the singular value decomposition. The inequality is saturated and becomes an equality when

\langlewj|aj\rangle=1

, 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

|\psi\rho\rangle

and

|\psi\sigma\rangle

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\operatorname|\sqrt\rho\sqrt\sigma|=\max|\langle\psi_\rho|\psi_\sigma\rangle|.

Consequences

Some immediate consequences of Uhlmann's theorem are

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

\rho

and

\sigma

. It follows from the above properties that

\theta\rho\sigma

is non-negative, symmetric in its inputs, and is equal to zero if and only if

\rho=\sigma

. 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

  1. R. Jozsa, Fidelity for Mixed Quantum States, J. Mod. Opt. 41, 2315--2323 (1994). DOI: http://doi.org/10.1080/09500349414552171
  2. Book: Nielsen . Michael A. . Quantum Computation and Quantum Information . Chuang . Isaac L. . Cambridge University Press . 2000 . 978-0521635035 . 10.1017/CBO9780511976667.
  3. Book: Bengtsson, Ingemar . Geometry of Quantum States: An Introduction to Quantum Entanglement . Cambridge University Press . 2017 . 978-1-107-02625-4 . Cambridge, United Kingdom New York, NY.
  4. Book: Walls . D. F. . Quantum Optics . Milburn . G. J. . Springer . 2008 . 978-3-540-28573-1 . Berlin.
  5. Book: Jaeger, Gregg . Quantum Information: An Overview . Springer . 2007 . 978-0-387-35725-6 . New York London.
  6. C. A. Fuchs, C. M. Caves: "Ensemble-Dependent Bounds for Accessible Information in Quantum Mechanics", Physical Review Letters 73, 3047(1994)
  7. Baldwin . Andrew J. . Jones . Jonathan A. . Efficiently computing the Uhlmann fidelity for density matrices . . 2023 . 107 . 012427 . 10.1103/PhysRevA.107.012427. 2211.02623 .
  8. M. Hübner, Explicit Computation of the Bures Distance for Density Matrices, Phys. Lett. A 163, 239--242 (1992). DOI: https://doi.org/10.1016/0375-9601%2892%2991004-B
  9. Book: Watrous, John . The Theory of Quantum Information . 2018-04-26 . Cambridge University Press . 10.1017/9781316848142 . 978-1-316-84814-2.
  10. Nielsen. M. A.. 1996-06-13. The entanglement fidelity and quantum error correction. quant-ph/9606012. 1996quant.ph..6012N.
  11. C. A. Fuchs and J. van de Graaf, "Cryptographic Distinguishability Measures for Quantum Mechanical States", IEEE Trans. Inf. Theory 45, 1216 (1999). arXiv:quant-ph/9712042
  12. Uhlmann . A. . 1976 . The "transition probability" in the state space of a ∗-algebra . Reports on Mathematical Physics . 9 . 2 . 273–279 . 1976RpMP....9..273U . 10.1016/0034-4877(76)90060-4 . 0034-4877.
  13. K. Życzkowski, I. Bengtsson, Geometry of Quantum States, Cambridge University Press, 2008, 131