Fidelity of quantum states - AbsoluteAstronomy.com

In quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It is not a metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

on the space of density matrices, but it can be used to define the Bures metric on this space.

Motivation

In probability theory, given two random variables p = (p₁...p_n) and q = (q₁...q_n) on the probability space X = {1,2...n}. The fidelity of p and q is defined to be the quantity

F(p,q) = \sum _i \sqrt{p_i q_i}

. In other words, the fidelity F(p,q) is the inner product of

(\sqrt{p_1}, \cdots ,\sqrt{p_n})

and

(\sqrt{q_1}, \cdots ,\sqrt{q_n})

viewed as vectors in Euclidean space. Notice that if and only if p = q, F(p,q) = 1. In general,

0 \leq F(p,q) \leq 1

. This measure is known classically as the Bhattacharyya distance

Bhattacharyya distance

In statistics, the Bhattacharyya distance measures the similarity of two discrete or continuous probability distributions. It is closely related to the Bhattacharyya coefficient which is a measure of the amount of overlap between two statistical samples or populations. Both measures are named after A...

. Making the appropriate modification for the matricial notion of square root and mimicking the above definition give the fidelity of two quantum state.

Definition

Given two density matrices ρ and σ, the fidelity is defined by

F(\rho, \sigma) = \operatorname{Tr} \left[\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}\right].

By M^½ of a positive semidefinite matrix M, we mean its unique positive square root given by the spectral theorem

Spectral theorem

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

. The Euclidean inner product from the classical definition is replaced by the Hilbert-Schmidt inner product. When the states are classical, i.e. when ρ and σ commute, the definition coincides with that for probability distributions. An equivalent definition is given by

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

where the norm is the trace norm (sum of the singular values). This definition has the advantage that it clearly shows that the fidelity is symmetric in its two arguments. Notice by definition F is non-negative, and F(ρ,ρ) = 1. In the following section it will be shown that it can be no larger than 1. In the original 1994 paper of Jozsa the name 'fidelity' was used for the quantity

F\;'=F^2

and this convention is often used in the literature. According to this convention 'fidelity' has a meaning of probability.

Pure states

Consider pure states

\rho = | \phi \rangle \langle \phi |

and

\sigma = | \psi \rangle \langle \psi |

. Their fidelity is

F(\rho, \sigma) = \operatorname{Tr} [\rho \; \sigma \rho]^{\frac{1}{2}} \operatorname{Tr} \; [ | \phi \rangle \langle \phi | \psi \rangle \langle \psi |\phi \rangle \langle \phi | ]^{\frac{1}{2}} | \langle \phi | \psi \rangle |.

This is sometimes called the overlap between two states. If, say,

|\phi\rangle

is an eigenstate of an observable, and the system is prepared in

| \psi \rangle

, then F(ρ, σ)² is the probability of the system being in state

|\phi\rangle

after the measurement.

Commuting states

Let ρ and σ be two density matrices that commute. Therefore they can be simultaneously diagonalized by unitary matrices, and we can write

\rho = \sum_i p_i | i \rangle \langle i |

and

\sigma = \sum_i q_i | i \rangle \langle i |

for some orthonormal basis

\{ | i \rangle \}

. Direct calculation shows the fidelity is

F(\rho, \sigma) = \sum_i \sqrt{p_i q_i}.

This shows that, heuristically, fidelity of quantum states is a genuine extension of the notion from probability theory.

Unitary invariance

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

\; F(\rho, \sigma) = F(U \rho \; U^*, U \sigma U^*)

for any unitary operator U.

Uhlmann's theorem

We saw that for two pure states, their fidelity coincides with the overlap. Uhlmann's theorem generalizes this statement to mixed states, in terms of their purifications: Theorem Let ρ and σ be density matrices acting on Cⁿ. Let ρ^½ be the unique positive square root of ρ and

| \psi _{\rho} \rangle = \sum_{i=1}^n (\rho^{\frac{1}{2}} | e_i \rangle) \otimes | e_i \rangle \in \mathbb{C}^n \otimes \mathbb{C}^n

be a purification

Purification of quantum state

In quantum mechanics, especially quantum information, purification refers to the fact that every mixed state acting on finite dimensional Hilbert spaces can be viewed as the reduced state of some pure state....

of ρ (therefore

\textstyle \{|e_i\rangle\}

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

F(\rho, \sigma) = \max_{|\psi_{\sigma} \rangle} | \langle \psi _{\rho}| \psi _{\sigma} \rangle |

where

| \psi _{\sigma} \rangle

is a purification of σ. Therefore, in general, the fidelity is the maximum overlap between purifications. Proof: A simple proof can be sketched as follows. Let

\textstyle |\Omega\rangle

denote the vector

| \Omega \rangle= \sum_{i=1}^n | e_i \rangle \otimes | e_i \rangle

and σ^½ 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^{\frac{1}{2}} V_1 \otimes V_2 ) | \Omega \rangle

where V_i's are unitary operators. Now we directly calculate

| \langle \psi _{\rho}| \psi _{\sigma} \rangle | | \langle \Omega | ( \rho^{\frac{1}{2}} \otimes I) ( \sigma^{\frac{1}{2}} V_1 \otimes V_2 ) | \Omega \rangle | | \operatorname{Tr} ( \rho^{\frac{1}{2}} \sigma^{\frac{1}{2}} V_1 V_2^T )|.

But in general, for any square matrix A and unitary U, it is true that |Tr(AU)| ≤ Tr (A^*A)^½. Furthermore, equality is achieved if U^* is the unitary operator in the polar decomposition of A. From this follows directly Uhlmann's theorem.

Consequences

Some immediate consequences of Uhlmann's theorem areNEWLINE

Fidelity is symmetric in its arguments, i.e. F (ρ,σ) = F (σ,ρ). Notice this is not obvious from the definition.
F (ρ,σ) lies in [0,1], by the Cauchy-Schwarz inequality.
F (ρ,σ) = 1 if and only if ρ = σ, since Ψ_ρ = Ψ_σ implies ρ = σ.

NEWLINE So we can see that fidelity behaves almost like a metric. This can be formalised and made useful by defining

\cos \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

Fubini-Study metric

In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study....

Relationship to Trace Distance

We can define the trace distance

Trace distance

In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices...

between two matrices A and B in terms of the trace norm by

D(A,B) = \frac{1}{2}\| A-B\|_{\rm tr} \, .

When A and B are both density operators, this is a quantum generalization of the statistical distance

Statistical distance

In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two samples, two random variables, or two probability distributions, for example.-Metrics:...

. This is relevant because the trace distance provides upper and lower bounds on the fidelity.

1-F(\rho,\sigma) \le D(\rho,\sigma) \le\sqrt{1-F(\rho,\sigma)^2} \, .

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)^2 \le D(\psi,\rho) \, .

Fidelity of quantum measurements

The fidelity of a measurement with a projective measurement is defined as the overlap between their pre-measurement states

Quantum tomography

Quantum tomography or quantum state tomography is the process of reconstructing the quantum state for a source of quantum systems by measurements on the systems coming from the source. The source may be any device or system which prepares quantum states either consistently into quantum pure states...

\mathcal{F}_{n}\left(\psi_{tar}\right)=\langle\psi_{tar}\vert\hat{\rho}_{retr}^{[n]}\vert\psi_{tar}\rangle,

where

\hat{\rho}_{retr}^{[n]}

and

\vert\psi_{tar}\rangle

are respectively the pre-measurement state corresponding to the result "n" and the target state in which we would like measuring the system before its interaction with the measurement apparatus. The pre-measurement state

Quantum tomography

is the main tool of the retrodictive approach of quantum physics in which we make predictions about state preparations leading to a certain measurement result. In such an approach, this fidelity has an interesting meaning: this is nothing else than the retrodictive probability of preparing the system in the target state

\vert\psi_{tar}\rangle

when we read the result "n". Thus, when a measurement is sufficiently faithful

\mathcal{F}_{n}\left(\psi_{tar}\right)\simeq 1

, the most probable state in which the system was prepared before the measurement giving the result "n" is this target state

\vert\psi_{tar}\rangle

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