Fidelity of quantum states

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

Motivation

In probability theory, given two random variables "p" = ("p"₁..."p_n") and "q" = ("q"₁..."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 when "p" = "q", "F(p,q)" = 1. In general, $0\; leq\; F(p,q)\; leq\; 1$.

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

:$F(\; ho,\; sigma)\; =\; operatorname\{Tr\}\; (sqrt\{sqrt\{\; ho\}\; sigma\; sqrt\{\; ho).$

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

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.

Simple examples

Pure states

Consider pure states $ho\; =\; |\; phi\; angle\; langle\; phi\; |$ and $sigma\; =\; |\; psi\; angle\; langle\; psi\; |$. Their fidelity is

:$F(\; ho,\; sigma)\; =\; operatorname\{Tr\}\; [\; ho\; ;\; sigma\; ho]\; ^\{frac\{1\}\{2=\; operatorname\{Tr\}\; ;\; [\; |\; phi\; angle\; langle\; phi\; |\; psi\; angle\; langle\; psi\; |phi\; angle\; langle\; phi\; |\; ]\; ^\{frac\{1\}\{2=\; |\; langle\; phi\; |\; psi\; angle\; |.$

This is sometimes called the "overlap" between two states. If, say, $|phi\; angle$ is an eigenstate of an observable, and the system is prepared in $|\; psi\; angle$, then "F(ρ, σ)"² is the probability of the system being in state $|phi\; angle$ 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

:$ho\; =\; sum\_i\; p\_i\; |\; i\; angle\; langle\; i\; |$ and $sigma\; =\; sum\_i\; q\_i\; |\; i\; angle\; langle\; i\; |$

for some orthonormal basis $\{\; |\; i\; angle\; \}$. Direct calculation shows the fidelity is

:$F(\; ho,\; sigma)\; =\; sum\_i\; sqrt\{p\_i\; q\_i\}.$

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

Some properties

Unitary invariance

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

:$;\; F(\; ho,\; sigma)\; =\; F(U\; ho\; ;\; U^*,\; U\; sigma\; U^*)$

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 _{ ho} angle = sum_{i=1}^n ho^{frac{1}{2 | e_i angle otimes | e_i angle in mathbb{C}^n otimes mathbb{C}^n

be a purfication of ρ (therefore {|"e"_i >} is an orthonormal basis), then the following equality holds:

:$F(\; ho,\; sigma)\; =\; max\_\{|psi\_\{sigma\}\; angle\}\; |\; langle\; psi\; \_\{\; ho\}|\; psi\; \_\{sigma\}\; angle\; |$

where $|\; psi\; \_\{sigma\}\; angle$ 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 |Ω > denote the vector

:$|\; Omega\; angle=\; sum\; |\; e\_i\; angle\; otimes\; |\; e\_i\; angle$

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

:$|\; psi\_\{sigma\}\; angle\; =\; (\; sigma^\{frac\{1\}\{2\; V\_1\; otimes\; V\_2\; )\; |\; Omega\; angle$

where "V"_i's are unitary operators. Now we directly calculate

:$$
langle psi _{ ho}| psi _{sigma} angle | = langle Omega | ( ho^{frac{1}{2 otimes I) ( sigma^{frac{1}{2 V_1 otimes V_2 ) | Omega angle = operatorname{Tr} ( ho^{frac{1}{2 sigma^{frac{1}{2 V_1 V_2 ).

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 are
* 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.

References

* A. Uhlmann "The "Transition Probability" in the State Space of a *-Algebra". Rep. Math. Phys. 9 (1976) 273 - 279. [http://www.physik.uni-leipzig.de/~uhlmann/PDF/Uh76a.pdf PDF]
* R. Jozsa, "Fidelity for mixed quantum states", Journal of Modern Optics, 1994, vol. 41, 2315-2323.