The great promise of quantum computers is their ability to solve problems that are not possible within a reasonable time frame for classical computers, or even impossible for them [1–7]. As our understanding of building quantum computers continues to advance, there is a growing desire for these systems to incorporate a greater number of qubits [8–11]. Furthermore, there is a recognition of the need for subsystems to have more than two levels, introducing the concept of qudits [12–16]. This quest is driven by the promise of solving more complex problems, achieving greater computational efficiency, and unlocking new frontiers in quantum information processing. Whether through significant leaps in scale or the increase in dimensions, the expansion of quantum systems also introduces a greater complexity for the understanding of these systems. This inherent complexity in large systems provide a fertile ground for quantify performance, optimizing error correction, and exploring the unique advantages offered by the quantum realm.
One way to assess the effectiveness of adjustments made in the quantum computers is by measuring the quantum fidelity between quantum states which are given by density operators. Quantum states describe the information that we have about the quantum system, once it is needed to calculate the probabilities of obtaining a specific measurement outcome of a given observable. Classically, one way to measure the similarity between the probabilities distributions p and q is using the classical fidelity, which is given by [2, 17]:
$\begin{eqnarray}F({\boldsymbol{p}},{\boldsymbol{q}})={\left(\displaystyle \sum _{j=1}^{d}\sqrt{{p}_{j}{q}_{j}}\right)}^{2},\end{eqnarray}$
where ${\sum }_{j=1}^{d}\sqrt{{p}_{j}{q}_{j}}$ is the Bhattacharyya coefficient. This expression quantifies how ‘close’ two classical probability distributions are from each other. It also provides a motivation in how to quantify the ‘closeness’ of two quantum states by making appropriate modifications.In equation (1 ), the classical fidelity is introduced as a measure of similarity between probability distributions p and q. Transitioning from classical to quantum settings involves the replacement of the classical probabilities pj and qj by the quantum mechanical probabilities $\mathrm{Tr}(\rho {{E}}_{j})$ and $\mathrm{Tr}(\sigma {{E}}_{j})$, respectively, where ρ and σ are density operators and {Ej} are the elements of a positive operator-valued measure (POVM) satisfying ${\sum }_{j}{E}_{j}={\mathbb{I}}$, with ${\mathbb{I}}$ being the identity operator [18]. It is worth mentioning that density operators can be interpreted as the quantum version of classical probability distributions. Therefore, in quantum mechanics, wherein the states are inherently more complex due to phenomena such as quantum superposition and entanglement, the natural extension for the fidelity F(ρ, σ) between two density operators, as delineated in [19], is given by
$\begin{eqnarray}F(\rho ,\sigma )=\mathop{\min }\limits_{\{{E}_{j}\}}{\left(\displaystyle \sum _{j}\sqrt{\mathrm{Tr}(\rho {{E}}_{j})}\sqrt{\mathrm{Tr}(\sigma {{E}}_{j})}\right)}^{2},\end{eqnarray}$
where the minimization is performed over all possible POVMs.Analogous to the classical quantification of the similarity between probabilities distributions, in quantum mechanics the task is commonly accomplished through quantum fidelity, which can be used for the quantification of the proximity between a theoretically ideal quantum state and the state prepared by a quantum computer. However, it is interesting to notice that the first quantum fidelity measure for density operators was proposed by Jozsa [20] to describe the transition probability as introduced by Uhlmann [21] and it is defined as
$\begin{eqnarray}F(\rho ,\sigma )\,:= {\left(\mathrm{Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^{2}.\end{eqnarray}$
This expression holds for any quantum states ρ and σ.The process to show that the definitions given by equations (2 ) and (3 ) are equivalent can be summarized in the following steps: (i) the first step is to consider the unitary invariance a cyclic property of the trace function operation, i.e., $\mathrm{Tr}(\rho {{E}}_{j})=\mathrm{Tr}({U}\sqrt{\rho }{{E}}_{j}\sqrt{\rho }{{U}}^{\dagger });$ (ii) the second step is the use of the Cauchy–Schwarz inequality, i.e, $\mathrm{Tr}({{A}}^{\dagger }{A})\mathrm{Tr}({{B}}^{\dagger }{B})\,\geqslant | \mathrm{Tr}({{A}}^{\dagger }{B}){| }^{2}$ for an appropriate choice of the operators A and B, which implies that $\sqrt{\mathrm{Tr}(\rho {{E}}_{j})}\sqrt{\mathrm{Tr}(\sigma {{E}}_{j})}\geqslant \mathrm{Tr}({U}\sqrt{\rho }{{E}}_{j}\sqrt{\sigma });$ (iii) the final step consists in summing over j and choosing an appropriate form for the unitary operator U in such way that the equality is attained, which is achieved by choosing $U\,=\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}{\rho }^{-1/2}{\sigma }^{-1/2}$. In [19], this demonstration is done rigorously and in more details.
Although there are various algorithms available for fidelity estimation [22–24], the direct method for quantifying fidelity involves obtaining the quantum state through quantum state tomography (QST) [25–27] to acquire the prepared state σ. Subsequently, matrix arithmetic operations are performed to compare the similarity between the prepared state σ and the target theoretical state ρ. The higher the fidelity, the better the quantum computer can reproduce the desired theoretical state. As we increase the number of subsystems of interest, the matrices involved in describing the quantum system grow exponentially. To calculate fidelity, we need to perform the diagonalization of a density matrix, calculate the product between them, and subsequently diagonalize the resulting matrix. Especially for very large density operators, obtaining diagonalization can be a demanding task since it is necessary to find the eigenvectors and eigenvalues numerically. Therefore, methods that approximate these values require careful consideration to ensure the desired precision. The authors in [28] demonstrated a method to reduce the number of diagonalizations required for calculating the Uhlmann–Jozsa fidelity [20, 21], thereby improving computational efficiency. Their alternative form for fidelity relies on the fact that $\sqrt{\rho }\sigma \sqrt{\rho }$ is similar to ρσ and, therefore, shares the same set of eigenvalues, as noted in the work of Miszczak et al [29].
The quantification of fidelity remains one of the best ways to measure quantum similarity and assess the advancements achieved in quantum computers. Alternative methods include algorithms or integral forms utilizing matrix algebraic operations. However, the complexity of this quantification can pose a challenge, particularly with high-dimensional systems. Therefore, finding ways to simplify this process is desirable. In accordance to the [28], we show in an alternative way the simplified expression obtained in their work. For that, our alternative proof relies on power series expansion and the properties of the trace function. By solely leveraging the properties of the trace function and assuming it is feasible to express a function composed of three linear operators in a series of polynomials, we will demonstrate that ${\rm{Tr}}\{f({ABC})\}={\rm{Tr}}\{f({CAB})\}={\rm{Tr}}\{f({BCA})\}$.
Let us consider three linear operators A, B, C defined on a Hilbert space ${ \mathcal H }$. We begin by writing a function of the product of these three linear operators that can be expanded as a power series:
$\begin{eqnarray}f({ABC})=\displaystyle \sum _{j=0}^{\infty }{c}_{j}{\left({ABC}\right)}^{j}.\end{eqnarray}$
Using the cyclic property of the trace function, we obtain $\begin{eqnarray}\begin{array}{rcl}\mathrm{Tr}\left\{{\left({ABC}\right)}^{j}\right\} & = & \mathrm{Tr}({ABCABC}\cdots {ABC})\\ & = & \mathrm{Tr}({CABCAB}\cdots {CAB})\\ & = & \mathrm{Tr}\left\{{\left({CAB}\right)}^{j}\right\}.\end{array}\end{eqnarray}$
With this, using the linearity of the trace function, we can write $\begin{eqnarray}\begin{array}{rcl}\mathrm{Tr}\{{f}({ABC})\} & = & \mathrm{Tr}\left\{\displaystyle \sum _{{\text{}}{\rm{j}}=0}^{\infty }{{c}}_{j}{\left({ABC}\right)}^{j}\right\}\\ & = & \displaystyle \sum _{j=0}^{\infty }{{c}}_{j}\mathrm{Tr}\{{\left({ABC}\right)}^{j}\}\\ & = & \displaystyle \sum _{j=0}^{\infty }{{c}}_{j}\mathrm{Tr}\{{\left({CAB}\right)}^{j}\}\\ & = & \mathrm{Tr}\left\{\displaystyle \sum _{{\rm{j}}=0}^{\infty }{{c}}_{j}{\left({CAB}\right)}^{j}\right\}\\ & = & \mathrm{Tr}\{{f}({CAB})\}.\end{array}\end{eqnarray}$
As a first application of this general result, we regard the Uhlmann–Jozsa fidelity between two density operators ρ and σ, that is defined as in equation (3 ). So, by applying equation (6 ) with
$\begin{eqnarray}f(x)=\sqrt{x},A=C=\sqrt{\rho },B=\sigma ,\end{eqnarray}$
we will have $\begin{eqnarray}\begin{array}{rcl}F(\rho ,\sigma ) & = & {\left(\mathrm{Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^{2}={\left(\mathrm{Tr}\sqrt{\sqrt{\rho }\sqrt{\rho }\sigma }\right)}^{2}\\ & = & {\left(\mathrm{Tr}\sqrt{\rho \sigma }\right)}^{2},\end{array}\end{eqnarray}$
which is the result obtained in [28].If the power series considered in equation (4 ) is the Taylor series, then this alternative derivation of equation (8 ) is valid only for positive-definite hermitian density operators, once the derivatives of the square root function are not defined for null eigenvalues. However, this derivation can be extended for positive semi-definite hermitian density operators by using the Chebyshev polynomials [30, 31], as done similarly in [32] for the von-Neumann entropy. The Chebyshev polynomials are presented in the appendix , where we also describe the expansion of an arbitrary function in terms of the Chebyshev polynomials as a power series in order for the derivations of equations (6 ) and (8 ) to remain valid. Moreover, since $\sqrt{\rho }\sigma \sqrt{\rho }$ is hermitian and also positive semi-definite, its eigenvalues are real and are defined in the interval [0, b], where $b\in {\mathbb{R}}$. This interval can be mapped in the Chebyshev interval as shown in appendix A.1 .
Now, let us consider another application of our result expressed by equation (6 ) regarding alternative quantum states dissimilarity functions. For instance, the α-z-relative Rényi entropy [33–35] is defined as:
$\begin{eqnarray}{D}_{\alpha ,z}(\rho | | \sigma )=\displaystyle \frac{1}{\alpha -1}\mathrm{logTr}{\left[{\sigma }^{(1-\alpha )/2z}{\rho }^{\alpha /z}{\sigma }^{(1-\alpha )/2z}\right]}^{z},\end{eqnarray}$
with $\alpha \in {\mathbb{R}}$ and $z\in {{\mathbb{R}}}^{+}$. Let us notice that Dα,1(ρ∣∣σ) = Dα(ρ∣∣σ) and ${D}_{\alpha ,\alpha }(\rho | | \sigma )={\tilde{D}}_{\alpha }(\rho | | \sigma ),$ i.e., Dα,z(ρ∣∣σ) generalizes the Rényi divergence of order α, ${D}_{\alpha }(\rho | | \sigma )\,=\tfrac{1}{\alpha -1}\mathrm{logTr}({\rho }^{\alpha }{\sigma }^{1-\alpha })$, and the sandwiched Rényi divergence of order α, ${\tilde{D}}_{\alpha }(\rho | | \sigma )=\tfrac{1}{\alpha -1}\mathrm{logTr}{\left[{\sigma }^{(1-\alpha )/2\alpha }\rho {\sigma }^{(1-\alpha )/2\alpha }\right]}^{\alpha }$. Besides, the relative entropy is obtained as the following limit: ${\mathrm{lim}}_{\alpha \to 1}{D}_{\alpha }(\rho | | \sigma )=D(\rho | | \sigma )=\mathrm{Tr}(\rho (\mathrm{log}\rho -\mathrm{log}\sigma )).$ Now, by applying our result $\mathrm{Tr}({f}({ABC}))=\mathrm{Tr}({f}({CAB}))$ with $\begin{eqnarray}f(x)={x}^{z},A=C={\sigma }^{(1-\alpha )/2z},B={\rho }^{\alpha /z},\end{eqnarray}$
we can write $\begin{eqnarray}\begin{array}{rcl}{D}_{\alpha ,z}(\rho | | \sigma ) & = & \displaystyle \frac{1}{\alpha -1}\mathrm{logTr}{\left[{\sigma }^{(1-\alpha )/2z}{\rho }^{\alpha /z}{\sigma }^{(1-\alpha )/2z}\right]}^{z}\\ & = & \displaystyle \frac{1}{\alpha -1}\mathrm{logTr}{\left[{\sigma }^{(1-\alpha )/2z}{\sigma }^{(1-\alpha )/2z}{\rho }^{\alpha /z}\right]}^{z}\\ & = & \displaystyle \frac{1}{\alpha -1}\mathrm{logTr}{\left[{\sigma }^{(1-\alpha )/z}{\rho }^{\alpha /z}\right]}^{z}.\end{array}\end{eqnarray}$
Let us notice that we did not avoid any matrix diagonalization, however we need one less matrix multiplication calculation for obtaining Dα,z(ρ∣∣σ).On the other hand, regarding
$\begin{eqnarray}\begin{array}{rcl}{D}_{\alpha ,\alpha }(\rho | | \sigma ) & = & \displaystyle \frac{1}{\alpha -1}\mathrm{logTr}{\left[{\sigma }^{(1-\alpha )/\alpha }{\rho }^{\alpha /\alpha }\right]}^{\alpha }\\ & = & \displaystyle \frac{1}{\alpha -1}\mathrm{logTr}{\left[{\sigma }^{(1-\alpha )/\alpha }\rho \right]}^{\alpha },\end{array}\end{eqnarray}$
we can avoid one matrix diagonalization in the following particular case $\begin{eqnarray}\begin{array}{rcl}{D}_{\tfrac{1}{2},\tfrac{1}{2}}(\rho | | \sigma ) & = & \displaystyle \frac{1}{1/2-1}\mathrm{logTr}{\left[{\sigma }^{(1-1/2)/(1/2)}\rho \right]}^{1/2}\\ & = & -2\mathrm{logTr}\sqrt{\sigma \rho }\\ & = & -2\mathrm{log}(F(\rho ,\sigma )).\end{array}\end{eqnarray}$
We observe that this equality was already reported in [33].To sum up, in this article we gave an alternative derivation of the simplified expression obtained by [28] for the Uhlmann–Jozsa fidelity. Our result relies solely on the expansion in power series of a function of the product of three linear operators and the cyclic and linearity properties of the trace function. Moreover, we also obtained an alternative expression for other quantum states’ dissimilarity functions. Finally, we hope that our method will enable further simplifications of other expressions that may arise in the study and development of quantum information science.
Finally, we observe that after the completion of this manuscript, we became aware of the related work reported in [36].