Introduction
One of the central objects in the field of quantum geometry is the Berry curvature [1] and the integration of the Berry curvature over the Brillouin zone leads to the Chern number to characterize topological phases. Berry curvature can induce the spin Hall effect [2, 3] and can be considered as a kind of magnetic field in momentum space, leading to anomalous Hall effect [4] and anomalous Nernst effect [5].
In single-flavor color superconductor systems, the interplay between Berry curvature and topological properties was investigated recently [6]. By reconstructing Berry curvature through Hall drift, the Chern number can be measured experimentally [7] and through resonant infrared magnetic circular dichroism, the Berry curvature was probed in magnetic topological insulators [8]. Gritsev and Polkovnikov used the result from adiabatic perturbation theory to study a slowly driven system, and the Berry curvature emerges due to a certain quench [9].
Geometric phases for pure states were generalized to the case of mixed states [10]. However, much less attention was devoted to the Berry curvature for mixed states. In the field of quantum multi-parameter estimation, a trade-off relation was derived between the regrets of Fisher information about different parameters. The inequality is given by [11]
$\begin{eqnarray}{{\rm{\Delta }}}_{\alpha }^{2}+{{\rm{\Delta }}}_{\beta }^{2}+2\sqrt{1-{C}_{\alpha \beta }^{2}}{{\rm{\Delta }}}_{\alpha }{{\rm{\Delta }}}_{\beta }\geqslant {C}_{\alpha \beta }^{2},\end{eqnarray}$
where the regret for parameter α is $\begin{eqnarray}{{\rm{\Delta }}}_{\alpha }^{2}=({{ \mathcal F }}_{\alpha }-{F}_{\alpha })/{{ \mathcal F }}_{\alpha },\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{r}{C}_{\alpha \beta }^{2}=| {\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle {| }^{2}/({{ \mathcal F }}_{\alpha }{{ \mathcal F }}_{\beta }),\end{array}\end{eqnarray}$
with the notation $\langle X\rangle ={\rm{tr}}(\rho X)$ being used for the quantum expectation value of any operator X on the state ρ. The quantity ${{ \mathcal F }}_{\alpha }$ is the quantum Fisher information (QFI) and Fα is the classical Fisher information with respect to a special quantum measurement. The Hermitian operator Lα is the symmetric logarithmic derivative (SLD) operator with respect to parameter α and for a density operator ρ depending on the parameter α, it is defined as $\begin{eqnarray}{\partial }_{\alpha }\rho =\frac{1}{2}({L}_{\alpha }\rho +\rho {L}_{\alpha }).\end{eqnarray}$
No matter the density operator is pure or mixed, the QFI is given by a uniform expression [12, 13] $\begin{eqnarray}\begin{array}{r}{{ \mathcal F }}_{\alpha }=\,{\rm{Tr}}\,(\rho {L}_{\alpha }^{2}).\end{array}\end{eqnarray}$
Since the expectation value of the SLD operator on the state ρ is zero, the QFI is the variance of the SLD operator. It should be noted that practical schemes were given to measure the QFI [14, 15]. In addition, the trade-off relation given by the inequality was further investigated from experiments [16, 17].We observe that, except for the two Fisher information in inequality (1 ), there appears a key quantity ${\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle $, which can be written as 5 ), the above inequality can be rewritten as
$\begin{eqnarray}{\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle =\frac{1}{2}{\rm{Im}}\langle [{L}_{\alpha },{L}_{\beta }]\rangle =-\frac{{\rm{i}}}{2}\langle [{L}_{\alpha },{L}_{\beta }]\rangle .\end{eqnarray}$
Actually, considering two SLDs, Lα and Lβ, we have the Heisenberg uncertainty relation $\begin{eqnarray}\langle {({\rm{\Delta }}{L}_{\alpha })}^{2}\rangle \langle {({\rm{\Delta }}{L}_{\beta })}^{2}\rangle \geqslant \frac{| \langle [{L}_{\alpha },{L}_{\beta }]\rangle {| }^{2}}{4}.\end{eqnarray}$
In terms of the QFI given by equation ( $\begin{eqnarray}{{ \mathcal F }}_{\alpha }{{ \mathcal F }}_{\beta }\geqslant | {\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle {| }^{2}={({\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle )}^{2}.\end{eqnarray}$
We see that the multiplication of two Fisher information is bounded below by the square of the quantity ${\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle $.We note that the reduction of the regrets of Fisher information about different parameters is restricted by a nonzero value of ${\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle $. When ${\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle =0$, it is easy to see that inequality (1 ) becomes trivial, i.e., there are no restrictions on the two regrets. Similarly, from equation (8 ), there is a restriction on the two QFIs only when ${\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle $ is not zero. Therefore, we conclude that the quantity plays a key role in the field of quantum multi-parameter estimation and it is just the quantum curvature given below up to a multiplicative constant.
Quantum curvature for mixed states
For a state ∣ψ⟩ with two parameters α and β, the Berry curvature is defined as 4 ). Furthermore, the SLDs for parameters α, β are given by 9 ), one finds
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}_{\alpha \beta }(\psi ) & = & {\rm{i}}(\langle {{\rm{\partial }}}_{\alpha }\psi |{{\rm{\partial }}}_{\beta }\psi \rangle -\langle {{\rm{\partial }}}_{\beta }\psi |{{\rm{\partial }}}_{\alpha }\psi \rangle )\\ & = & -2{\rm{Im}}(\langle {{\rm{\partial }}}_{\alpha }\psi |{{\rm{\partial }}}_{\beta }\psi \rangle ).\end{array}\end{eqnarray}$
We now try to give a new form of the Berry curvature for pure states in terms of SLDs. Let us consider a quantum pure state written in the form of a density matrix ρ = ∣ψ⟩⟨ψ∣. It follows ρ2 = ρ that $\begin{eqnarray}{\partial }_{\alpha }\rho ={\partial }_{\alpha }{\rho }^{2}=\left({\partial }_{\alpha }\rho \right)\rho +\rho \left({\partial }_{\alpha }\rho \right),\end{eqnarray}$
implying that the SLD operator can be written as $\begin{eqnarray}{L}_{\alpha }=2{\partial }_{\alpha }\rho ,\end{eqnarray}$
where we have used equation ( $\begin{eqnarray}\begin{array}{r}{L}_{\alpha }=2(| {\partial }_{\alpha }\psi \rangle \langle \psi | +| \psi \rangle \langle {\partial }_{\alpha }\psi | ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{L}_{\beta }=2(| {\partial }_{\beta }\psi \rangle \langle \psi | +| \psi \rangle \langle {\partial }_{\beta }\psi | ).\end{array}\end{eqnarray}$
From these two SLDs, a straightforward calculation leads to $\begin{eqnarray}\begin{array}{rcl}\langle \psi | {L}_{\alpha }{L}_{\beta }| \psi \rangle & = & 4(\langle \psi | {\partial }_{\beta }\psi \rangle \langle \psi | {\partial }_{\alpha }\psi \rangle +\langle {\partial }_{\alpha }\psi | {\partial }_{\beta }\psi \rangle ),\\ \langle \psi | {L}_{\beta }{L}_{\alpha }| \psi \rangle & = & 4(\langle \psi | {\partial }_{\alpha }\psi \rangle \langle \psi | {\partial }_{\beta }\psi \rangle +\langle {\partial }_{\beta }\psi | {\partial }_{\alpha }\psi \rangle ).\end{array}\end{eqnarray}$
Combining the above equation and equation ( $\begin{eqnarray}{{\rm{\Omega }}}_{\alpha \beta }(\psi )=\frac{{\rm{i}}}{4}\langle \psi | [{L}_{\alpha },{L}_{\beta }]| \psi \rangle .\end{eqnarray}$
The Berry curvature is written as an expectation value of the commutator of two SLDs.It is natural to define the quantum curvature for a mixed state ρ as
$\begin{eqnarray}\begin{array}{r}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=\frac{{\rm{i}}}{4}{\rm{Tr}}(\rho [{L}_{\alpha },{L}_{\beta }])=-\frac{1}{2}{\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle ,\end{array}\end{eqnarray}$
which differs from the quantity ${\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle $ by a multiplicative factor −1/2. This quantum curvature directly reduces to the Berry curvature when the mixed state becomes pure. Here the two SLDs were defined for any state ρ with two parameters. Thus, it is general and applicable for any quantum system. Note that the quantum curvature is the expectation value of the Hermitian operator i[Lα, Lβ]/4. Similarly, the elements of the Fisher information matrix $\begin{eqnarray}{{ \mathcal F }}_{\alpha \beta }=\langle {[{L}_{\alpha },{L}_{\beta }]}_{+}/2\rangle ,\end{eqnarray}$
is the expectation of anticommutator ${[{L}_{\alpha },{L}_{\beta }]}_{+}/2$.Carollo et al called the quantum curvature as the mean Uhlmann curvature defined as the Uhlmann geometric phase [18, 19] per unit area of a density matrix evolving along an infinitesimal loop in the parameter space and the curvature was used to investigate non-equilibrium steady-state quantum phase transitions [20–22]. Candeloro et al adopted this quantity as an asymptotic incompatibility measure in multi-parameter quantum estimation [23]. However, here we see that the quantum curvature is a natural generalization of the Berry curvature from pure to mixed states. Thus, we call it mixed-state Berry curvature.
Now we write the trade-off relation (1 ) in the following form as 16 ), we have 8 ) can be written as 20 ) or the two QFIs in equation (21 ). Thus, the quantum curvature plays a key role in the field of multi-parameter estimations.
$\begin{eqnarray}\begin{array}{l}{{ \mathcal F }}_{\beta }{\delta }_{\alpha }^{2}+{{ \mathcal F }}_{\alpha }{\delta }_{\beta }^{2}+2{\delta }_{\alpha }{\delta }_{\beta }\sqrt{{{ \mathcal F }}_{\alpha }{{ \mathcal F }}_{\beta }-| {\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle {| }^{2}}\\ \quad \geqslant | {\rm{Im}}\langle {L}_{\alpha }{L}_{\beta }\rangle {| }^{2},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\delta }_{\alpha }^{2}={{ \mathcal F }}_{\alpha }-{F}_{\alpha }.\end{eqnarray}$
Then, from the definition of the quantum curvature ( $\begin{eqnarray}\begin{array}{r}{{ \mathcal F }}_{\beta }{\delta }_{\alpha }^{2}+{{ \mathcal F }}_{\alpha }{\delta }_{\beta }^{2}+2{\delta }_{\alpha }{\delta }_{\beta }\sqrt{{{ \mathcal F }}_{\alpha }{{ \mathcal F }}_{\beta }-2{{\rm{\Omega }}}_{\alpha \beta }^{2}}\geqslant 2{{\rm{\Omega }}}_{\alpha \beta }^{2}.\end{array}\end{eqnarray}$
Every entry in the above equation is clear. It includes two QFIs, two classical Fisher information and one quantum curvature. Similarly, the inequality ( $\begin{eqnarray}{{ \mathcal F }}_{\alpha }{{ \mathcal F }}_{\beta }\geqslant 4{{\rm{\Omega }}}_{\alpha \beta }^{2}.\end{eqnarray}$
We see that the multiplication of two Fisher information is bounded below by four times the square of the Berry curvature. For the case of zero quantum curvature, there are no trade-off relations between the two regrets in equation (Quantum geometric tensor and spectral decomposition
The quantum curvature given above is closely related to the quantum geometric tensor (QGT) [24–27]. Let us first consider the QGT of a pure state ∣ψ⟩ with parameters α and β. It is defined as 9 ). For mixed states ρ, the QGT is given by as 16 ), we obtain
$\begin{eqnarray}\begin{array}{rcl}{Q}_{\alpha \beta }(\psi ) & = & \langle {\partial }_{\alpha }\psi | {\partial }_{\beta }\psi \rangle -\langle {\partial }_{\alpha }\psi | \psi \rangle \langle \psi | {\partial }_{\beta }\psi \rangle \\ & = & \langle {\partial }_{\alpha }\psi | {\partial }_{\beta }\psi \rangle +\langle \psi | {\partial }_{\alpha }\psi \rangle \langle \psi | {\partial }_{\beta }\psi \rangle .\end{array}\end{eqnarray}$
Noting that ⟨∂αψ∣ψ⟩ is purely imaginary, we have $\begin{eqnarray}{{\rm{\Omega }}}_{\alpha \beta }(\psi )=-2{\rm{Im}}{Q}_{\alpha \beta }(\psi ),\end{eqnarray}$
where we have used equation ( $\begin{eqnarray}{Q}_{\alpha \beta }(\rho )=\frac{1}{4}{\rm{Tr}}\left(\rho {L}_{\alpha }{L}_{\beta }\right).\end{eqnarray}$
Therefore, from the definition of quantum curvature for mixed states ( $\begin{eqnarray}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=-\frac{1}{2}{\rm{Im}}\,{\rm{Tr}}\left(\rho {L}_{\alpha }{L}_{\beta }\right)=-2{\rm{Im}}{Q}_{\alpha \beta }(\rho ).\end{eqnarray}$
The quantum curvature is the imaginary part of the QGT multiplied by minus two.For a density matrix with full-rank, the spectral decomposition of density matrix ρ is written as 4 ) with the above spectral decomposition, one can obtain the element of the SLD operator as 27 ) into the definition of QGT for mixed states, we can obtain the explicit expression of the QGT for full-rank density matrices as [28]28 ), the QGT can be rewritten as 25 ) and the form of QGT equation (30 ), we finally obtain
$\begin{eqnarray}\rho =\displaystyle \sum _{i=1}^{N}{p}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| ,\end{eqnarray}$
where N is the dimension of the Hilbert space. Solving the SLD equation ( $\begin{eqnarray}\begin{array}{r}\langle {\psi }_{i}| {L}_{\alpha }| {\psi }_{j}\rangle =\left\{\begin{array}{ll}\frac{{\partial }_{\alpha }{p}_{i}}{{p}_{i}},\quad & i=j,\\ \frac{2({p}_{i}-{p}_{j})}{{p}_{i}+{p}_{j}}\langle {\partial }_{\alpha }{\psi }_{i}| {\psi }_{j}\rangle ,\quad & i\ne j,\end{array}\right.\end{array}\end{eqnarray}$
where we have used the relation ⟨∂αψi∣ψj⟩ + ⟨ψi∣∂αψj⟩ = 0. Substituting equation ( $\begin{eqnarray}\begin{array}{rcl}{Q}_{\alpha \beta }(\rho ) & = & \frac{1}{4}\displaystyle \sum _{i,j=1}^{N}{p}_{i}\langle {\psi }_{i}| {L}_{\alpha }| {\psi }_{j}\rangle \langle {\psi }_{j}| {L}_{\beta }| {\psi }_{i}\rangle \\ & = & \frac{1}{4}\displaystyle \sum _{i=1}^{N}\frac{({\partial }_{\alpha }{p}_{i})({\partial }_{\beta }{p}_{i})}{{p}_{i}}+\displaystyle \sum _{i\ne j}^{N}\frac{{p}_{i}{({p}_{i}-{p}_{j})}^{2}}{{({p}_{i}+{p}_{j})}^{2}}{{ \mathcal A }}_{ij}^{\alpha \beta },\end{array}\end{eqnarray}$
where ${{ \mathcal A }}_{ij}^{\alpha \beta }$ is the double Wilczek–Zee connection defined in [28] as $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{ij}^{\alpha \beta } & = & \langle {\psi }_{i}| {\partial }_{\alpha }{\psi }_{j}\rangle {\langle {\psi }_{i}| {\partial }_{\beta }{\psi }_{j}\rangle }^{* }\\ & = & \langle {\partial }_{\alpha }{\psi }_{i}| {\psi }_{j}\rangle \langle {\psi }_{j}| {\partial }_{\beta }{\psi }_{i}\rangle .\end{array}\end{eqnarray}$
It is easy to verify that ${{ \mathcal A }}_{ji}^{\alpha \beta }={({{ \mathcal A }}_{ij}^{\alpha \beta })}^{* }$, which implies that the real part of ${{ \mathcal A }}_{ij}^{\alpha \beta }$ is symmetric while the imaginary part is antisymmetric with respect to the indices i and j. Using the above property and symmetrizing the summation over indices i and j in equation ( $\begin{eqnarray}\begin{array}{rcl}{Q}_{\alpha \beta }(\rho ) & = & \frac{1}{4}\displaystyle \sum _{i=1}^{N}\frac{\left({\partial }_{\alpha }{p}_{i}\right)\left({\partial }_{\beta }{p}_{i}\right)}{{p}_{i}}\\ & & +\frac{1}{2}\displaystyle \sum _{i\ne j}^{N}\left[\frac{{({p}_{i}-{p}_{j})}^{2}}{{p}_{i}+{p}_{j}}{{ \mathcal R }}_{ij}^{\alpha \beta }+{\rm{i}}\frac{{({p}_{i}-{p}_{j})}^{3}}{{({p}_{i}+{p}_{j})}^{2}}{{ \mathcal I }}_{ij}^{\alpha \beta }\right],\end{array}\end{eqnarray}$
where ${{ \mathcal R }}_{ij}^{\alpha \beta }$ and ${{ \mathcal I }}_{ij}^{\alpha \beta }$ are respectively the real and imaginary parts of ${{ \mathcal A }}_{ij}^{\alpha \beta }$. Then from relation equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}_{\alpha \beta }(\rho ) & = & -\displaystyle \sum _{i\ne j}\frac{{\left({p}_{i}-{p}_{j}\right)}^{3}}{{\left({p}_{i}+{p}_{j}\right)}^{2}}{{ \mathcal I }}_{ij}^{\alpha \beta }\\ & = & -\displaystyle \sum _{i\ne j}\frac{{\left({p}_{i}-{p}_{j}\right)}^{3}}{{\left({p}_{i}+{p}_{j}\right)}^{2}}\,{\rm{Im}}\,\left(\langle {\partial }_{\alpha }{\psi }_{i}| {\psi }_{j}\rangle \langle {\psi }_{j}| {\partial }_{\beta }{\psi }_{i}\rangle \right).\end{array}\end{eqnarray}$
Moreover, due to the fact ${{ \mathcal I }}_{ij}^{\alpha \beta }=-{{ \mathcal I }}_{ji}^{\alpha \beta }$, the above equation can be written as $\begin{eqnarray}\begin{array}{r}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=-2\displaystyle \sum _{i\lt j}\frac{{\left({p}_{i}-{p}_{j}\right)}^{3}}{{\left({p}_{i}+{p}_{j}\right)}^{2}}{{ \mathcal I }}_{ij}^{\alpha \beta }.\end{array}\end{eqnarray}$
We see that the quantum curvature is only related to the imaginary part of the double Wilczek–Zee connection.We now study the quantum curvature of a non-full-rank density matrix, i.e., the rank M of the density operator is less than the dimension N of the Hilbert space. The density matrix is written as 22 ) and (29 ), the QGT can be written in the following form
$\begin{eqnarray}\rho =\displaystyle \sum _{i=1}^{M}{p}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| .\end{eqnarray}$
With the above spectral decomposition, the QGT is given as $\begin{eqnarray}\begin{array}{rcl}{Q}_{\alpha \beta }(\rho ) & = & \frac{1}{4}\displaystyle \sum _{i=1}^{M}\displaystyle \sum _{j=1}^{N}{p}_{i}\langle {\psi }_{i}| {L}_{\alpha }| {\psi }_{j}\rangle \langle {\psi }_{j}| {L}_{\beta }| {\psi }_{i}\rangle \\ & = & \frac{1}{4}\displaystyle \sum _{i=1}^{M}\frac{\left({\partial }_{\alpha }{p}_{i}\right)\left({\partial }_{\beta }{p}_{i}\right)}{{p}_{i}}+\displaystyle \sum _{i=1}^{M}{p}_{i}(\langle {\partial }_{\alpha }{\psi }_{i}| {\partial }_{\beta }{\psi }_{i}\rangle -{{ \mathcal R }}_{ii}^{\alpha \beta })\\ & & -2\displaystyle \sum _{i\ne j}^{M}\frac{{p}_{i}{p}_{j}}{{p}_{i}+{p}_{j}}\left[{{ \mathcal R }}_{ij}^{\alpha \beta }+{\rm{i}}\frac{{p}_{i}-{p}_{j}}{{p}_{i}+{p}_{j}}{{ \mathcal I }}_{ij}^{\alpha \beta }\right].\end{array}\end{eqnarray}$
By using equations ( $\begin{eqnarray}\begin{array}{rcl}{Q}_{\alpha \beta }(\rho ) & = & \frac{1}{4}\displaystyle \sum _{i=1}^{M}\frac{\left({\partial }_{\alpha }{p}_{i}\right)\left({\partial }_{\beta }{p}_{i}\right)}{{p}_{i}}+\displaystyle \sum _{i=1}^{M}{p}_{i}{Q}_{\alpha \beta }({\psi }_{i})\\ & & -2\displaystyle \sum _{i\ne j}^{M}\frac{{p}_{i}{p}_{j}}{{p}_{i}+{p}_{j}}\left[{{ \mathcal R }}_{ij}^{\alpha \beta }+{\rm{i}}\frac{{p}_{i}-{p}_{j}}{{p}_{i}+{p}_{j}}{{ \mathcal I }}_{ij}^{\alpha \beta }\right].\end{array}\end{eqnarray}$
Therefore, for the density operators whose rank is less than the dimension of the Hilbert space, we have
$\begin{eqnarray}\begin{array}{r}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=\displaystyle \sum _{i=1}^{M}{p}_{i}{{\rm{\Omega }}}_{\alpha \beta }({\psi }_{i})+4\displaystyle \sum _{i\ne j}^{M}\frac{{p}_{i}{p}_{j}\left({p}_{i}-{p}_{j}\right)}{{\left({p}_{i}+{p}_{j}\right)}^{2}}{{ \mathcal I }}_{ij}^{\alpha \beta }.\end{array}\end{eqnarray}$
The first term in the above equation is just the average of Berry curvature of all pure states ∣ψi⟩ in the spectral decomposition of the density matrix. One can also simply define the quantum curvature for mixed states as $\begin{eqnarray}\begin{array}{r}{{\rm{\Omega }}}_{\alpha \beta }^{{\prime} }(\rho )=\displaystyle \sum _{i=1}^{M}{p}_{i}{{\rm{\Omega }}}_{\alpha \beta }({\psi }_{i}).\end{array}\end{eqnarray}$
It is clear that both these quantum curvatures reduces to the Berry curvature of the pure state ∣ψ1⟩ when the rank of the density matrix becomes one.Let us consider the continuity of quantum curvature at the boundary where the rank of the density matrix changes, e.g., between the full-rank and non-full-rank density matrices. It was shown that the real part of the QGT, which is equivalent to the QFI, might be discontinuous at such a boundary [29–31]. This discontinuity means that the limit of QFI for full-rank density matrices when some eigenvalues approach zero is not equal to the QFI of the non-full-rank density matrix obtained by setting these eigenvalues to zero. The quantum curvature may also exhibit similar discontinuity. Assume that both eigenstates and eigenvalues of the density matrix are smooth functions of the parameters such that ${{ \mathcal A }}_{ij}^{\alpha \beta }$ are continuous. It can be seen from equation (31 ) that the quantum curvature for full-rank density matrices may be discontinuous only when two eigenvalues of the density matrix simultaneously approach zero. When there is only one eigenvalue approaching zero, the quantum curvature for full-rank density matrices converges to that for non-full-rank density matrices, but the QFI may be discontinuous in this case due to the first summation term in equation (31 ).
Example
As an example, we consider the pure state of a qubit with the following Bloch presentation 11 ) leads to
$\begin{eqnarray}\rho =\frac{1}{2}\left(I+{\boldsymbol{n}}\cdot {\boldsymbol{\sigma }}\right),\end{eqnarray}$
with n being a unit vector and σ be the Pauli matrix vector. Substituting the above equation into equation ( $\begin{eqnarray}{L}_{\alpha }=\left({\partial }_{\alpha }{\boldsymbol{n}}\right)\cdot {\boldsymbol{\sigma }}.\end{eqnarray}$
For two SLDs Lα and Lβ, the expectation of the commutator [Lα, Lβ] on state ρ can be obtained as $\begin{eqnarray}\langle [{L}_{\alpha },{L}_{\beta }]\rangle ={\rm{Tr}}\left[\rho [{L}_{\alpha },{L}_{\beta }]\right]=2\,{\rm{i}}\,{\boldsymbol{n}}\cdot {\partial }_{\alpha }{\boldsymbol{n}}\times {\partial }_{\beta }{\boldsymbol{n}},\end{eqnarray}$
where we have used the identity $\begin{eqnarray}{\rm{Tr}}(({\boldsymbol{C}}\cdot {\boldsymbol{\sigma }})[({\boldsymbol{A}}\cdot {\boldsymbol{\sigma }}),({\boldsymbol{B}}\cdot {\boldsymbol{\sigma }})])=4\,{\rm{i}}\,{\boldsymbol{C}}\cdot ({\boldsymbol{A}}\times {\boldsymbol{B}}).\end{eqnarray}$
Then, the Berry curvature is given in the following form
$\begin{eqnarray}{{\rm{\Omega }}}_{\alpha \beta }=\frac{{\rm{i}}}{4}\langle [{L}_{\alpha },{L}_{\beta }]\rangle =-\frac{1}{2}{\boldsymbol{n}}\cdot {\partial }_{\alpha }{\boldsymbol{n}}\times {\partial }_{\beta }{\boldsymbol{n}}.\end{eqnarray}$
The geometric meaning is clear. The quantum curvature is the volume of the parallelepiped spanned by the three vectors n, ∂αn, and ∂βn up to a constant.We now investigate a mixed state of a qubit with the Bloch representation 32 ), the quantum curvature is given by 9 ) and equation (16 ), we get
$\begin{eqnarray}\rho =\frac{1}{2}\left(I+{\boldsymbol{r}}\cdot {\boldsymbol{\sigma }}\right),\end{eqnarray}$
where r is a 3-dimensional real vector whose length r is less than 1. From equation ( $\begin{eqnarray}\begin{array}{r}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=-2{\left({p}_{1}-{p}_{2}\right)}^{3}\,{\rm{Im}}\,\left(\langle {\partial }_{\alpha }{\psi }_{1}| {\psi }_{2}\rangle {\langle {\partial }_{\beta }{\psi }_{1}| {\psi }_{2}\rangle }^{* }\right),\end{array}\end{eqnarray}$
where we have used the fact p1 + p2 = 1. Then one step further leads to $\begin{eqnarray}\begin{array}{l}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=-2{\left({p}_{1}-{p}_{2}\right)}^{3}\,{\rm{Im}}\,\left(\langle {\partial }_{\alpha }{\psi }_{1}| {\psi }_{2}\rangle \langle {\psi }_{2}| {\partial }_{\beta }{\psi }_{1}\rangle \right)\\ \,=-2{\left({p}_{1}-{p}_{2}\right)}^{3}\times \,{\rm{Im}}\,\left(\langle {\partial }_{\alpha }{\psi }_{1}| {\partial }_{\beta }{\psi }_{1}\rangle -\langle {\partial }_{\alpha }{\psi }_{1}| {\psi }_{1}\rangle \langle {\psi }_{1}| {\partial }_{\beta }{\psi }_{1}\rangle \right),\end{array}\end{eqnarray}$
where the completeness relation was used. As ⟨∂αψ1∣ψ1⟩ and ⟨∂βψ1∣ψ1⟩ are all purely imaginary, we finally have a simple form $\begin{eqnarray}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=-2{\left({p}_{1}-{p}_{2}\right)}^{3}\,{\rm{Im}}\,\left(\langle {\partial }_{\alpha }{\psi }_{1}| {\partial }_{\beta }{\psi }_{1}\rangle \right).\end{eqnarray}$
With the Bloch representation of density operators of a qubit, we have p1 = (1 − r)/2 and p2 = (1 + r)/2 and thus $\begin{eqnarray}{\left({p}_{1}-{p}_{2}\right)}^{3}=-{r}^{3}.\end{eqnarray}$
Then, from definition ( $\begin{eqnarray}{{\rm{\Omega }}}_{\alpha \beta }(\rho )=-{r}^{3}{{\rm{\Omega }}}_{\alpha \beta }({\psi }_{1}).\end{eqnarray}$
The quantum curvature is proportional to the Berry curvature for pure state ∣ψ1⟩. When r = 0, the quantum curvature vanishes as the corresponding state is the completely mixed state.Conclusions
In conclusion, we have proposed a mixed-state quantum curvature based on the SLDs in the field of quantum precision measurement. For full-rank and non-full-rank density matrices, we provided explicit expressions of quantum curvature after the spectral decomposition. Our work extended the quantum curvature to the case of mixed states and the quantum curvature was found to be a key role in the field of multi-parameter estimations. And this concept is expected to play an important role in the study of fundamental problems of quantum physics.