Error trade-off relations for two-parameter unitary model with commuting generators

Let us consider arbitrary finite dimensional system. We consider the two-parameter unitary transformation with the generators $X$ and $Y$ , i.e.,

$$U(\theta^{1},\,\theta^{2})=\mathrm{e}^{-\mathrm{i}X\theta^{1}-\mathrm{i}Y\theta^{2}}.$$

(1)

We denote the two-parameter family of states generated from the state $\rho_{0}$ as $\rho_{\theta}$.

$$\rho_{\theta}=U(\theta^{1},\,\theta^{2})\,\rho_{0}U^{\dagger}(\theta^{1},\,\theta^{2}).$$

(2)

The state $\rho_{0}$ is called as a reference state. In this paper, we mainly consider the case of the commuting generators, $[X,\,Y]=0$ unless stated explicitly.

In the remaining of the paper, we consider unitary models only. We shall drop the parameter $\theta$ to denote the quantum Fisher information matrices, since they are independent of the parameter. To derive a trade-off relation between the diagonal components of the mean square error (MSE) matrix $V=[V_{ij}]$, suppose we have a quantum CR inequality

$$V\geq J_{\mathrm{Q}}^{\>-1},$$

with $J_{\mathrm{Q}}$ a quantum Fisher information matrix. In Ref. sf , we derived a trade-off relation based on the inequality below,

$$\left[V_{11}-J_{\mathrm{Q}}^{\>11}\right]\left[V_{22}-J_{\mathrm{Q}}^{\>22}\right]>\left|\mathrm{Im}\,J_{\mathrm{Q}}^{\>12}\right|^{\,2}.$$

(3)

where $J_{\mathrm{Q}}^{-1}=[J_{\mathrm{Q}}^{\>ij}]$. $\mathrm{Im}$ denotes the imaginary part of a complex number. From this expression, we see that there exists a non-trivial error trade-off relation when $\mathrm{Im}\,J_{\mathrm{Q}}^{\>12}\neq 0$. Note, however, that this inequality alone does not give a conclusive argument whether an error trade-off relation exists or not. This is because the quantum CR inequality is not tight unless certain special conditions are satisfied. The central idea of this paper is to consider two different CR inequalities set by the SLD and RLD Fisher information matrices. When combining two error trade-off relations, we can determine the shape of an error trade-off relation more accurately.

From the discussion above, we do not have the trade-off relation given by Eq. (3) when the SLD Fisher information matrix $J_{\mathrm{S}}$ is used. This is because it is a real symmetric matrix. The other candidate for giving rise to an error trade-off relation is the RLD Fisher information matrix $J_{\mathrm{R}}$. In this case, the necessary condition to have an error trade-off relation is

$$\left|\mathrm{Im}\,J_{\mathrm{R}}^{\>12}\right|^{\,2}\neq 0.$$

(4)

as in Eq. (3). By defining $\delta:=J_{\mathrm{R},\,12}-J_{\mathrm{R},\,21}$, we have an equivalent condition,

$$\mathrm{Condition\,1}:\qquad\delta=J_{\mathrm{R},\,12}-J_{\mathrm{R},\,21}\neq 0,$$

(5)

where $J_{\mathrm{R}}=[J_{\mathrm{R},\,ij}]$.

$(i,\,j)$ component of the RLD Fisher information matrix, $J_{\mathrm{R},\,ij}$ is defined as follows.

$$J_{\mathrm{R},\,ij}=\mathrm{tr}\left(\rho_{0}L_{\mathrm{R},\,j}\,L^{\dagger}_{\mathrm{R},\,i}\right),$$

(6)

where $\partial_{i}\rho_{\theta}\,|_{\theta=0}=\rho_{0}L_{\mathrm{R},\,i}$. By using $\partial_{i}\rho_{\theta}\,|_{\theta=0}=L^{\dagger}_{\mathrm{R},\,i}\,\rho_{0}$, Eqs (1), and  (2), we obtain

$$J_{\mathrm{R},\,ij}=-\mathrm{tr}\left([X_{j}\,\rho_{0}][X_{i},\,\rho_{0}]\,\rho_{0}^{-1}\right),$$

(7)

where $X_{1}=X$ and $X_{2}=Y$. With this, Condition 1 is

$$\delta=\mathrm{tr}\left(\big{[}[X,\,\rho_{0}]\,,\,[Y,\,\rho_{0}]\big{]}\,\rho_{0}^{-1}\right),$$

(8)

and thus, it is relatively easy to check this condition analytically. We stress that having commuting generators, $[X,Y]=0$ does not immediately imply $\delta=0$.

Now suppose Condition 1 is satisfied.

$$\left[V_{11}-J_{\mathrm{R}}^{\>11}\right]\left[V_{22}-J_{\mathrm{R}}^{\>22}\right]>\left|\mathrm{Im}\,J_{\mathrm{R}}^{\>12}\right|^{\,2}.$$

(9)

To give a conclusive argument for the existence of an error trade-off relation, we also consider consider the SLD CR inequality. Since the SLD Fisher information matrix is real, the diagonal components of the MSE matrix obey

$$V_{11}-J_{\mathrm{S}}^{\>11}\geq 0,\ V_{22}-J_{\mathrm{S}}^{\>22}\geq 0.$$

(10)

Note that from the general relationship between the SLD and RLD Fisher information matrices, we have petz

$$J_{\mathrm{S}}^{\>-1}\geq\mathrm{Re}\,(J_{\mathrm{R}}^{\>-1}),$$

where $\mathrm{Re}$ denotes the real part of a matrix. Since there exists a locally unbiased estimator such that its MSE for $\theta^{1}$ is arbitrary close to $J_{\mathrm{S}}^{\>11}$, (The same statement holds for $\theta^{2}$ as well.) intersections of two inequalities (9) and (10) imply the existence of the error trade-off relation. See Fig. 5 for the occurrence of intersections of the two bounds and the error trade-off relation as an example. Working out elementary algebra, we find that the following condition needs to be satisfied in order to have a non-trivial error trade-off relation sf .

$$\mathrm{Condition\,2}:\quad\Delta:=\left|\mathrm{Im}\,J_{\mathrm{R}}^{\>12}\right|^{2}-\left[J_{\mathrm{R}}^{\>11}-J_{\mathrm{S}}^{\>11}\right]\left[J_{\mathrm{R}}^{\>22}-J_{\mathrm{S}}^{\>22}\right]>0.$$

(11)

And hence, we have a trade-off relation between $V_{11}$ and $V_{22}$ if these two conditions (5) and (11) are satisfied.

Next, let us make a remark about D-invariant models. It is known that, the RLD CR inequality is saturated when the model is D-invariant kahn ; hayashi ; kahn2 ; yamagata ; suzuki ; yang . This is true at least in the asymptotic setting. There is no intersection of the RLD and SLD CR bounds in the D-invariant models, because the RLD CR bound is dominant over the SLD CR bound. If the model is D-invariant and if the imaginary part of the off-diagonal elements of RLD Fisher information matrix are not zero, there is a trade-off relation that results from Condition 1 only. In the following, we mainly investigate the non-asymptotic setting unless stated explicitly. This is in contrast to the previous study kull , where the authors focused on the D-invariant model.

We consider the case of a single qubit in a mixed state. We first consider the general two-parameter unitary model to get insight into the problem. By using the Bloch vector, we can express the reference state $\rho_{0}$ as

$$\rho_{0}=\frac{1}{2}(\mathrm{I}+\vec{s}_{0}\cdot\vec{\sigma}).$$

(18)

where $|\vec{s}_{0}|<1$. $\rho_{\theta}$ is given by Eq. (2). The generators $X,\,Y$ can also be expanded with using Pauli matrices.

	$\displaystyle X$	$\displaystyle=x_{0}\,\mathrm{I}+\vec{x}\cdot\vec{\sigma},$		(19)
	$\displaystyle Y$	$\displaystyle=y_{0}\,\mathrm{I}+\vec{y}\cdot\vec{\sigma}.$		(20)

The inverse of SLD and RLD Fisher information matrices, $J_{\mathrm{S}}^{\>-1}$ and $J_{\mathrm{R}}^{\>-1}$ are explicitly written as

	$\displaystyle J_{\mathrm{S}}^{\>-1}$	$\displaystyle=\frac{4}{\mathrm{det}\,J_{\mathrm{S}}}\begin{pmatrix}(\vec{y}\times\vec{s}_{0})^{2}&-(\vec{x}\times\vec{s}_{0})\cdot(\vec{y}\times\vec{s}_{0})\\ -(\vec{x}\times\vec{s}_{0})\cdot(\vec{y}\times\vec{s}_{0})&(\vec{x}\times\vec{s}_{0})^{2}\\ \end{pmatrix},$		(21)
	$\displaystyle J_{\mathrm{R}}^{\>-1}$	$\displaystyle=J_{\mathrm{S}}^{-1}$
		$\displaystyle+\frac{4}{\mathrm{det}\,J_{\mathrm{S}}}\begin{pmatrix}0&-\mathrm{i}\vec{s}_{0}^{\>2}[\vec{s}_{0}\cdot(\vec{x}\times\vec{y})]\\ \mathrm{i}\vec{s}_{0}^{\>2}[\vec{s}_{0}\cdot(\vec{x}\times\vec{y})]&0\end{pmatrix},$		(22)

where $\mathrm{det}\,J_{\mathrm{S}}$ is the determinant of $J_{\mathrm{S}}$, and it is

$$\mathrm{det}\,J_{\mathrm{S}}={16\,\vec{s}_{0}^{\>2}[\vec{s}_{0}\cdot(\vec{x}\times\vec{y})]^{2}}.$$

(23)

As shown in Eqs. (21) and (22), $J_{\mathrm{S}}^{\>-1}=\mathrm{Re}\,J_{\mathrm{R}}^{\>-1}$ holds. It follows that our qubit model is D-invariant. (See Lemma III-3 in Ref.suzuki .) Therefore, the RLD CR bound is asymptotically achievable and gives a trade-off relation. In this case, as explained earlier, the SLD and RLD CR bounds do not have intersections, but the trade-off relation exists in the asymptotic setting.

As for the Nagaoka bound, or the Gill-Massar (GM) bound for a two-parameter qubit model, which is known to be achievable nagaoka2 ; RDGill in the non-asymptotic setting, an inequality regarding the diagonal components of the MSE matrix can be derived. The inequality of the Nagaoka band is written as

$$\left[V_{11}-J_{\mathrm{S}}^{\>11}\right]\left[V_{22}-J_{\mathrm{S}}^{\>22}\right]>\frac{1}{\mathrm{det}\,J_{\mathrm{S}}}.$$

(24)

From Eqs. (10) and (22), we obtain the inequality of the RLD CR bound.

$$\left[V_{11}-J_{\mathrm{S}}^{\>11}\right]\left[V_{22}-J_{\mathrm{S}}^{\>22}\right]>\frac{\vec{s}_{0}^{\>2}}{\mathrm{det}\,J_{\mathrm{S}}}.$$

(25)

We used $J_{\mathrm{R}}^{\>11}=J_{\mathrm{S}}^{\>11}$ and $J_{\mathrm{R}}^{\>22}=J_{\mathrm{S}}^{\>22}$. Since $\vec{s}_{0}^{\>2}<1$, the Nagaoka bound is tighter than the RLD CR bound. This is because the Nagaoka bound is achieved by a separable measurement.

Next, we derive a relationship between $X$ and $Y$, or $\vec{x}$ and $\vec{y}$ when $X$ and $Y$ commute. From Eqs. (19) and (20), the commuting relation of $X$ and $Y$ is given as

$$[X,\,Y]=[\vec{x}\cdot\vec{\sigma},\,\vec{y}\cdot\vec{\sigma}]=2\mathrm{i}(\vec{x}\times\vec{y})\cdot\vec{\sigma}.$$

It immediately follows that the necessary and sufficient condition for $X$ and $Y$ to commute is $\vec{x}\times\vec{y}=\vec{0}$. There is no trade-off relation because the unitary transformation is no longer two-parameter model, because $\vec{x}$ and $\vec{y}$ are parallel when $X$ and $Y$ commute.

As one of the simplest examples, we pick an example with pure imaginary off-diagonal components as a reference state $\rho_{0}$ with five reference state parameters $v_{1}$, $v_{2}$, $v_{3}$, $u_{1}$, $u_{2}$, and $u_{3}$. ($v_{1}+v_{2}+v_{3}=1$)

$$\rho_{0}=\frac{1}{3}\begin{pmatrix}v_{1}&-\mathrm{i}\sqrt{u_{1}}&\mathrm{i}\sqrt{u_{2}}\\ \mathrm{i}\sqrt{u_{1}}&v_{2}&-\mathrm{i}\sqrt{u_{3}}\\ -\mathrm{i}\sqrt{u_{2}}&\mathrm{i}\sqrt{u_{3}}&v_{3}\\ \end{pmatrix}.$$

(31)

We choose the reference state $\rho_{0}$ as above, because imaginary parts of the off-diagonal components of the reference state $\rho_{0}$ are important to satisfy Condition 1 as seen in Eq. (29).

We calculate $\Delta$ in Condition 2 with using the reference state $\rho_{0}$ defined by Eq. (31) of which reference state parameters are generated by random numbers. We pick those which satisfy $\mathrm{tr}\,\rho_{0}=1$ and $\rho_{0}>0$ and calculate the RLD and SLD Fisher information matrices $J_{\mathrm{S}}$ and $J_{\mathrm{R}}$. The RLD Fisher information matrix is obtained by using Eq. (7). The SLD Fisher information calculation is done in the standard method. (See for example, Refs. paris ; Liu .) The number of samples generated is on the order of $10^{6}$. Figure 1 shows $\Delta$ as a function of $u_{max}$, the maximum of $u_{1}$, $u_{2}$, and $u_{3}$. There exists a region $\Delta>0$. The ratio of obtaining $\Delta>0$ out of all of the samples generated is 3.0%. Figures 2 and 3 show $\Delta$ as a function of $\lambda_{min}$ and $\lambda_{max}$, respectively. $\lambda_{min}$ and $\lambda_{max}$ are the minimum and maximum of eigenvalues of $\rho_{0}$, respectively. For $\Delta$ to be positive, $\lambda_{min}$ and $\lambda_{max}$ must be in a certain range. $\lambda_{min}$ is more than about 0.13 and $\lambda_{max}$ is less than about 0.58.

Next, we set the reference state parameters in Eq. (31) as $v_{1}=v_{2}=v_{3}=1$ and $u_{1}=u_{2}=u_{3}=u$ in order to investigate the model more in detail analytically. We pick the reference state parameters as above, because the result of Section V.1 indicates that the eigenvalues of the reference state Eq. (31) be roughly in the range $1/3\pm 0.2$ to exhibit the non-trivial trade-off relation. The reference state $\rho_{0}$ is, then explicitly written as

$$\rho_{0}=\frac{1}{3}I+\frac{1}{3}\sqrt{u}\begin{pmatrix}0&-\mathrm{i}&\mathrm{i}\\ \mathrm{i}&0&-\mathrm{i}\\ -\mathrm{i}&\mathrm{i}&0\\ \end{pmatrix},$$

(32)

where $I$ denotes 3$\times$3 identity matrix. The reference state $\rho_{0}$ is a sum of the completely mixed state of the qutrit system and a perturbation with one parameter $u$. The parameter $u$ must be in the range, $0<u<1/3$ for the reference state $\rho_{0}$ to be positive. We exclude $u=0$, because $\rho_{0}=\mathrm{I}/3$ at $u=0$.

In the following, we show that the reference state $\rho_{0}$ Eq. (31) always gives a non-trivial trade-off relation with a certain choice of the reference state parameter $u$ and that the possibility of seeing the non-trivial trade-off relation is not small.

Unlike a qubit reference state or a pure state reference state, there exists a non-trivial trade-off relation for some qutrit reference states even when the generators commute. We show analytically that a non-trivial trade-off relation always exists in a certain range of the reference state parameter $u$ when the reference state $\rho_{0}$ is defined by Eq. (32) that is a sum of the completely mixed state and a perturbation.

Furthermore, the strengths of trade-off relation $\Delta_{1}$ and $\Delta_{2}$ increase as $u$ approaches 0. This looks counterintuitive, because we can regard $u$ as a small perturbation from 3x3 identity matrix when $u\ll 1$ by the definition of $\rho_{0}$, Eq. (32). This reflects the fact that $\partial_{i}\rho_{\theta}$ is not necessarily small when the perturbation itself is small. Since the $(i,j)$ component of the RLD Fisher information matrix is $J_{\mathrm{R},\,ij}=\mathrm{tr}[\partial_{j}\rho_{\theta}L_{R,\,i}^{\dagger}]$, the component $g_{\mathrm{R},\,ij}$ may not be small if $\partial_{i}\rho_{\theta}$ is not small.

In a more general case when $\rho_{0}$ is expressed by Eq. (31), we conducted numerical analysis. In this case also, there exists a non-trivial trade-off relation. Furthermore, in the case of four dimensional system with pure imaginary off-diagonal components, we also see a non-trivial trade-off relation by the same numerical analysis as well. With these, we conclude that the error trade-off relation is a generic phenomenon in the sense that it occurs with a finite volume in the spate space.