Bayesian uncertainty relation for a joint measurement of canonical variables

TL;DR

This paper introduces a Bayesian uncertainty relation for joint measurements of canonical variables in quantum optics, allowing for practical assessment of measurement devices without the unbiasedness assumption.

Contribution

It develops a Bayesian framework for joint-measurement uncertainty relations that applies to Gaussian states and does not require unbiasedness, enhancing experimental quantification.

Findings

Reproduces the optimal bound for phase-space displacement estimation with pure Gaussian states

Provides a practical method to evaluate joint-measurement devices using coherent states

Offers a Bayesian approach that is experimentally feasible and theoretically robust

Abstract

We present a joint-measurement uncertainty relation for a pair of mean square deviations of canonical variables averaged over Gaussian distributed quantum optical states. Our Bayesian formulation is free from the unbiasedness assumption, and enables us to quantify experimentally implemented joint-measurement devices by feeding a moderate set of coherent states. Our result also reproduces the most informative bound for quantum estimation of phase-space displacement in the case of pure Gaussian states.

Equations

$$⟨(\hat{x}-\sqrt{G_x}{x}_{\text{in}}{)}^2⟩⟨(\hat{p}-\sqrt{G_p}{p}_{\text{in}}{)}^2⟩\ge \frac{{\hslash }^2}{4}{\left(G+|G-1|\right)}^2,$$

$$⟨(\hat{x}-\sqrt{G_x}{x}_{\text{in}}{)}^2⟩⟨(\hat{p}-\sqrt{G_p}{p}_{\text{in}}{)}^2⟩\ge \frac{{\hslash }^2}{4}{\left(2G+1\right)}^2.$$

$$⟨({\hat{Q}}_1-x_{\text{in}}{)}^2⟩⟨({\hat{P}}_2-p_{\text{in}}{)}^2⟩\ge {\hslash }^2,$$

$$x_{\alpha }:=\text{Tr}(\hat{x}{\rho }_{\alpha })=\frac{\alpha +{\alpha }^{\ast }}{\sqrt{2}},p_{\alpha }:=\text{Tr}(\hat{p}{\rho }_{\alpha })=\frac{\alpha -{\alpha }^{\ast }}{\sqrt{2}i}.$$

$${\bar{V}}_M^{(x)}({\eta }_x,\lambda )$$

$${\bar{V}}_N^{(p)}({\eta }_p,\lambda )$$

$$p_{\lambda }(\alpha ):=\frac{\lambda }{\pi }\exp (-\lambda |\alpha {|}^2)$$

$${\bar{V}}_x^{(x)}(G_x,\lambda ){\bar{V}}_{\pm p}^{(p)}(G_p,\lambda )\ge \frac{1}{4}{\left(\frac{G}{1+\lambda }+\left|\frac{G}{1+\lambda }\mp 1\right|\right)}^2,$$

$${\bar{V}}_x^{(x)}(G_x,\lambda ){\bar{V}}_p^{(p)}(G_p,\lambda )\ge \frac{1}{4}{\left(\frac{2G}{1+\lambda }+1\right)}^2.$$

$${\bar{V}}_M^{(x)}(G_x,\lambda ){\bar{V}}_N^{(p)}(G_p,\lambda )\ge {\left(\frac{G}{1+\lambda }\right)}^2.$$

$$J=$$

$${\bar{V}}_M^{(x)}({\eta }_x,\lambda )=$$

$${\bar{V}}_N^{(p)}({\eta }_p,\lambda )=$$

$$({\tau }_x,{\tau }_p):=({\eta }_x,{\eta }_p)/(1+\lambda ).$$

$$\text{Tr}[({\hat{M}}_A-\sqrt{{\tau }_x}{\hat{x}}_B{)}^{2}J]\text{Tr}[({\hat{N}}_A+\sqrt{{\tau }_p}{\hat{p}}_B{)}^{2}J]$$

$$\ge$$

$$\left({\bar{V}}_M^{(x)}({\eta }_x,\lambda )-\frac{{\eta }_x}{2(1+\lambda )}\right)\left({\bar{V}}_N^{(p)}({\eta }_p,\lambda )-\frac{{\eta }_p}{2(1+\lambda )}\right)$$

$$\ge \frac{1}{4}{\left|⟨[\hat{M},\hat{N}]{⟩}_{{\text{Tr}}_B[J]}-\frac{\sqrt{{\eta }_x{\eta }_p}}{(1+\lambda )}\right|}^2,$$

$$\hat{M}={\hat{Q}}_A\otimes {\text{openone}}_{A^{\prime }},\hat{N}={\text{openone}}_A\otimes {\hat{P}}_{A^{\prime }},$$

$$\left({\bar{V}}_M^{(x)}(\eta s,\lambda )-\frac{1}{2}\frac{\eta s}{1+\lambda }\right)$$

$$\times \left({\bar{V}}_N^{(p)}(\eta /s,\lambda )-\frac{1}{2}\frac{\eta /s}{1+\lambda }\right)\ge \frac{1}{4}{\left(\frac{\eta }{1+\lambda }\right)}^2,$$

$$\mathcal{E}({\rho }_{\alpha })={\rho }_{\alpha /\sqrt{2}}\otimes {\rho }_{\alpha /\sqrt{2}}.$$

$$(\hat{M},\hat{N})$$

$$\left({\bar{V}}_M(\eta s,\lambda ),{\bar{V}}_N(\eta /s,\lambda )\right)$$

$$V_X:=\sum _{\theta \in \Theta }\sum _i{p}_{\theta }(X_{\theta }-X_i{)}^2\text{Tr}({\hat{m}}_i{\rho }_{\theta }),$$

$$V_X{V}_P\ge {\left(\frac{G}{1+\lambda }\right)}^2$$

$$V_X:=$$

$$V_P:=$$

$$\mathcal{E}(\rho )=\sum _i\text{Tr}({\hat{m}}_i\rho )|u_i⟩⟨u_i|\otimes |v_i⟩⟨v_i|,$$

$$\hat{M}=\sum _i(X_i|u_i⟩⟨u_i|)\otimes \text{openone},\hat{N}=\text{openone}\otimes \sum _i{P}_i|v_i⟩⟨v_i|.$$

$${\bar{V}}_M^{(x)}({\eta }_M,\lambda )=$$

$${\bar{V}}_N^{(p)}({\eta }_N,\lambda )=$$

$$g_x{V}_x+g_p{V}_p\ge g_x{\sigma }_x^2+g_p{\sigma }_p^2+\sqrt{g_x{g}_p}.$$

Full text

The canonical commutation relation $[\hat{x},\hat{p}]=i\hbar$ determines a primary noise property on quantum states, and it is highly fascinating if one can characterize the penalty on controlling or measuring a pair of canonical variables in a unified manner Busch et al. (2014); Arthurs and Goodman (1988); Clerk et al. (2010). The amplification uncertainty relation represents the penalty of the linear amplification by a gain dependent uncertainty relation Caves (1982); Clerk et al. (2010)

[TABLE]

where $(G_{x},G_{p})$ represents the gain of the physical process with $G=\sqrt{G_{x}G_{p}}$, and $(x_{\textrm{in}},p_{\textrm{in}})$ stands for the mean value of $(\hat{x},\hat{p})$ for an arbitrary input state. The amplification map is assumed to satisfy the covariance condition, $\langle\hat{x}\rangle=\sqrt{G_{x}}x_{\textrm{in}}$ and $\langle\hat{p}\rangle=\sqrt{G_{p}}p_{\textrm{in}}$. Under the covariance condition, a class of quantum channels composed of a state-preparation process after a measurement process referred to as entanglement breaking channels is suggested to fulfill Braunstein and Kimble (1998); Holevo (2008); Namiki and Azuma (2015)

[TABLE]

The right-hand value of Eq. (2) is two units of the shot noise $2\times(\hbar/2)$ larger than the right-hand side of Eq. (1) in the case of a unit gain $G=1$ Braunstein and Kimble (1998); Holevo (2008); Namiki and Azuma (2015). For a measurement process, the famous joint-measurement uncertainty relation Arthurs and Kelly (1965) can be expressed as

[TABLE]

where the variables of two meter systems $\hat{Q}_{1}$ and $\hat{P}_{2}$ are assumed to satisfy the unbiasedness condition, $\langle\hat{Q}_{1}\rangle=x_{\textrm{in}}$ and $\langle\hat{P}_{2}\rangle=p_{\textrm{in}}$. The right-hand term of Eq. (3) becomes larger-than the right-hand term of Eq. (1) with $G=1$ by a single unit of the shot noise $\hbar/2$. As a standard interpretation, this extra noise originates from an introduction of the auxiliary system, and another extra unit for entanglement breaking channels in Eq. (2) is due to the state-preparation process.

Unfortunately, neither the covariance condition nor the unbiasedness condition for continuous-variable (CV) systems is realized in experiments since such a condition implies devices executing a perfectly linear response to arbitrary field amplitude. Hence, there is no reason to consider that the inequalities in Eqs. (1), (2), and (3) hold for real physical systems. In addition, the amplitude of input states is practically bounded due to physical conditions in experimental systems, and it is impossible to confirm the linearity. This motivate us to search for theoretical limitations on physical maps assuming a moderate set of input states Appleby (1998). For optimization of the joint measurement, a Bayesian approach for input-state ensembles of coherent states with Gaussian prior had been studied in depth in the seminal results of quantum estimation Holevo (1973); Yuen and Lax (1973); Holevo (1975, 2011); Helstrom (1976). Due to the Bayesian framework, the measurement devices need not to be unbiased, and the Gaussian prior enable us to avoid the contribution from unrealistically higher input amplitude. Such Bayesian state ensembles have also been employed to determine classical limits or fundamental quantum limits for continuous-variable (CV) quantum gates mainly in terms of the average gate fidelity Hammerer et al. (2005); Namiki et al. (2008); Owari et al. (2008); Namiki (2011a, b); Namiki et al. (2011); Chiribella and Xie (2013); Chiribella and Adesso (2014); Yang et al. (2014); Namiki (2016a); Zhao and Chiribella (2017); Namiki (2016b). Recently, the uncertainty-relation type inequalities essentially the same form as Eqs. (1) and (2) have been proven in terms of the mean-square deviations (MSD) for canonical variables in Refs. Namiki (2015) and Namiki and Azuma (2015), respectively.

In this report, we derive a joint-measurement uncertainty relation for Bayesian MSDs which coincides with the form of the Arthurs-Kelly relation in Eq. (3). It turns out that the three uncertainty relations in Eqs. (1), (2), and (3) can be reformulated to be applicable for a wide range of physical processes realized in experiments. The inequality is proven as a straightforward consequence of commutation relations and the property of quantum channels. Further, our measurement model is proven to be equivalent to the model of parameter estimation based on the positive operator valued measure (POVM). Our result also reproduces the most informative bound for quantum estimation of phase-space displacement in the case of pure Gaussian states Holevo (2011).

In our approach Namiki (2015); Namiki and Azuma (2015), we consider the MSDs for a set of coherent states instead of the deviations for an arbitrary state in the left-hand side of Eqs. (1), (2) and (3). In what follows we set $\hbar=1$ and the canonical commutation relation indicates $[\hat{x},\hat{p}]=i$ as a standard notation. The mean input quadrature $(x_{\textrm{in}},p_{\textrm{in}})$ for a coherent state $\rho_{\alpha}:=\left|\alpha\right\rangle\!\left\langle\alpha\right|$ is specified by

[TABLE]

Let $\hat{M}$ and $\hat{N}$ be self-adjoint operators, and $\mathcal{E}$ be a quantum channel. We define a pair of the MSDs as

[TABLE]

where $(\eta_{x},\eta_{p})$ is a pair of non-negative number, and

[TABLE]

is a Gaussian prior distribution with $\lambda>0$. If we choose the canonical pair $(\hat{M},\hat{N})=(\hat{x},\pm\hat{p})$, we can reach the Bayesian amplification uncertainty relation for quantum channels Namiki (2015)

[TABLE]

where the lower sign corresponds to the case of phase-conjugate amplification and attenuation. Moreover, if $\mathcal{E}$ is entanglement breaking, the minimum of the product $\bar{V}_{x}\bar{V}_{p}$ has to satisfy a more restricted condition Namiki and Azuma (2015):

[TABLE]

Notably, Eqs. (7) and (8) reproduce the forms of Eqs. (1) and (2) in the limit of the uniform prior $\lambda\to 0$. In what follows, we will find that a general joint measurement, which is described by a quantum channel $\mathcal{E}$ and commutable observables $[\hat{M},\hat{N}]=0$, has to fulfill

[TABLE]

This relation will establish a joint-measurement uncertainty relation free from the unbiasedness assumption. The inequality in Eq. (9) also reproduces the form of Eq. (3) in the limit $\lambda\to 0$ for $G=1$.

We will find a bound on the pair of the MSDs based on the method in Refs. Namiki (2015); Namiki and Azuma (2015). Let be $\left|\psi_{\lambda}\right\rangle=\sqrt{\lambda/(1+\lambda)}\sum_{n=0}^{\infty}(1+\lambda)^{-n/2}\left|n\right\rangle_{A}\left|n\right\rangle_{B}$ a two-mode squeezed state, and consider the state after an action of the quantum channel $\mathcal{E}$ on subsystem $A$,

[TABLE]

where $I$ denotes the identity map. We can observe that the MSD represents the correlation between a measurement observable on $A$ and a canonical variable on $B$ for the state $J$. To be concrete, a straightforward calculation Namiki (2015); Namiki and Azuma (2015) from Eqs. (5), (6), and (10), leads to

[TABLE]

where the rightmost terms are responsible for the vacuum fluctuation due to the mapping procedure from the q-numbers to c-numbers $(\hat{x},\hat{p})\to(x_{\alpha},p_{\alpha})$, and we defined

[TABLE]

A physical limitation in a product form is directly imposed by using the preparation uncertainty relation for the state $J$:

[TABLE]

**Lemma.— ** Let $\hat{M}$ and $\hat{N}$ be a pair of self-adjoint operators, and $\mathcal{E}$ be a quantum channel. For any given positive numbers $(\lambda,\eta_{x},\eta_{p})$, the following relation holds:

[TABLE]

where the MSDs $(\bar{V}_{M}^{(x)},\bar{V}_{N}^{(p)})$ is defined in Eq. (5), and the state $J$ is given in Eq. (10). Note that we can readily prove a similar relation when $\mathcal{E}$ is a stochastic quantum channel (See Namiki (2015); Namiki and Azuma (2015); Namiki (2016b)).

Proof.— Substituting Eqs. (11) and (12) into Eq. (Bayesian uncertainty relation for a joint measurement of canonical variables) with the help of the canonical commutation relation $[\hat{x}_{B},\hat{p}_{B}]=i$, we obtain the inequality of Eq. (Bayesian uncertainty relation for a joint measurement of canonical variables). $\blacksquare$

When we set $(\hat{M},\hat{N})=(\hat{x},\pm\hat{p})$, Lemma leads to the Bayesian amplification uncertainty relation in Eq. (7) Namiki (2015). Similar setting enables us to derive Eq. (8) where a separable inequality for Einstein-Podolsky-Rosen like operators Giovannetti et al. (2003); Namiki (2013) is employed instead of Eq. (Bayesian uncertainty relation for a joint measurement of canonical variables) Namiki and Azuma (2015).

Our interest here is to address the joint-measurement uncertainty relation in the form of Eq. (9). In a general setup of joint measurements, the signal state in system $A$ is interacted with an ancilla system $A^{\prime}$. Then, a projective measurement $\hat{Q}$ concerning the original, or true position of the signal is performed on the system $A$, and another projection $\hat{P}$ concerning the momentum of the original signal is carried out on system $A^{\prime}$. The measurement observables are typically written as

[TABLE]

and thus commutable $[\hat{M},\hat{N}]=0$.

Here, we describe the interaction with possible ancillary systems by a quantum channel $\mathcal{E}$, in which an input state in a single mode system could be transformed into an output state in any physically allowable system of an arbitrary size. As for the measurement observables we only assume that they are commutable $[\hat{M},\hat{N}]=0$.

Proposition.— Let $(\hat{M},\hat{N})$ be a pair of commutable observables satisfying $[\hat{M},\hat{N}]=0$ and $\mathcal{E}$ be a quantum channel. For any given positive numbers $(\lambda,\eta_{x},\eta_{p})$, the following trade-off relation holds

[TABLE]

where the MSDs $(\bar{V}_{M}^{(x)},\bar{V}_{N}^{(p)})$ are defined in Eq. (5). Moreover, the equality of Eq. (16) can be achieved by a joint-measurement setup using a beam splitter and quadrature measurements.

**Proof of Proposition.—**Substituting $(\eta_{x},\eta_{p})=\eta(s,s^{-1})$ and $[\hat{M},\hat{N}]=0$ into our Lemma of Eq. (Bayesian uncertainty relation for a joint measurement of canonical variables), we obtain Eq. (16). In order to prove the attainability, let us consider a half-beam splitter with a vacuum field as an ancilla that transforms the coherent state as

[TABLE]

Let us set the pair of observables as

[TABLE]

It clearly fulfills $[\hat{M},\hat{N}]=0$. Substituting Eqs. (17) and (18) into Eq. (5), we obtain

[TABLE]

This saturates the inequality in Eq. (16). Note that the ratio $s=\sqrt{\eta_{x}/\eta_{p}}$ can be interpreted as a consequence that either the state is squeezed or the measured value is rescaled. $\blacksquare$

Proposition implies that the product $\bar{V}_{M}\bar{V}_{N}$ is lower bounded as in Eq. (9) when the gain $\eta=\sqrt{\eta_{x}\eta_{p}}$ is fixed and the ratio $s=\sqrt{\eta_{x}/\eta_{p}}$ is tweaked as in Fig. 1 (See also Sec. IIB of Namiki (2015)). Moreover, the inverse proportional curve due to Eq. (9) can be swept by Eq. (19). Therefore, this completes the final step to reformulate the three uncertainty relations of Eqs. (1), (2), and (3) into experimentally testable relations of Eqs. (7), (8), and (9) based on a unified framework without imposing the linearity assumptions on the physical maps. We can observe that the minimum penalty curve for the measurement process is located in the middle of the curves for the quantum channel and the entanglement breaking channel (See Fig. 1). This is a rigorous example that shows a hierarchy on the trade-off relations of controlling canonical variables for general quantum channels, joint measurements, and entanglement breaking channels suggested from the uncertainty relations in Eqs. (1), (2), and (3).

Thus far, we have described the measurement process by using a quantum channel $\mathcal{E}$ and a pair of observables, $\{\mathcal{E},\hat{M},\hat{N}\}$. In the following part, we rewrite our measurement model by using the POVM. This will establish a link between our measurement uncertainty relation and a major result of the quantum estimation theory Holevo (1975). Let us start with recalling the framework of parameter estimation. We consider a set of quantum states called parametric family $\{\rho_{\theta}\}_{\theta\in\Theta}$, where ${\theta}$ is an unknown parameter belongs to a set $\Theta$. A prior probability distribution $\{p_{\theta}\}_{\theta\in\Theta}$ is assigned to specify an ensemble of states $\{p_{\theta},\rho_{\theta}\}_{\theta\in\Theta}$ in Bayesian approach. The true value of an operator $\hat{X}$ for an unknown state $\rho_{\theta}$ is defined as $X_{\theta}={\textrm{Tr}}(\rho_{\theta}\hat{X})$. An estimation process is described by a POVM $\{\hat{m}_{i}\}$ and a set of real quantities $\{X_{i}\}$; The estimator $\{\hat{m}_{i},X_{i}\}$ an estimation value $X_{i}$ for each measurement outcome $i$ of the POVM. An estimator is said to be optimal if it minimizes the mean square error (MSE)

[TABLE]

where both the set $\{X_{i}\}$ and the POVM $\{\hat{m}_{i}\}$ are optimized. Roughly speaking, the estimator $\{\hat{m}_{i},X_{i}\}$ is more favorable if the loss function $V_{X}$ is smaller. For the multi-parameter estimation, we may consider a set of operators $\{\hat{X},\hat{Y},\hat{Z},\cdots\}$, and the problem is to minimize a wighted sum of the mean square errors for a set of operators such as $g_{X}V_{X}+g_{Y}V_{Y}+g_{Z}V_{Z}+...$, where $g_{X,Y,Z,...}\geq 0$ denotes the wighting factors, and a multi-parameter estimator can be specified by the sequence $\{\hat{m}_{i},X_{i},Y_{i},Z_{i},...\}$.

Our Proposition leads to a bound for a product of the MSEs for the canonical observables $(\hat{x},\hat{p})$.

**Corollary.— ** Let be $\lambda>0$, $G>0$, and $R\in\mathbb{R}$. Let $\{\hat{m}_{i}\}$ be a POVM. For any multi-parameter estimator described by $\{\hat{m}_{i},X_{i},P_{i}\}$, the following trade-off relation holds

[TABLE]

where the pair of the MSEs are given by

[TABLE]

Note that one can drop the parameters $(G,R)$ by considering a rescaled estimation with $(\tilde{X}_{i},\tilde{P}_{i})=G^{-1/2}(X_{i}e^{R},P_{i}e^{-R})$. In such a scenario we have $\tilde{V}_{X}\tilde{V}_{P}\geq({1+\lambda})^{-2}$ with $(\tilde{V}_{X},\tilde{V}_{P})=G^{-1}(V_{X}e^{2R},V_{P}e^{-2R})$, instead of Eq. (21). This is indeed in the form of Eq. (3).

Proof.— Let us define an entanglement breaking channel associated with the POVM $\{\hat{m}_{i}\}$ as

[TABLE]

where $\{\left|u_{i}\right\rangle\}_{i}$ and $\{\left|v_{i}\right\rangle\}_{i}$ are orthonormal bases. Let us choose a pair of commutable observables as

[TABLE]

By substituting Eqs. (23) and (24) into Eq. (5) with $(\eta_{x},\eta_{p})=(Ge^{-2R},Ge^{2R})$, we obtain $V_{X}$ and $V_{P}$ in Eqs. (22), i.e., we can write $(V_{X},X_{P})=(\bar{V}_{M}(Ge^{-2R},\lambda),\bar{V}_{N}(Ge^{2R},\lambda))$. Hence, the pair $(V_{X},X_{P})$ satisfies Eq. (9) as well as Eq. (16). This implies Eq. (21). $\blacksquare$

In the proof of Corollary we have shown that any multi-parameter estimator $\{m_{i},X_{i},P_{i}\}$ can be described by the measurement model $\{\mathcal{E},\hat{M},\hat{N}\}$.We can show that the converse direction is also true, and the two descriptions are equivalent as a type of Naimark’s theorem.

Theorem.— For any joint measurement described by $\{\mathcal{E},\hat{M},\hat{N}\}$ with $[\hat{M},\hat{N}]=0$, there exists a multi-parameter estimator $\{\hat{m}_{i}^{\prime},a_{i},b_{i}\}$ that gives the MSDs $(\bar{V}_{M},\bar{V}_{N})$. The converse direction also holds.

Proof.— Since $[\hat{M},\hat{N}]=0$, there exists an orthonormal basis $\{\left|\omega_{i}\right\rangle\}$ that simultaneously diagonalizes $(\hat{M},\hat{N})$ such that $\hat{M}=\sum_{i}a_{i}\left|\omega_{i}\right\rangle\!\left\langle\omega_{i}\right|$ and $\hat{N}=\sum_{i}b_{i}\left|\omega_{i}\right\rangle\!\left\langle\omega_{i}\right|$ with the sets of eigenvalues $\{a_{i}\}$ and $\{b_{i}\}$. Since $\mathcal{E}$ is a quantum channel, we have a Kraus representation $\mathcal{E}(\rho)=\sum_{j}K_{j}\rho K_{j}^{\dagger}$ with $\sum_{j}K_{j}^{\dagger}K_{j}=\openone$. Hence, we can write

[TABLE]

where $\hat{m}_{i}^{\prime}:=\sum_{j}K_{j}\left|\omega_{i}\right\rangle\!\left\langle\omega_{i}\right|K_{j}^{\dagger}\geq 0$. From $\sum_{j}K_{j}^{\dagger}K_{j}=\openone$ and $\sum_{i}\left|\omega_{i}\right\rangle\!\left\langle\omega_{i}\right|=\openone$ we can readily check that $\{\hat{m}_{i}^{\prime}\}$ fulfills the condition for a POVM, $\sum_{i}\hat{m}_{i}^{\prime}=\openone$. This confirms that the pair $(\bar{V}_{M},\bar{V}_{N})$ represents the MSEs in the form of Eq. (20). Therefore, the pair $(\bar{V}_{M},\bar{V}_{N})$ can be determined by the property of an estimator $\{\hat{m}_{i}^{\prime},a_{i},b_{i}\}$. Conversely, for any estimator $\{\hat{m}_{i},X_{i},P_{i}\}$ we can find a set $\{\mathcal{E},\hat{M},\hat{N}\}$ that gives the pair of the MSEs $(V_{X},V_{P})$ as shown in the proof of Corollary. $\blacksquare$

Finally, we will reproduce one of the central outcomes in the quantum estimation theory. For the mean-value estimation for Gaussian states with the variances $(\sigma_{x}^{2},\sigma_{p}^{2})$, the most informative bound for weighted MSEs Holevo (1975) is given by (see Eq. (6.6.65) of Ref. Holevo (2011))

[TABLE]

We can immediately reach this relation from Eq. (16) for the case of pure Gaussian states, i.e., $\sigma_{x}^{2}\sigma_{p}^{2}=1/4$, as follows. Let us set $(\eta,\lambda)=(1,0)$ and take the square root of Eq. (16). Then, by using the relation $(at+bt^{-1})/2\geq\sqrt{ab}$ for positive numbers $(a,b,t)$, we obtain

[TABLE]

We can see that Eq. (27) coincides with Eq. (26) by applying the following replacement: $(\bar{V}_{M}(s,0),\bar{V}_{M}(s^{-1},0))\to(V_{x},V_{p})$, $(s,s^{-1})\to 2(\sigma_{x}^{2},\sigma_{p}^{2})$, and $t\to\sqrt{g_{x}/g_{p}}$ as long as $\sigma_{x}^{2}\sigma_{p}^{2}=1/4$ holds. To this end, our approach reveals that the origin of the bound rather directly comes from commutation relations. The case for mixed Gaussian states should be addressed elsewhere. From a geometrical point of view, Eq. (26) corresponds to tangent lines of the curve of Eq. (16) (See Fig. 1).

We have presented a joint-measurement uncertainty relation based on a Bayesian input ensemble of optical states. It reproduces the form of the Arthurs-Kelly relation in Eq. (3) in the uniform prior limit, and the most informative bound for quantum estimation of phase-space displacement in the case of pure Gaussian states. Our measurement model is equivalent to the parameter estimation based on the POVM. Our uncertainty relation is applicable to such a general measurement process and, one can determine to what extend the performance of a given joint-measurement device is close to the theoretical limit in terms of the MSDs by using a moderate set of input states within realistic assumptions.

This work was supported by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).