Bayesian Logarithmic Derivative Type Lower Bounds for Quantum Estimation ^††thanks: This work was supported in part by JSPS KAKENHI Grant Numbers JP21K11749, JP24K14816.

In this section, we recall the Bayesian classical case at first. The notations referring to the Bayesian statistics are listed as bellows:

• •

  $p_{\theta}(x)=p(x|\theta)$: A parametric model (likelihood) = A family of probability distribution on $\mathcal{X}$.

• •

  $\pi(\theta)$: A prior distribution for the parameter space $\Theta$.

• •

  $q(\theta|x)$: A posterior distribution for a datum $x\in{\cal X}$.

• •

  $p_{x}(x)$: A marginal distribution (evidence).

The expectation value of a random variable $X$ with respect to the model $p_{\theta}$ is denoted by $E_{\theta}[X]=\int_{x}x\,p_{\theta}(x)dx$, whereas the expectation value with respect to the joint distribution $\pi p_{\theta}$ is expressed without subscript, and the expectation value with respect to the marginal $p_{X}$ is denoted as $E_{X}[X]$. An estimator that returns values on the set $\Theta$ is denoted as $\hat{\theta}(x)=(\hat{\theta}_{1}(x),\hat{\theta}_{2}(x),\cdots,\hat{\theta}_{n}(x))$. The main objective of this proceedings is to minimize the average of the weighted trace of the MSE matrix. We consider the most general problem with a parameter-dependent weight matrix, which is $n\times n$ positive definite matrix: $W(\theta)=[W_{jk}(\theta)]>0$. The Bayesian risk is then defined by

where $V_{\theta}[\hat{\theta}]$ is the MSE matrix with the $j,k$ elements

Throughout the paper, $\mathrm{Tr}[\cdot]$ is the trace over $n$-dimensional parameter space.

The optimal estimator is known to be given as follows. First define an inner product on the space of $n$-dimensional vectors of random variables $F(X)=(f_{1}(X),f_{2}(X),\ldots,f_{n}(X))$ and $G=(g_{1}(X),g_{2}(X),\ldots,g_{n}(X))$ by $\langle F|G\rangle:=\sum_{j=1}^{n}E_{X}[f_{j}(X)g_{j}(X)]$. The Bayes risk is minimized by completing the square as

where the first equality follows by defining the following quantities.

Furthermore, the optimal estimator is identified as

However, this result cannot be extended in the quantum case directly because a quantum version of ${d}^{\mathrm{post}}$ is not uniquely defined due to non-commutativity.

We aim at deriving lower bounds in another direction. Before that, we need to define the problem in quantum Bayesian setting. Let $\mathcal{H}$ be a finite dimensional Hilbert space and $\left\{S_{\theta}~{}|~{}\theta\in\Theta\right\}$ be an $n$-parameter quantum model with $\theta=(\theta_{1},\theta_{2},\cdots,\theta_{n})$. In the following, we consider $S_{\theta}$ is full rank for all $\theta\in\Theta$ and the map $\theta\rightarrow S_{\theta}$ is one-to-one. Let a set of positive semidefinite matrices $\{\Pi_{x}\}$ be a measurement with the measurement outcome $x\in{\cal X}$. The quantum measurement is normally referred to as a positive operator-valued measure (POVM), which is defined by

where $I$ is the identity operator on $\mathcal{H}$.

Measurement outcome is described by a random variable $X$ that obeys the conditional probability distribution:

where $\mathrm{tr}[\cdot]$ denotes the trace on the Hilbert space $\mathcal{H}$. The performance of the estimator in this study is quantified by the quantum Bayes risk:

The main objective is to find the best pair $(\Pi,\hat{\theta})$ that minimizes the Bayes risk.

In a recent work, we have proposed a new class of Bayesian lower bound, called the Bayesian Nagaoka-Hayashi bound [11]. Before we recall this bound, we obtain the alternative form of the Bayes risk. We first introduce a new set of variables for quantum measurement and estimator by

In the following, we will omit the argument $[\Pi,\hat{\theta}]$ when it is clear. The key idea is to regard the above quantities as the operator-valued matrix and vector [12, 7]. We introduce the extended Hilbert space by $\mathbb{H}={\mathbb{C}}^{n}\otimes{\cal H}$, and then $\mathbb{L}$ is identified as a matrix on $\mathbb{H}$, whose block matrix representation is given by $[\mathbb{L}_{jk}]$.

Next, define the following quantities defined by the model and prior distribution.

Then the next lemma is immediate.

In this lemma, $\overline{\cdot}$ denotes the averaged operators with respect to the prior distribution. $X^{T_{1}}$ means the transpose over the parameter space and $\mathds{T}\mathrm{r}[\cdot]$ is the trace over both the Hilbert space and the parameter space.

The fundamental theorem of Ref. [11] is the lower bound for the Bayes risk $R_{\mathrm{B}}[\Pi,\hat{\theta}]$.

In Ref. [11], a Bayesian version of the Holevo bound was also proposed. We recall that the Bayesian Holevo-type bound is lower than the Bayesian Nagaoka-Hayashi bound in general.

This lower bound is only expressed in terms of minimization over the set of Hermitian matrices $X=(X_{1},X_{2},\ldots,X_{n})$. However, it is still hard to analyze it further, since the second term is expressed as an integration of functions containing variable $X$.

Firstly, we consider to generalize the Bayesian classically optimum bound in quantum state with $\theta$-dependent weight matrix $W(\theta)$. This is done by apply the constraint $\mathbb{L}\geq{X}X^{T_{1}}$ in the definition of the Bayesian Nagaoka-Hayashi bound.

The proof of this bound is straightforward. However, this cannot be tight because $X_{j},({\overline{\mathbb{S}}}^{-1})_{ij}$ ($i,j$th element of $(\overline{\mathbb{S}})^{-1}$), and $\overline{D_{i}}$ are Hermitian, but $\overline{\mathbb{S}}^{-1}_{ij}$ and $\overline{D_{i}}$ are not commute. This means that optimal $X_{\mathrm{opt}}$ cannot be Hermitian. This bound is the direct generalization of the Bayesian classical one. If we set the state as classical we can immediately degenerate in the previous bound.

Next, we derive a new one-parameter family of lower bounds which is tighter than the direct generalization. The key idea in this proceedings is to define a map $\mathcal{E}$ from $\mathcal{L}(\mathds{C}^{n}\otimes\mathcal{H})$ to $\mathcal{L}(\mathds{C}^{n})$ such that satisfies the condition.

Here $\mathcal{L}_{\mathrm{sym}}(\mathds{C}^{n}\otimes\mathcal{H})$ denotes the set of all Hermitian $\mathbb{L}$ which is invariant under the partial transposition $\cdot^{T_{1}}$. Using the block-matrix representation, this is equivalent to $\forall i,j,\,\mathbb{L}_{ij}=\mathbb{L}_{ji}$.

Given a positive operator $\mathbb{S}$ on $\mathbb{H}=\mathds{C}^{n}\otimes\mathcal{H}$, we define one-parameter family of maps for $\mathds{X}\in\mathcal{L}(\mathds{C}^{n}\otimes\mathcal{H})$.

where

We can show the next property.

After identifying a real symmetric map, we can derive a family of lower bounds, which has a similar structure as the Holevo bound in point estimation.

We now discuss the advantages of the proposed bound:

• •

  The two Bayesian Holevo-type bounds are similar; both are expressed as minimization over $X$ operators. However as an optimization problem, the new one is easier to calculate since the previous bound (Theorem 3) requires to compute the integral over a complicated function of $X$, namely $\int d\theta\pi(\theta)\mathrm{Tr}|\mathrm{Im}\,Z_{\theta}(X)|$. In contrast, the new bound refers to the averaged state $\overline{\mathbb{S}}=\int d\theta W(\theta)\otimes S_{\theta}$ only. Thus, what we need is to calculate the averaged state in advance.

• •

  This new bound is tighter than the direct quantum version of the optimal Bayesian bound (Theorem 4).

  $$\mathcal{C}_{\mathrm{BH}}^{(\lambda)}\geq\mathcal{C}_{\mathrm{direct}}.$$

  (24)

  The reason is that we can show the real part of the bound (22) is equal to the first term of function in the proof (11) and the imaginary part (23) is non-negative.

  $$\mathrm{Tr}[\mathrm{Re}\,Z^{(\lambda)}(X)]+\mathrm{Tr}|\mathrm{Im}\,Z^{(\lambda)}(X)|\\ \geq\mathrm{Tr}[\mathrm{Re}\,Z^{(\lambda)}(X)]=\mathrm{tr}[X^{T_{1}}\overline{\mathbb{S}}X].$$

• •

  Lastly, the family of our new bounds immediately gives one-parameter family generalization of the previously established Bayesian lower bounds when the weight matrix is parameter independent. This will be explored in the next subsection.

In this subsection, we take the $\theta$-independent weight matrix. The Bayesian MSE is defined as

The goal here is to derive a family of quantum Bayesian lower bounds in a closed form by approximating the optimization over $X$ operators. The key ingredient in our approach is to utilize the logarithmic derivative (LD) type equation which was shown to be useful for point estimation [13, 14]. For convenience, we call this family as the Bayesian $\lambda$LD bound, even though this is different from the quantum Cramér-Rao bound based on quantum LDs.

We substitute parameter-independent weight matrix in the quantum Bayesian bound of Theorem 7 to get the following theorem.

We have two remarks before proving this theorem. First, note that the last form of the bound is a Bayesian version of the maximum logarithmic derivative Cramé-Rao bound proposed by Yamagata in point estimation [13]. Second, due to the process of deriving the Bayesian $\lambda$LD bound, it is tighter than the Personick bound [8] and its generalization [15, 16], and the Bayesian bound proposed by Holevo [17, 18]. In other words, the above one-parameter family of bounds includes well-known bounds as special cases. This will be shown later.

By the above proof, we can immediately show that the Personick bound is recovered when $\lambda=0$, whereas the Holevo’s result corresponds to $\lambda=1$. To do this, we can set $\theta$-independent $W$ as the rank-1 projector $W=cc^{\dagger}$ with $c\in\mathds{C}^{n}$. Next, we obtain a lower bound for $\mathcal{C}_{\mathrm{BLD}}^{(\lambda)}$. This gives the desired result.

where we set $L_{c}^{(\lambda)}=\sum_{j}c_{j}L_{j}^{(\lambda)}$. Since this is true for any choice of $c$, we prove the matrix inequality $V_{\mathrm{B}}\geq M-K^{(\lambda)}$. If we choose $\lambda=0$, thereby, we recover the Personick bound [8].