Evaluating the Holevo Cram\'er-Rao bound for multi-parameter quantum metrology

TL;DR

This paper develops a method to evaluate the Holevo Cramér-Rao bound for multi-parameter quantum estimation, revealing conditions under which it can be attained and exploring its implications in optical and magnetometry systems.

Contribution

It recasts the evaluation of the Holevo Cramér-Rao bound as a semidefinite program for finite-dimensional systems and applies this to practical quantum metrology scenarios.

Findings

HCRB can be attained with optimal measurements in certain regimes.

Numerical evidence shows projective measurements attain HCRB in noiseless two-qubit systems.

Interplay between HCRB and measurement incompatibility is demonstrated.

Abstract

Only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed. This is due to the incompatibility of observables. The fundamental bound for multi-parameter quantum estimation is the Holevo Cram\'er-Rao bound (HCRB) whose evaluation has so far remained elusive. For finite-dimensional systems we recast its evaluation as a semidefinite program, with reduced size for rank-deficient states. We use this result to study phase and loss estimation in optical interferometry and three-dimensional magnetometry with noisy multi-qubit systems. For the former, we show that, in some regimes, it is possible to attain the HCRB with the optimal (single-copy) measurement for phase estimation. For the latter, we show a non-trivial interplay between the HCRB and incompatibility, and provide numerical evidence that projective single-copy measurements attain the HCRB in the noiseless two-qubit case.

Figures

Equations

$$\bm{\Sigma}_{\bm{\theta}}\mathopen{}\mathclose{{}\left(\bm{\Pi},\tilde{\bm{\theta}}}\right)=\sum_{\omega\in\Omega}p(\omega|\bm{\theta})\mathopen{}\mathclose{{}\left(\tilde{\bm{\theta}}(\omega)-\bm{\theta}}\right)\mathopen{}\mathclose{{}\left(\tilde{\bm{\theta}}(\omega)-\bm{\theta}}\right)^{\mathsf{T}},$$

$$\sum _{\omega \in \Omega }({\tilde{\theta }}_i(\omega )-{\theta }_i)p(\omega |\mathbf{\theta })=0,\sum _{\omega \in \Omega }{\tilde{\theta }}_i(\omega )\frac{\partial p(\omega |\mathbf{\theta })}{\partial {\theta }_j}={\delta }_{ij}.$$

$$\bm{\Sigma}_{\bm{\theta}}\mathopen{}\mathclose{{}\left(\bm{\Pi},\tilde{\bm{\theta}}}\right)\succeq F(\rho_{\bm{\theta}},\bm{\Pi})^{-1}\,$$

$$F(\rho_{\bm{\theta}},\bm{\Pi})=\sum_{\omega\in\Omega}p(\omega|\bm{\theta})\mathopen{}\mathclose{{}\left(\frac{\partial\log p(\omega|\bm{\theta})}{\partial\bm{\theta}}}\right)\mathopen{}\mathclose{{}\left(\frac{\partial\log p(\omega|\bm{\theta})}{\partial\bm{\theta}}}\right)^{\mathsf{T}},$$

$$\operatorname{Tr}\mathopen{}\mathclose{{}\left[W\bm{\Sigma}_{\bm{\theta}}\mathopen{}\mathclose{{}\left(\bm{\Pi},\tilde{\bm{\theta}}}\right)}\right]\geq C^{\text{H}}_{\bm{\theta}}\geq\max\mathopen{}\mathclose{{}\left\{C^{\text{S}}_{\bm{\theta}},C^{\text{R}}_{\bm{\theta}}}\right\}.$$

$$C_{\mathbf{\theta }}^{\text{H}}({\rho }_{\mathbf{\theta }};W)=C_{\mathbf{\theta }}^{\text{S}}({\rho }_{\mathbf{\theta }};W)⟺D_{\mathbf{\theta }}=0_n,$$

$$C_{\bm{\theta}}^{\text{H}}\mathopen{}\mathclose{{}\left(\rho_{\bm{\theta}};W}\right)=\min_{V\in\mathbb{S}^{n},\bm{X}\in\mathcal{X}_{\bm{\theta}}}\mathopen{}\mathclose{{}\left(\operatorname{Tr}\mathopen{}\mathclose{{}\left[WV}\right]\;|\;V\succeq Z\mathopen{}\mathclose{{}\left[\bm{X}}\right]}\right)\,,$$

$$\mathcal{X}_{\bm{\theta}}=\mathopen{}\mathclose{{}\left\{\bm{X}=\mathopen{}\mathclose{{}\left(X_{1},\dots,X_{n}}\right)\,|\,\operatorname{Tr}\mathopen{}\mathclose{{}\left[X_{i}\partial_{j}\rho_{\bm{\theta}}}\right]=\delta_{ij}}\right\}\,.$$

$$Z\mathopen{}\mathclose{{}\left[\bm{X}}\right]_{ij}=\operatorname{Tr}\mathopen{}\mathclose{{}\left[X_{i}X_{j}\rho_{\bm{\theta}}}\right]=\bm{x}_{i}^{\mathsf{T}}S_{\bm{\theta}}\bm{x}_{j},$$

$$V-B^{†}B⪰0⟺\left(\begin{array}{cc}V& B^{†}\\ B& 1\end{array}\right)⪰0,$$

$$\begin{aligned} &\underset{V\in\mathbb{S}^{n},\bm{X}\in\mathbb{R}^{\tilde{d}{\times}n}}{\text{minimize}}&&\operatorname{Tr}\mathopen{}\mathclose{{}\left[WV}\right]\\ &\quad\text{subject to}&&\begin{pmatrix}V&\bm{X}^{\mathsf{T}}R_{\bm{\theta}}^{\dagger}\\ R_{\bm{\theta}}\bm{X}&\mathbb{1}_{\tilde{r}}\end{pmatrix}\succeq 0\\ &&&\bm{X}^{\mathsf{T}}\,\frac{\partial\bm{s}_{\bm{\theta}}}{\partial\bm{\theta}}=\mathbb{1}_{n}\,\end{aligned}\;,$$

$$h_{\mathbf{\theta }}(\mathbf{X})=$$

$$=$$

$$\operatorname{Im}\mathopen{}\mathclose{{}\left(\operatorname{Tr}\mathopen{}\mathclose{{}\left[L_{\phi}L_{\eta}\rho_{\phi,\eta}}\right]}\right)=-\frac{J^{\text{S}}_{\phi\phi}}{2\eta}\,.$$

$$|{\psi }_M^{\text{3D-GHZ}}⟩=\frac{1}{\mathcal{N}}\sum _{k=1}^3{|{\varphi }_k^{+}⟩}^{\otimes M}+{|{\varphi }_k^{-}⟩}^{\otimes M},$$

Full text

Introduction—Measuring physical quantities with ever increasing precision underlies both technological and scientific progress. Quantum mechanics plays a central role in this challenge. On the one hand, the unavoidable statistical uncertainty due to quantum fluctuations is a fundamental limitation to high precision metrology. On the other hand, quantum-enhanced metrological schemes that take advantage of nonclassical features, such as entanglement, coherence, or squeezing, have been proposed and implemented experimentally Giovannetti et al. (2011); Demkowicz-Dobrzański et al. (2015); Degen et al. (2017); Pezzè et al. (2018); Braun et al. (2018); Pirandola et al. (2018); Berchera and Degiovanni (2019). Myriad metrological applications are intrinsically multi-parameter Szczykulska et al. (2016), e.g., sensing electric, magnetic or gravitational fields Baumgratz and Datta (2016), force sensing Tsang et al. (2011); Ang et al. (2013), imaging Humphreys et al. (2013); Gagatsos et al. (2016), superresolution Tsang et al. (2016); Chrostowski et al. (2017); Řehaček et al. (2017); Yu and Prasad (2018); Napoli et al. (2019). As a consequence, the field of multi-parameter quantum metrology has been growing rapidly, both theoretically D’Ariano and Paris (1997); D’Ariano et al. (1998); Macchiavello (2003); Ballester (2004); Young et al. (2009); Vaneph et al. (2013); Genoni et al. (2013); Zhang and Fan (2014); Yuan and Fung (2015); Berry et al. (2015); Liu and Yuan (2017); Liu and Cable (2017); Pezzè et al. (2017); Eldredge et al. (2018); Nair (2018); Nichols et al. (2018); Gessner et al. (2018); Kura and Ueda (2018); Yang et al. (2018); Chen and Yuan (2019); Rubio and Dunningham (2019) and experimentally Vidrighin et al. (2014); Roccia et al. (2018a, b); Parniak et al. (2018); Polino et al. (2019).

The mathematical framework behind quantum metrology is quantum estimation theory Paris (2009), pioneered by Helstrom and Holevo Helstrom (1967, 1968, 1976); Holevo (1973, 1976, 2011). In particular, multi-parameter quantum estimation highlights a defining trait of quantum theory, absent in single parameter estimation: incompatibility of observables Heinosaari et al. (2016); Zhu (2015). Because of this, multi-parameter quantum estimation is much more challenging, but also serves as a testbed for understanding quantum measurements.

Precision bounds for multi-parameter estimation are given in terms of matrix inequalities for the mean square error matrix (MSEM) $\bm{\Sigma}$, see Eq. (1). However, matrix bounds are in general not tight for multi-parameter quantum estimation. Instead, the Holevo Cramér-Rao bound (HCRB) Holevo (2011); Hayashi (2005) is the most fundamental scalar lower bound imposed by quantum mechanics on the weighted mean square error (WMSE) \operatorname{Tr}\mathopen{}\mathclose{{}\left[W\bm{\Sigma}}\right] (for a positive definite $W$). The HCRB represents the best precision attainable with global measurements on an asymptotically large number of identical copies of a quantum state Guţă and Kahn (2006); Hayashi and Matsumoto (2008); Kahn and Guţă (2009); Yamagata et al. (2013); Yang et al. (2019). Implementing such collective measurements is exceptionally challenging Roccia et al. (2018a); Hou et al. (2018), but in some cases the HCRB is attained by single-copy measurements: for pure states Matsumoto (2002) and for displacement estimation with Gaussian states Holevo (2011).

Despite its importance, the HCRB has been used more as a mathematical object in asymptotic quantum statistics Gill and Guţă (2013) than applied to concrete metrological problems. Indeed, the HCRB is considered hard to evaluate, even numerically, being defined through a constrained minimization over a set of operators. Closed form results for non-trivial cases are known only for qubits Suzuki (2016), two-parameter estimation with pure states [][.ReprintedinChapter20ofRef.[54]; anupdatedshorterversionispublishedasRef.[61].]Matsumoto1997 and two-parameter displacement estimation with two-mode Gaussian states Bradshaw et al. (2017, 2018), while a numerical investigation has been attempted for pure states and Hamiltonian parameters Gorecki et al. (2019). The evaluation of the HCRB thus remains a major roadblock in the development of multi-parameter quantum metrology.

This Letter removes this roadblock by providing a recipe for evaluating the HCRB numerically for finite-dimensional systems. Our main result recasts the optimization required for evaluating the HCRB as a semidefinite program (SDP). This was shown only for displacement estimation with Gaussian states Bradshaw et al. (2018). We present a SDP whose complexity grows with the rank of the state instead of a naïve dependence on the Hilbert space dimension. The application of our recipe to evaluate the HCRB for two well-known metrological problems provides new insights. In particular, we provide numerical evidence that single-copy attainability of the HCRB with projective measurements is possible in non-trivial cases.

Multi-parameter quantum estimation—We consider a generic finite-dimensional quantum system with Hilbert space $\mathcal{H}\cong\mathbb{C}^{d}$, denote the space of linear operators ($d{\times}d$ matrices) on $\mathcal{H}$ as \mathcal{L}\mathopen{}\mathclose{{}\left(\mathcal{H}}\right)\cong\mathbb{C}^{d{\times}d} and the space of observables (Hermitian matrices) as \mathcal{L}_{\mathsf{h}}\mathopen{}\mathclose{{}\left(\mathcal{H}}\right).

The state of the system \rho_{\bm{\theta}}\in\mathcal{L}_{\mathsf{h}}\mathopen{}\mathclose{{}\left(\mathcal{H}}\right) is parametrized by a real vector $(\theta_{1},\dots,\theta_{n})^{\mathsf{T}}=\bm{\theta}\in\Theta\subset\mathbb{R}^{n}$ 111 We assume a sufficiently regular parametrization, in particular the state has a fixed rank for all $\bm{\theta}$., the collection $\{\rho_{\bm{\theta}}\}$ for all the values of $\bm{\theta}$ is called the quantum statistical model. The goal is to simultaneously estimate all $n$ parameters by measuring possibly multiple copies of $\rho_{\bm{\theta}}.$ After measurement, classical data is processed with an estimator $\tilde{\bm{\theta}}$, a function from the space of measurement outcomes $\Omega$ to the space of parameters $\Theta$. The MSEM of the estimator

[TABLE]

quantifies the precision of the estimation. The probability of observing the outcome $\omega$ is given by the Born rule $p(\omega|\bm{\theta})=\operatorname{Tr}(\rho_{\bm{\theta}}\Pi_{\omega})$; the measurement is described by a positive operator valued measure (POVM): \bm{\Pi}=\mathopen{}\mathclose{{}\left\{\Pi_{\omega}\succeq 0,\omega\in\Omega\,|\,\sum_{\omega\in\Omega}\Pi_{\omega}=\mathbb{1}_{d}}\right\}, without loss of generality we consider $\Omega$ to be a finite set Chiribella et al. (2007).

We consider locally unbiased estimators that satisfy

[TABLE]

For this class of estimators the matrix Cramér-Rao bound (CRB) on the MSEM is Cramér (1946)

[TABLE]

($A\succeq 0$ iff $A$ is positive semidefinite); the classical Fisher information matrix (CFIM) $F(\rho_{\bm{\theta}},\bm{\Pi})$ is defined as

[TABLE]

where $\partial f(\bm{\theta})/\partial\bm{\theta}$ is the gradient of the function $f$. For locally unbiased estimators the MSEM is the covariance matrix (CM) and the bound is attainable: there is always an estimator in this class with a CM equal to the inverse FIM [][.ReprintedasChapter8inRef.~[54].]Nagaoka1989; Gill and Guţă (2013). To meaningfully compare the precision of different multi-parameter estimators, it is customary to consider a scalar cost function, the WMSE \operatorname{Tr}\mathopen{}\mathclose{{}\left[W\bm{\Sigma}_{\bm{\theta}}\mathopen{}\mathclose{{}\left(\bm{\Pi},\tilde{\bm{\theta}}}\right)}\right], with $0\prec W\in\mathbb{S}^{n}$ ($\mathbb{S}^{n}$ is the set of real symmetric $n$-dimensional matrices).

Well-known bounds for multi-parameter quantum estimation are obtained from the symmetric logarithmic derivatives (SLDs) $L_{i}\in\mathcal{L}_{\mathsf{h}}(\mathcal{H})$, satisfying $2\partial\rho_{\bm{\theta}}/\partial\theta_{i}=L_{i}\rho_{\bm{\theta}}+\rho_{\bm{\theta}}L_{i}$, and from the right logarithmic derivatives (RLDs) $\tilde{L}_{i}\in\mathcal{L}(\mathcal{H})$ satisfying $\partial\rho_{\bm{\theta}}/\partial\theta_{i}=\rho_{\bm{\theta}}\tilde{L}_{i}$. The (real symmetric) quantum Fisher information matrix (QFIM) is defined as J^{\text{S}}_{ij}=\operatorname{Re}\mathopen{}\mathclose{{}\left(\operatorname{Tr}\mathopen{}\mathclose{{}\left[\rho_{\bm{\theta}}L_{i}L_{j}}\right]}\right) Helstrom (1967, 1968); an analogous (complex Hermitian) matrix in terms of the RLDs is J^{\text{R}}_{ij}=\operatorname{Tr}\mathopen{}\mathclose{{}\left[\rho_{\bm{\theta}}\tilde{L}_{i}\tilde{L}_{j}}\right] Yuen and Lax (1973); Belavkin (1976). Both matrices give lower bounds for the MSEM: \bm{\Sigma}_{\bm{\theta}}\mathopen{}\mathclose{{}\left(\bm{\Pi},\tilde{\bm{\theta}}}\right)\succeq\mathopen{}\mathclose{{}\left(J^{\text{S/R}}}\right)^{-1} and the corresponding scalar bounds are C^{\text{S}}_{\bm{\theta}}(\rho_{\bm{\theta}};W)=\operatorname{Tr}\mathopen{}\mathclose{{}\left[W(J^{\text{S}})^{-1}}\right] and C^{\text{R}}_{\bm{\theta}}(\rho_{\bm{\theta}};W)=\operatorname{Tr}\mathopen{}\mathclose{{}\left[W\operatorname{Re}(J^{\text{R}})^{-1}}\right]+\mathopen{}\mathclose{{}\left\|\sqrt{W}\operatorname{Im}(J^{\text{R}})^{-1}\sqrt{W}}\right\|_{1}, where \|A\|_{1}=\operatorname{Tr}\mathopen{}\mathclose{{}\left[\sqrt{A^{\dagger}A}}\right] is the trace norm Holevo (2011); Nagaoka (1989); Hayashi (2017).

Holevo introduced a tighter bound, the HCRB $C^{\text{H}}_{\bm{\theta}}$ Holevo (1976, 2011):

[TABLE]

Both inequalities can be tight. In particular Ragy et al. (2016):

[TABLE]

where (D_{\bm{\theta}})_{ij}\equiv\operatorname{Im}\mathopen{}\mathclose{{}\left(\operatorname{Tr}\mathopen{}\mathclose{{}\left[L_{j}L_{i}\rho_{\bm{\theta}}}\right]}\right) is a skew-symmetric matrix Nagaoka (1989). Condition (6) is called weak commutativity and quantum statistical models satisfying it asymptotically classical Suzuki (2018).

Computing the bound with a SDP—The HCRB is obtained as the result of the following minimization Holevo (2011); Hayashi (2017):

[TABLE]

with the Hermitian $n{\times}n$ matrix Z\mathopen{}\mathclose{{}\left[\bm{X}}\right]_{ij}=\operatorname{Tr}\mathopen{}\mathclose{{}\left[X_{i}X_{j}\rho_{\bm{\theta}}}\right] and the collection $\bm{X}$ of operators $X_{i}\in\mathcal{L}_{\mathsf{h}}(\mathcal{H})$ in the set

[TABLE]

For a density matrix with rank $r<d$ we can restrict the operators $X_{i}$ to the quotient space \mathcal{L}^{r}_{\mathsf{h}}(\mathcal{H})=\mathcal{L}_{\mathsf{h}}(\mathcal{H})/\mathcal{L}_{\mathsf{h}}\mathopen{}\mathclose{{}\left(\mathrm{ker}(\rho_{\bm{\theta}})}\right), with dimension $\tilde{d}=2dr-r^{2}$. For any $X\in\mathcal{L}_{\mathsf{h}}(\mathcal{H}),$ any scalar quantity evaluated in the eigenbasis of $\rho_{\bm{\theta}}$ is independent of the diagonal block of $X$ corresponding to the kernel of $\rho_{\bm{\theta}}$ Holevo (2011); Fujiwara and Nagaoka (1995) (see Sec. I of SM for details).

We introduce a basis $\lambda_{i}$ of Hermitian operators for $\mathcal{L}_{\mathsf{h}}^{r}(\mathcal{H})$, orthonormal w.r.t. the Hilbert-Schmidt inner product \operatorname{Tr}\mathopen{}\mathclose{{}\left[\lambda_{i}\lambda_{j}}\right]=\delta_{ij}. Using such a basis, each operator $X_{i}\in\mathcal{L}^{r}_{\mathsf{h}}(\mathcal{H})$ corresponds to a real valued vector $\bm{x}_{i}\in\mathbb{R}^{\tilde{d}}$. With some abuse of notation we use $\bm{X}$ to denote also the collection of these real vectors, i.e., the $\tilde{d}{\times}n$ real matrix with $\bm{x}_{i}$ as columns. The quantum state also belongs to $\mathcal{L}^{r}_{\mathsf{h}}(\mathcal{H})$ and therefore corresponds to a vector $\bm{s}_{\bm{\theta}}$ in the chosen basis. This corresponds to the generalized Bloch vector Bertlmann and Krammer (2008); Watanabe (2014) when working in the full space $\mathcal{L}_{\mathsf{h}}(\mathcal{H})$.

A quantum state induces an inner product on \mathcal{L}^{r}_{\mathsf{h}}\mathopen{}\mathclose{{}\left(\mathcal{H}}\right) via

[TABLE]

where $S_{\bm{\theta}}\succeq 0$ is the Hermitian matrix representing the inner product in the chosen basis. With this choice we can write Z\mathopen{}\mathclose{{}\left[\bm{X}}\right]=\bm{X}^{\mathsf{T}}S_{\bm{\theta}}\bm{X} so that the matrix inequality on the r.h.s of Eq. (7) reads $V\succeq\bm{X}^{\mathsf{T}}S_{\bm{\theta}}\bm{X}$. Crucially, this last matrix inequality can be converted to a linear matrix inequality (LMI) by using the Schur complement condition for positive semidefiniteness Zhang (2005):

[TABLE]

for any matrix $B$ and identity matrix $\mathbb{1}$ of appropriate size. Thus, we can rewrite the minimization problem in Eq. (7) as

[TABLE]

where the matrix $R_{\bm{\theta}}$ can be any $\tilde{r}{\times}\tilde{d}$ matrix (with rd=\operatorname{rank}\mathopen{}\mathclose{{}\left(S_{\bm{\theta}}}\right)\leq\tilde{r}\leq\tilde{d}) satisfying $S_{\bm{\theta}}=R_{\bm{\theta}}^{\dagger}R_{\bm{\theta}}$, e.g. a Cholesky-like decomposition. Here $\partial\bm{s}_{\bm{\theta}}/\partial\bm{\theta}$ is a matrix with the vector components of the operators $\partial\rho_{\bm{\theta}}/\partial\theta_{i}=\partial_{i}\rho_{\bm{\theta}}$ as columns; this is the Jacobian matrix of $s_{\bm{\theta}}$ only if the basis \mathopen{}\mathclose{{}\left\{\lambda_{i}}\right\} is parameter independent. The program (11) can be readily recognized as a convex minimization problem Boyd and Vandenberghe (2004), since the set of solutions to LMI is convex and the objective function is linear. It can be converted to a SDP (see Sec. II of SM for details), which can be solved numerically using efficient and readily available algorithms with a guarantee of global optimality. In practice, the program (11) can be fed directly to a numerical modeling framework, such as CVX Grant and Boyd (2019) or YALMIP Löfberg (2004).

This convex optimization satisfies Slater’s condition for strong duality Boyd and Vandenberghe (2004), as long as $J^{\text{S}}\succ 0$, i.e. a non-singular quantum statistical model. Strong duality holds when the minimum is finite and coincides with the maximum of the Lagrangian dual problem. Qualitatively, Slater’s condition means that the interior of the set of feasible points for (11) must not be empty. We denote by $\bm{L}$ the matrix with the real vectors representing the SLDs as columns. Upon noticing that (\bm{L}^{\mathsf{T}}\partial\bm{s}_{\bm{\theta}}/\partial\bm{\theta})_{ij}=\operatorname{Tr}\mathopen{}\mathclose{{}\left[L_{i}\partial_{j}\rho_{\bm{\theta}}}\right]=(J^{\text{S}})_{ij} it is easy to show that the matrices $\bm{X}=\bm{L}\,(J^{\text{S}})^{-1}$ and $V=(J^{\text{S}})^{-1}+V^{\prime}$, with an arbitrary $V^{\prime}\succ 0$, satisfy both constraints in (11). For this choice of $V$ and $\bm{X}$ the matrix inequality in (7) and (11) are strict and Slater’s condition is satisfied.

An analytical optimization over $V$ in  (7) leads to Nagaoka (1989):

[TABLE]

so that C_{\bm{\theta}}^{\text{H}}\mathopen{}\mathclose{{}\left(\rho_{\bm{\theta}};W}\right)=\min_{\bm{X}\in\mathcal{X}_{\bm{\theta}}}h_{\bm{\theta}}(\bm{X}) and usually this last optimization is solved numerically. From our previous convexity argument we also infer that $h_{\bm{\theta}}(\bm{X})$ is a convex function of $\bm{X}$; being a partial minimization of an affine function over a convex set Boyd and Vandenberghe (2004). This may not be apparent from (13) since the second term is not convex; the sum of the two terms is convex as long as the matrix $Z[\bm{X}]$ is positive semidefinite and the identity (10) can be used.

Optical interferometry with loss—Optical interferometry, where the goal is to measure a phase difference between two optical paths, is a prime example of quantum metrology Demkowicz-Dobrzański et al. (2015). In some instances, one may wish to estimate both the phase and the loss induced by a sample in one arm of a Mach-Zehnder interferometer Crowley et al. (2014).

We consider initial states with a fixed photon number $N$ across two modes $|\psi_{\text{in}}\rangle=\sum_{k=0}^{N}c_{k}|k,N-k\rangle.$ These include for example N00N states and Holland-Burnett states Holland and Burnett (1993). The evolved state after the lossy interferometer, with one arm characterized by a transmissivity $\eta$ and a phase shift $\phi$, has a direct sum form $\rho_{\phi,\eta}=\bigoplus_{k=0}^{N}p_{l}|\psi_{l}\rangle\langle\psi_{l}|$, where each $|\psi_{l}\rangle$ corresponds to $l$ lost photons Demkowicz-Dobrzański et al. (2009) (see Sec. III of SM for details). For this problem it is possible to obtain the SLDs $L_{\phi}$ and $L_{\eta}$ analytically, as well as the QFIM $J^{\text{S}}=\mathrm{diag}(J^{\text{S}}_{\phi\phi},J^{\text{S}}_{\eta\eta})$. Crucially, this multi-parameter estimation problem is never asymptotically classical, since Crowley et al. (2014)

[TABLE]

Hence the weak commutativity condition (6) never holds if the model is non-singular; thus we get $C_{\bm{\theta}}^{\text{H}}>C_{\bm{\theta}}^{\text{S}}>C_{\bm{\theta}}^{\text{R}}=0$ (the RLD bound is completely uninformative Crowley et al. (2014)). Eq. (14) also means that phase and loss cannot be jointly estimated with the same precision obtainable by estimating each parameter individually and there exists a trade-off between precisions. Following Crowley et al. (2014) we focus on a strategy to estimate $\phi$ with the best possible precision and still get an estimate of $\eta$, by considering the projective POVM $\bm{\Pi}_{\phi}$ obtained from the spectral decomposition of the SLD $L_{\phi}$.

More concretely, we study Holland-Burnett states, a family of states particularly resilient to imperfections Datta et al. (2011); we also fix $W=\mathbb{1}_{2}$. Fig. (1) shows the classical CRB C^{\text{C}}_{\phi}=\operatorname{Tr}\mathopen{}\mathclose{{}\left[F\mathopen{}\mathclose{{}\left(\rho_{\phi,\eta},\bm{\Pi}_{\phi}}\right)^{-1}}\right], along with the HCRB (computed by solving the SDP numerically) and the SLD-CRB $C^{\text{S}}_{\phi,\eta}$, as a function of $\eta$ for $N$ up to $14$. Panel (a) shows that the HCRB is over 30% tighter than the SLD bound, especially for intermediate transmissivities. Panel (b) shows that the measurement we consider attains the HCRB for certain values of $N$ and $\eta$, i.e. the relative difference is zero up to numerical noise. Even when the bound is not attained, the relative difference remains small at around 4% for $N=14$.

For generic 1-photon states ($N=1$) we have found the analytical conditions for the HCRB to be attained by $\bm{\Pi}_{\phi}$. For $|\psi_{\text{in}}\rangle=c_{0}|0,1\rangle+c_{1}|1,0\rangle$ (with $|c_{0}|^{2}+|c_{1}|^{2}=1$) we have C^{\text{C}}_{\phi}=C^{\text{H}}\mathopen{}\mathclose{{}\left(\rho_{\phi,\eta};\mathbb{1}_{2}}\right) as long as $|c_{1}|^{2}\geq 1/2$ or \mathopen{}\mathclose{{}\left(1-|c_{1}|^{2}/|c_{0}|^{2}}\right)/2\leq\eta\leq 1. The relative difference $1-C^{\text{H}}/C^{\text{C}}_{\phi}$ is at most 4.9% and always zero for $\eta\geq 1/2$ (see Sec. III.A of SM for details). A numerical analysis on random states for higher values of $N$ suggests that there is indeed a threshold value of $\eta$, increasing with $N$, above which $\bm{\Pi}_{\phi}$ attains the HCRB.

Finally, we remark that working in the space $\mathcal{L}_{\mathsf{h}}^{r}(\mathcal{H})$ provides a distinct advantage for the numerics, since the Hilbert space dimension is $(N^{2}+3N+2)/2$, while $\rho_{\phi,\eta}$ has rank $r=N+1,$whereby $\tilde{d}=(N+1)^{3}<(N+1)^{4}$.

3D magnetometry—Noiseless 3D magnetometry, another illustrative example of multi-parameter quantum metrology, has been studied terms of the QFIM Baumgratz and Datta (2016). Here, we highlight the necessity of using the HCRB for this problem and present results on 3D magnetometry using $M$ qubits in the presence of dephasing noise. The parameters to be estimated $\bm{\varphi}=(\varphi_{1},\varphi_{2},\varphi_{3})$ appear via the single-qubit Hamiltonian $H^{(j)}(\bm{\varphi})=\bm{\varphi}\cdot\bm{\sigma}^{(j)}$, where $\bm{\sigma}^{(j)}$ is a vector of Pauli operators acting on the $j^{\text{th}}$ qubit. The parameters are imprinted on the probe state via the unitary U_{\bm{\varphi}}=\bigotimes_{j}^{M}\exp\mathopen{}\mathclose{{}\left(-\mathrm{i}H^{(j)}(\bm{\varphi})}\right). This is followed by local dephasing along the $z-$axis described by the single qubit CPTP map $2\mathcal{E}_{\gamma}[\rho]=(1+\sqrt{1-\gamma})\rho+(1-\sqrt{1-\gamma})\sigma_{z}\rho\sigma_{z}$, with $\gamma\in[0,1]$; an approximation valid when the sensing time is short.

We use as probe states the family of 3D-GHZ states

[TABLE]

which was shown to present Heisenberg scaling in the noiseless case Baumgratz and Datta (2016); $\ket{\psi_{k}^{\pm}}$ are the eigenvectors corresponding to the $\pm 1$ eigenvalues of the $k^{\text{th}}$ Pauli matrix and $\mathcal{N}$ is the normalisation. The final state for which we compute the bound is $\rho_{\bm{\varphi}}=\mathcal{E}_{\gamma}^{\otimes M}[U_{\bm{\varphi}}|\psi_{M}^{\text{3D-GHZ}}\rangle\langle\psi_{M}^{\text{3D-GHZ}}|U_{\bm{\varphi}}^{\dagger}]$; for the numerical results we choose equal parameter values $\varphi_{i}=1\,\forall i$ and $W=\mathbb{1}_{3}$.

In Fig. (2) we show the non-trivial relationship between the HCRB, the SLD-CRB and incompatibility for this quantum statistical model, as a function of the dephasing strength $\gamma$. We quantify the incompatibility of the model with the magnitude of the matrix $D_{\bm{\theta}}$, capturing the violation of the weak commutativity condition (6); in particular, we use the Frobenius norm $\|D_{\bm{\varphi}}\|_{\text{F}}^{2}=\sum_{ij}|(D_{\bm{\varphi}})_{ij}|^{2}$. Panel (a) shows that the relative difference $1-C^{\text{S }}/C^{\text{H}}$ is monotonically decreasing for 2 and 3 qubits, while it has a non-monotonic behaviour for 5 or more qubits. Panel (b) shows that this behaviour is not always reflected at the level of incompatibility; this is remarkably different from the simple monotonic relationship found for 2-parameter pure state models Matsumoto (2002, 1997). Furthermore, whilst the matrices $D_{\bm{\theta}}$ have a comparable magnitude for different number of qubits, the relative differences do not, e.g., being around 0.2% for 5 qubits and around 30% for 2 qubits.

Our SDP formulation grants us a previously inaccessible ease in the evaluation of the HCRB. In turn, this enables us to get these glimpses into the non-commutative information geometry of 3-parameter mixed state problems of non-trivial dimension. From this example we see that the pair of matrices $J^{\text{S}}$ and $D_{\bm{\theta}}$ are not sufficient for a complete description of a given state’s performance for multi-parameter estimation.

Finally, we concentrate on the noiseless ($\gamma=0$) case with two qubits, noting that a single qubit does not allow to estimate all the components of $\bm{\varphi}$ Baumgratz and Datta (2016). In Fig. (2) we see that for $M=2$ the SLD bound is considerably looser than the HCRB, with a relative difference around 30% for $\gamma=0$. On the contrary, we conjecture that the HCRB is attainable with single-copy projective measurements. We base this on the numerical equality between the HCRB and a numerical minimization of the classical scalar CRB over all 2-qubit projective measurements. For 5000 random initial states with parameter values taken from five sets, the relative difference between the two quantities was always found to be smaller than $10^{-4}$ (see Sec. IV.A of SM for details). While for pure states the HCRB is always attainable with single-copy measurements, the optimal POVM needs not be projective Matsumoto (2002), making it harder to implement experimentally. This finding shows that optimal protocols for 3D-magnetometry with two qubits may be not too far from experimental reach.

Conclusions—We have shown how to evaluate the HCRB by solving a SDP, making it more easily accessible than previously believed. This enabled us to study two examples—optical interferometry and 3D magnetometry—and gather numerical evidence that the HCRB is attainable by single-copy projective measurements, whereas the SLD bound is not. These findings suggest that there may be further unstudied cases where the HCRB is easier to attain than naively expected. They also illustrate the potential of our formulation to enable new discoveries in multi-parameter quantum estimation, which should aid a deeper quantitative understanding of quantum measurements more generally.