Bures geodesics and quantum metrology

TL;DR

We address the problem of characterizing geodesics on the quantum state space endowed with the Bures metric and their role in quantum metrology. We derive explicit geodesics via purification and the Riemannian submersion formalism, showing their projection corresponds to physical system–ancilla evolutions generated by a geodesic Hamiltonian H_g_V. We prove that when the unknown parameter x is a phase, geodesic evolutions saturate the quantum Cramer-Rao bound without information loss to the ancilla, achieving Heisenberg scaling in the number of probes. These results provide a geometric framework for open-system metrology and indicate practical routes to circuit implementations and reservoir engineering for enhanced precision.

Abstract

We study the geodesics on the manifold of mixed quantum states for the Bures metric. It is shown that these geodesics correspond to physical non-Markovian evolutions of the system coupled to an ancilla. Furthermore, we argue that geodesics lead to optimal precision in single-parameter estimation in quantum metrology. More precisely, if the unknown parameter $x$ is a phase shift proportional to the time parametrizing the geodesic, the estimation error obtained by processing the data of measurements on the system is equal to the smallest error that can be achieved from joint detections on the system and ancilla, meaning that there is no information loss on this parameter in the ancilla. This error can saturate the Heisenberg bound. Reciprocally, assuming that the system-ancilla output and input states are related by a unitary $e^{-i x H}$ with $H$ a $x$-independent Hamiltonian, we show that if the error obtained from measurements on the system is equal to the minimal error obtained from joint measurements on the system and ancilla then the system evolution is given by a geodesic. In such a case, the measurement on the system bringing most information on $x$ is $x$-independent and can be determined in terms of the intersections of the geodesic with the boundary of quantum states. These results show that geodesic evolutions are of interest for high-precision detections in systems coupled to an ancilla in the absence of measurements on the ancilla.

Figures (3)

• Figure 1: The manifold of quantum states ${\cal E}={\cal E}_{\cal H}$ is the projection $\pi({\cal S})$ of the manifold ${\cal S}$ of pure states on an enlarged Hilbert space ${\cal H}\otimes {\cal H}_{\sf A}$, where $\pi$ is the partial trace over ${\cal H}_{\sf A}$. The horizontal subspaces at $| \Psi \rangle$ and $| \Phi_V \rangle$ are orthogonal to the orbits $\pi^{-1} (\rho)$ and $\pi^{-1} (\sigma)$ (red lines). A geodesic in ${\cal S}$ joining $| \Psi \rangle$ to $| \Phi_V \rangle$ (plain black curve) with a horizontal initial tangent vector $| {\dot{\Psi}}^{\rm{h}} \rangle$ projects out to a geodesic $\gamma_{{\mathrm{g}},V}$ on ${\mathcal{E}}$ (green plain curve). In contrast, if the geodesic in ${\cal S}$ (blue dashed curve) has a non horizontal initial tangent vector, its projection (green dashed line) is not a geodesic on ${\cal E}$. The differential ${\rm{d}} \pi$ maps the horizontal tangent vector $| {\dot{\Psi}}^{\rm{h}} \rangle$ to a tangent vector $\dot{\rho}$ of $\gamma_{{\mathrm{g}},V}$ having the same length $\| \dot{\rho}\|=\| {\dot{\Psi}}^{\rm{h}}\|$. A non horizontal vector $| {\dot{\Psi}} \rangle$ is mapped by ${\rm{d}} \pi$ to a vector $\dot{\rho}$ with a smaller length, given by $\| \dot{\rho}\|^2 = \| \dot{\Psi}\|^2 - \| {\dot{\Psi}}^{\rm{v}} \|^2$, where $| {\dot{\Psi}}^{\rm{v}} \rangle$ is the vertical component of $| {\dot{\Psi}} \rangle$ (Pythagorean theorem).
• Figure 2: Quantum circuit implementing the geodesic evolution. (a): General circuit for a $d$-qubit system; $R_y (\tau)= e^{-{\rm{i}} \tau \sigma_y}$ is a rotation around the $y$-axis and $U_{{\sf S} {\sf A}}$ a system-ancilla entangling unitary satisfying (\ref{['eq-def_entangling_unitary']}). (b), (c): examples of quantum circuits for $U_{{\sf S} {\sf A}}$; the unitaries $W$, $U_{\sf S}$, and $U_{\sf A}$ are such that $U_i | 0 \rangle_i = \sum_k \sqrt{p_k} | k \rangle_{i}$, $i= {\sf S},{\sf A}$, $U_{\sf S} | 1 \rangle_{\sf S} = \sum_k \alpha_k | k \rangle$, and $W | k \rangle_{\sf S} = | w_k \rangle_{\sf S}$, $k=0,\ldots,2^d-1$, where $\sqrt{p_k}$ and $\{ | w_k \rangle_{\sf S}\}$ are the coefficients and orthonormal basis in the Schmidt decomposition of $| \Psi \rangle$ and $\alpha_k \in {\mathbb{R}}\setminus\{0\}$, $\sum_k \alpha_k \sqrt{p_k}=0$.
• Figure 3: Quantum circuit implementing a geodesic transformation for the estimation of a phase shift $x$, with an error reaching the Heisenberg scaling. The single qubit unitaries $R_y$, $U_{\sf A}$ and $W$ are as in Fig. \ref{['fig-Q_circuit_geodesic_evol']} with $d=1$. In spite of the presence of the C-NOT gates entangling the probe qubits with the ancilla qubits, the maximal information on $x$ can be recovered from measurements on the probe qubits only, with a minimal error $(\Delta x)_{\rm QCRB} = (2 N \epsilon\sqrt{N_{\rm meas}})^{-1}$.

Theorems & Definitions (7)

• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6