Generalized Limits for Single-Parameter Quantum Estimation

We develop generalized bounds for quantum single-parameter estimation problems for which the coupling to the parameter is described by intrinsic multisystem interactions. For a Hamiltonian with $k$-system parameter-sensitive terms, the quantum limit scales as $1/N^k$, where $N$ is the number of systems. These quantum limits remain valid when the Hamiltonian is augmented by any parameter-independent interaction among the systems and when adaptive measurements via parameter-independent coupling to ancillas are allowed.

Figure 1

Quantum-circuit diagrams illustrating the principle of deferred measurement. In the circuit on the left, a measurement $M$ on the lower ancilla yields result $a$; this result controls a subsequent unitary $U_a$, applied to the probe and the upper ancilla, and determines a conditional measurement $M_a$ on the upper ancilla, which has result $b$. The two measurement results then control a unitary $U_{a,b}$ applied to the probe and a conditional measurement $M_{a,b}$ on the probe. The left-hand circuit is equivalent to the circuit on the right, in which the controls are applied coherently (boxed gates $U_A$ and $U_{A,B}$), and the measurements, deferred to the end of the circuit, tell one which unitary was applied. Without loss of generality, we can assume the measurements are described by orthogonal projectors $P_a$, $P_{b|a}$, and $P_{c|a,b}$, because any generalized measurement can be modeled by a projection-valued measurement on an extended system. The unitary transformations in the left-hand circuit, $U_a$ and $U_{a,b}$, are evolution operators generated by the Hamiltonians $\hslash \gamma h_0+{\tilde{H}}_a(t)$ and $\hslash \gamma h_0+{\tilde{H}}_{a,b}(t)$, whereas the corresponding coherent controlled unitaries in the circuit on the right, $U_A$ and $U_{A,B}$, are generated by the Hamiltonians $\hslash \gamma h_0+\sum _a{\tilde{H}}_a(t)\otimes P_a$ and $\hslash \gamma h_0+\sum _{a,b}{\tilde{H}}_{a,b}(t)\otimes P_{b|a}{P}_a$. It is easy to verify from the evolution equations that the controlled unitaries in the right-hand circuit are given by $U_A=\sum _a{U}_a\otimes P_a$ and $U_{A,B}=\sum _{a,b}{U}_{a,b}\otimes P_{b|a}{P}_a$. Thus the principle of deferred measurement can be rendered algebraically in the following way: if we use the left-hand circuit, the probability for obtaining results $a$, $b$, and $c$ takes the form $\mathrm{tr}(C_{a,b,c}{\rho }_0{C}_{a,b,c}^{†})$, where ${\rho }_0$ is the initial state of the probe and ancillas and $C_{a,b,c}=P_{c|a,b}{U}_{a,b}{P}_{b|a}{U}_a{P}_{a}U$; pulling the measurement projectors to the left in $C_{a,b,c}$ changes the unitaries to the corresponding coherent controlled operations, i.e., $C_{a,b,c}=P_{c|a,b}{P}_{b|a}{P}_a{U}_{A,B}{U}_{A}U$, which gives the form of the probability obtained from the right-hand circuit.