Quantum metrological gain

The metrological gain is, in general, the gain in sensitivity of a quantum state compared to a product state. Metrological gains up to 100 are reported in experiments. Let us consider a unitary dynamics with a parameter \theta from initial state \varrho_0, :\varrho(\theta)=\exp(-iA\theta)\varrho_0\exp(+iA\theta), the quantum Fisher information F_{\rm Q} constrains the achievable precision in statistical estimation of the parameter \theta via the quantum Cramér–Rao bound as :(\Delta \theta)^2 \ge \frac 1 {m F_{\rm Q}[\varrho,A]}, where m is the number of independent repetitions. For the formula, one can see that the larger the quantum Fisher information, the smaller can be the uncertainty of the parameter estimation. For a multiparticle system of N spin-1/2 particles :F_{\rm Q}[\varrho, J_z] \le N holds for separable states, where F_{\rm Q} is the quantum Fisher information, : J_z=\sum_{n=1}^N j_z^{(n)}, and j_z^{(n)} is a single particle angular momentum component. Thus, the metrological gain can be characterize by :\frac{F_{\rm Q}[\varrho, J_z]}{N}. The maximum for general quantum states is given by :F_{\rm Q}[\varrho, J_z] \le N^2. Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology. Moreover, for quantum states with an entanglement depth k, :F_{\rm Q}[\varrho, J_z] \le sk^2 + r^{2} holds, where s=\lfloor N/k \rfloor is the largest integer smaller than or equal to N/k, and r=N-sk is the remainder from dividing N by k. Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation. It is possible to obtain a weaker but simpler bound :F_{\rm Q}[\varrho, J_z] \le Nk. Hence, a lower bound on the entanglement depth is obtained as :\frac{F_{\rm Q}[\varrho, J_z]}{N} \le k. == Mathematical definition for a system of qudits ==

The situation for qudits with a dimension larger than d=2 is more complicated. In this more general case, the metrological gain for a given Hamiltonian is defined as the ratio of the quantum Fisher information of a state and the maximum of the quantum Fisher information for the same Hamiltonian for separable states \mathcal F_Q^{({\rm sep})}(\mathcal H)=\sum_{n=1}^N [ \lambda_{\max}(h_n)-\lambda_{\min}(h_n) ]^2, where \lambda_{\max}(X) and \lambda_{\min}(X) denote the maximum and minimum eigenvalues of X, respectively. We also define the metrological gain optimized over all local Hamiltonians as g(\varrho)=\max_{\mathcal H}g_{\mathcal H}(\varrho). The case of qubits is special. In this case, if the local Hamitlonians are chosen to be h_n=\sum_{l=x,y,z} c_{l,n}\sigma_l, where c_{l,n} are real numbers, and |\vec c_n|=1, then \mathcal F_Q^{({\rm sep})}(\mathcal H)=4N, independently from the concrete values of c_{l,n}. Thus, in the case of qubits, the optimization of the gain over the local Hamiltonian can be simpler. For qudits with a dimension larger than 2, the optimization is more complicated. == Relation to quantum entanglement ==

If the gain larger than one g(\varrho)>1, then the state is entangled, and it is more useful metrologically than separable states. In short, we call such states metrologically useful. If h_n all have identical lowest and highest eigenvalues, then g(\varrho)>k-1 implies metrologically useful k-partite entanglement. If for the gain g(\varrho)>N-1 holds, then the state has metrologically useful genuine multipartite entanglement. In general, for quantum states g(\varrho)\le N holds. == Properties of the metrological gain ==

The metrological gain cannot increase if we add an ancilla to a subsystem or we provide an additional copy of the state. The metrological gain g(\varrho) is convex in the quantum state. == Numerical determination of the gain ==

There are efficient methods to determine the metrological gain via an optimization over local Hamiltonians. They are based on a see-saw method that iterates two steps alternatively. == References ==