Draft:Quantum metrological gain

Thank you! Now I refer also to Trenyi et al. 2024 in that line. I changed the wording. The notion has already been defined in Toth el al. 2020. Quantum1956

Thank you! Is this OK now? Quantum1956

In quantum metrology in a multiparticle system, the quantum metrological gain for a quantum state is defined as the sensitivity of phase estimation achieved by that state divided by the maximal sensitivity achieved by separable states, i.e., states without quantum entanglement. In practice, the best separable state is the trivial fully polarized state, in which all spins point into the same direction. If the metrological gain is larger than one then the quantum state is more useful for making precise measurements than separable states. Clearly, in this case the quantum state is also entangled.

Background
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 experiements.

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


 * $$\frac{F_{\rm Q}[\varrho, J_z]}{N} \le k. $$

The situation for qudits with a dimension is larger than $$d=2$$ is more complicated.

Mathematical definition for a system of qudits
In general, 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

$$g_{\mathcal H}(\varrho)=\frac{\mathcal F_Q[\varrho,{\mathcal H}]}{\mathcal F_Q^{({\rm sep})}(\mathcal H)},$$

where the Hamiltonian is

$$\mathcal H=h_1+h_2+...+h_N,$$

and $$h_n$$ acts on the nth spin. The maximum of the quantum Fisher information for separable states is given as

$$ \mathcal F_Q^{({\rm sep})}(\mathcal H)=\sum_{n=1}^N [ \lambda_{\max}(h_n)-\lambda_{\min}(h_n) ]^2, $$

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

$$\mathcal F_Q^{({\rm sep})}(\mathcal H)=4N$$,

independtly from the concrete values of $$ c_{l,n}$$. Thus, in the case of qubits, the optmization of the gain over the local Hamiltonian can be simpler. For qudits with a dimension larger than 2, the optmization is more complicated.

Relation to quantum entanglement
If the gain larger than one

$$ g(\varrho)>1, $$ then the state is entangled, and its 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 alternatingly.