Quantum metrological gain

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Let us consider a unitary dynamics with a parameter ${\displaystyle \theta }$ from initial state ${\displaystyle \varrho _{0}}$,

${\displaystyle \varrho (\theta )=\exp(-iA\theta )\varrho _{0}\exp(+iA\theta ),}$

the quantum Fisher information ${\displaystyle F_{\rm {Q}}}$ constrains the achievable precision in statistical estimation of the parameter ${\displaystyle \theta }$ via the quantum Cramér–Rao bound as

${\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,A]}},}$

where ${\displaystyle m}$ 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 ${\displaystyle N}$ spin-1/2 particles^[1]

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N}$

holds for separable states, where ${\displaystyle F_{\rm {Q}}}$ is the quantum Fisher information,

${\displaystyle J_{z}=\sum _{n=1}^{N}j_{z}^{(n)},}$

and ${\displaystyle j_{z}^{(n)}}$ is a single particle angular momentum component. Thus, the metrological gain can be characterize by

${\displaystyle {\frac {F_{\rm {Q}}[\varrho ,J_{z}]}{N}}.}$

The maximum for general quantum states is given by

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N^{2}.}$

Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology. Moreover, for quantum states with an entanglement depth ${\displaystyle k}$,

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq sk^{2}+r^{2}}$

holds, where ${\displaystyle s=\lfloor N/k\rfloor }$ is the largest integer smaller than or equal to ${\displaystyle N/k,}$ and ${\displaystyle r=N-sk}$ is the remainder from dividing ${\displaystyle N}$ by ${\displaystyle k}$. Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.^[2][3] It is possible to obtain a weaker but simpler bound ^[4]

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq Nk.}$

Hence, a lower bound on the entanglement depth is obtained as

${\displaystyle {\frac {F_{\rm {Q}}[\varrho ,J_{z}]}{N}}\leq k.}$

These ideas have been used to measure the metrological gain in many physical systems such as cold atoms and photons.^[5] Metrological gains up to 100 are reported in experiments with cold atomic ensembles.^[6]

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The situation for qudits with a dimension larger than ${\displaystyle d=2}$ 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^[7][8]

${\displaystyle g_{\mathcal {H}}(\varrho )={\frac {{\mathcal {F}}_{Q}[\varrho ,{\mathcal {H}}]}{{\mathcal {F}}_{Q}^{({\rm {sep}})}({\mathcal {H}})}},}$

where the Hamiltonian is

${\displaystyle {\mathcal {H}}=h_{1}+h_{2}+...+h_{N},}$

and ${\displaystyle h_{n}}$ acts on the nth spin. The maximum of the quantum Fisher information for separable states is given as^{[9] [10] [7]}

${\displaystyle {\mathcal {F}}_{Q}^{({\rm {sep}})}({\mathcal {H}})=\sum _{n=1}^{N}[\lambda _{\max }(h_{n})-\lambda _{\min }(h_{n})]^{2},}$

where ${\displaystyle \lambda _{\max }(X)}$ and ${\displaystyle \lambda _{\min }(X)}$ denote the maximum and minimum eigenvalues of ${\displaystyle X,}$ respectively.

We also define the metrological gain optimized over all local Hamiltonians as

${\displaystyle g(\varrho )=\max _{\mathcal {H}}g_{\mathcal {H}}(\varrho ).}$

The case of qubits is special. In this case, if the local Hamitlonians are chosen to be

${\displaystyle h_{n}=\sum _{l=x,y,z}c_{l,n}\sigma _{l},}$

where ${\displaystyle c_{l,n}}$ are real numbers, and ${\displaystyle |{\vec {c}}_{n}|=1,}$ then

${\displaystyle {\mathcal {F}}_{Q}^{({\rm {sep}})}({\mathcal {H}})=4N}$,

independently from the concrete values of ${\displaystyle c_{l,n}}$.^[11] 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.

The quantum metrological gain is directly connected to multiparticle entanglement in many-particle systems.

If the gain larger than one

${\displaystyle g(\varrho )>1,}$

then the state is entangled, and it is more useful metrologically than separable states. In short, we call such states metrologically useful. If ${\displaystyle h_{n}}$ all have identical lowest and highest eigenvalues, then

${\displaystyle g(\varrho )>k-1}$

implies metrologically useful ${\displaystyle k}$-partite entanglement. If for the gain^[8]

${\displaystyle g(\varrho )>N-1}$

holds, then the state has metrologically useful genuine multipartite entanglement.^[7] In general, for quantum states ${\displaystyle g(\varrho )\leq N}$ holds.

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The metrological gain cannot decrease if we add an ancilla to a subsystem or we provide an additional copy of the state.^[7][8] Thus,

${\displaystyle g(\varrho _{AB}\otimes \varrho _{a})\geq g(\varrho _{AB})}$

holds, where ${\displaystyle A}$ and ${\displaystyle a}$ belong to the same party. Moreover,

${\displaystyle g(\varrho _{AB}\otimes \varrho _{A'B'})\geq g(\varrho _{AB})}$

also holds, where ${\displaystyle A}$ and ${\displaystyle A'}$ belong to the same party, and ${\displaystyle B}$ and ${\displaystyle B'}$ also belong to the same party.

It has been proven that for quantum states ${\displaystyle \varrho _{1}}$ and ${\displaystyle \varrho _{2}}$, and for ${\displaystyle 0\leq p\leq 1}$

${\displaystyle g(p\varrho _{1}+(1-p)\varrho _{2})\leq p_{1}g(\varrho _{1})+p_{2}g(\varrho _{2})}$

holds, thus the metrological gain is convex in the quantum state.^[7][8]

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The optimization of the metrological gain over local Hamiltonians is a hard task, since it is defined as a fraction of two quantities that both depend on the Hamiltonian. Moreover, in the denominator, the quantum Fisher information is a convex function of the Hamiltonian. Maximizing a convex function over a convex set is a difficult task.

There are still efficient methods to determine the metrological gain based on an optimization over local Hamiltonians. They are based on see-saw methods that iterate two steps alternatively.^[7] A possible see-saw method is based on

${\displaystyle \max _{H}F_{Q}[\varrho ,H]=\max _{H,M}{\frac {|\langle i[H,M]\rangle _{\varrho }|^{2}}{(\Delta M)_{\varrho }^{2}}}.}$

In the right-hand side, there is the error-propagation formula. Here, instead of a direct optmization over local Hamiltonians ${\displaystyle H,}$ we can alternatingly optimize of over ${\displaystyle H}$ and ${\displaystyle M.}$

An alternative is optimizing the Fisher matrix element using ^[12]

${\displaystyle \max _{H}F_{Q}[\varrho ,H]=\max _{H_{1},H_{2}}F_{Q}[\varrho ,H_{1}H_{2}].}$

Here, instead of a direct optimization over local Hamiltonians ${\displaystyle H,}$ we can alternatingly optimize of over ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}.}$ The Fisher matrix element is given as

${\displaystyle F_{\rm {Q}}[\varrho ,H_{1},H_{2}]=2\sum _{k,l}{\frac {(\lambda _{k}-\lambda _{l})^{2}}{(\lambda _{k}+\lambda _{l})}}\langle k\vert H_{1}\vert l\rangle \langle l\vert H_{2}\vert k\rangle .}$

Clearly, we have

${\displaystyle F_{\rm {Q}}[\varrho ,H,H]=F_{\rm {Q}}[\varrho ,H].}$