Review waiting, please be patient.
This may take 4 months or more, since drafts are reviewed in no specific order. There are 3,277 pending submissions waiting for review.
Where to get help
How to improve a draft
You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Editor resources
Reviewer tools
|
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.
The metrological gain is, in general, the gain in sensitivity of a quantum state compared to a product state. [1] Metrological gains up to 100 are reported in experiements. [2]
Let us consider a unitary dynamics with a parameter from initial state ,
the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter via the quantum Cramér–Rao bound as
where 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 spin-1/2 particles [3]
holds for separable states, where is the quantum Fisher information,
and is a single particle angular momentum component. Thus, the metrological gain can be characterize by
The maximum for general quantum states is given by
Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology. Moreover, for quantum states with an entanglement depth ,
holds, where is the largest integer smaller than or equal to and is the remainder from dividing by . Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation. [4] [5] It is possible to obtain a weaker but simpler bound [6]
Hence, a lower bound on the entanglement depth is obtained as
The situation for qudits with a dimension is larger than is more complicated.
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 [7] [8]
where the Hamiltonian is
and acts on the nth spin. The maximum of the quantum Fisher information for separable states is given as [9] [10] [7]
where and denote the maximum and minimum eigenvalues of respectively.
We also define the metrological gain optimized over all local Hamiltonians as
The case of qubits is special. In this case, if the local Hamitlonians are chosen to be
where are real numbers, and then
,
independtly from the concrete values of . [11] 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.
If the gain larger than one
then the state is entangled, and its is more useful metrologically than separable states. In short, we call such states metrologically useful. If all have identical lowest and highest eigenvalues, then
implies metrologically useful -partite entanglement. If for the gain [8]
holds, then the state has metrologically useful genuine multipartite entanglement. [7] In general, for quantum states holds.
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 is convex in the quantum state. [7] [8]
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. [7]
Category:Quantum information science Category:Quantum optics
Review waiting, please be patient.
This may take 4 months or more, since drafts are reviewed in no specific order. There are 3,277 pending submissions waiting for review.
Where to get help
How to improve a draft
You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Editor resources
Reviewer tools
|
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.
The metrological gain is, in general, the gain in sensitivity of a quantum state compared to a product state. [1] Metrological gains up to 100 are reported in experiements. [2]
Let us consider a unitary dynamics with a parameter from initial state ,
the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter via the quantum Cramér–Rao bound as
where 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 spin-1/2 particles [3]
holds for separable states, where is the quantum Fisher information,
and is a single particle angular momentum component. Thus, the metrological gain can be characterize by
The maximum for general quantum states is given by
Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology. Moreover, for quantum states with an entanglement depth ,
holds, where is the largest integer smaller than or equal to and is the remainder from dividing by . Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation. [4] [5] It is possible to obtain a weaker but simpler bound [6]
Hence, a lower bound on the entanglement depth is obtained as
The situation for qudits with a dimension is larger than is more complicated.
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 [7] [8]
where the Hamiltonian is
and acts on the nth spin. The maximum of the quantum Fisher information for separable states is given as [9] [10] [7]
where and denote the maximum and minimum eigenvalues of respectively.
We also define the metrological gain optimized over all local Hamiltonians as
The case of qubits is special. In this case, if the local Hamitlonians are chosen to be
where are real numbers, and then
,
independtly from the concrete values of . [11] 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.
If the gain larger than one
then the state is entangled, and its is more useful metrologically than separable states. In short, we call such states metrologically useful. If all have identical lowest and highest eigenvalues, then
implies metrologically useful -partite entanglement. If for the gain [8]
holds, then the state has metrologically useful genuine multipartite entanglement. [7] In general, for quantum states holds.
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 is convex in the quantum state. [7] [8]
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. [7]
Category:Quantum information science Category:Quantum optics