The precision limit can be calculated using the Fisher information, defined as F = ∑— i Pτ( x i)( ∂P τ( x i)
∂τ) 2, where x i are the possible measurement outcomes of the experiment. Given a state and a parameter, there are a number of different measurements that can be implemented to infer the value of this parameter, leading to different probabilities P τ( x i). Using F, we see that the precision associated with the measurement of the parameter
Indeed, if we consider a system composed of K qubits, we have that each qubit can be prepared not only in states | 0 > and | 1 >, the analogous of the classical bits 0 and 1, but also in states as | Ψ > = 1— √ – 2(| 0 > + | 1 >), which
LIGHT AND COLOR FOCUS examined in greater details later in this article. This initial state serves as a probe, incorporating both modal and statistical properties.
In the“ Evolution” step, information about the parameter to be measured( denoted τ) is imprinted onto the state. For instance, in the case of time estimation, this can occur through the free evolution of the field, which depends on the parameter( time) and is governed by a Hamiltonian( specifically, the free-field Hamiltonian). Next, a measurement is performed, yielding an outcome x with probability P τ( x), which depends on the parameter to be estimated τ. From these outcomes, we define an estimator— a function that maps measurement results to the estimated parameter. The average value of this estimator over many experimental runs provides an estimate of the parameter’ s value, with a certain precision. Naturally, the higher the precision, the more accurate the measurement.
The precision limit can be calculated using the Fisher information, defined as F = ∑— i Pτ( x i)( ∂P τ( x i)
M
1—
∂τ) 2, where x i are the possible measurement outcomes of the experiment. Given a state and a parameter, there are a number of different measurements that can be implemented to infer the value of this parameter, leading to different probabilities P τ( x i). Using F, we see that the precision associated with the measurement of the parameter
1 τ is limited by δτ ≥
√—, where N is
NF the number of repetitions of the experiment. This precision limit is known as the Cramér-Rao bound.
Increasing the precision of a given parameter estimation protocol involves finding measurement strategies such that F is as large as possible. But how large can F be? By optimizing the precision over all possible measurement strategies, we can define the quantum Fisher information( QFI): = max F, leading to the quantum precision limit, ∆τ ≥ 1
√— N, called the quantum Cramér-Rao bound. Hence, the best possible precision occurs when F =. And what limits the QFI? For pure states undergoing Hamiltonian evolution( see Box 1), the QFI takes a relatively simple form: = 4∆ 2 Ĥ, where Ĥ is the Hamiltonian that generates the parameter dependent evolution into the state considered as a probe. Hence, given a Hamiltonian, the maximal achievable precision depends entirely on the choice of probe state. This is where quantum resources may offer advantages over classical ones.
Indeed, if we consider a system composed of K qubits, we have that each qubit can be prepared not only in states | 0 > and | 1 >, the analogous of the classical bits 0 and 1, but also in states as | Ψ > = 1— √ – 2(| 0 > + | 1 >), which
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