Experimental Advances in Phase Estimation with Photonic Quantum States
Abstract
1. Introduction
2. Theoretical Tools for Parameter Estimation
- (i)
- preparation of an initial quantum state , the so-called probe state;
- (ii)
- encoding of the parameter to be estimated into a quantum state by means of a quantum operation applied to the initial state;
- (iii)
- performing a measurement on the final state that is, in general, given by a positive-operator valued measure (POVM) with possible outcomes ;
- (iv)
- obtaining an estimator from the probabilities of the outcomes that are, according to the Born rule, .
2.1. Maximum Likelihood Estimation
- Consistency: the estimator converges (in probability) to the true value of the parameter;
- Unbiasedness: the mean value of the estimator’s distribution is equal to the true value of the parameter;
- Efficiency: the estimator has the smallest possible variance among all unbiased estimators.
2.2. Cramér–Rao Bound and Fisher Information
2.3. Quantum Cramér–Rao Bound and Quantum Fisher Information
- , where the equality holds if and only if is a real number :
- where we used the Cauchy–Schwarz inequality , taking and . The equality holds if and only if
3. Phase Estimation
3.1. Interferometric Setups
3.2. Optimized Probe States
3.2.1. Coherent States
3.2.2. Squeezed States
3.2.3. Twin Fock States
3.2.4. N00N States
4. Photonic Schemes for Phase Estimation: State of the Art
5. Conclusions and Perspectives
Funding
Data Availability Statement
Conflicts of Interest
References
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Knoll, L.T.; Magnoni, A.G.; Larotonda, M.A. Experimental Advances in Phase Estimation with Photonic Quantum States. Entropy 2025, 27, 712. https://doi.org/10.3390/e27070712
Knoll LT, Magnoni AG, Larotonda MA. Experimental Advances in Phase Estimation with Photonic Quantum States. Entropy. 2025; 27(7):712. https://doi.org/10.3390/e27070712
Chicago/Turabian StyleKnoll, Laura T., Agustina G. Magnoni, and Miguel A. Larotonda. 2025. "Experimental Advances in Phase Estimation with Photonic Quantum States" Entropy 27, no. 7: 712. https://doi.org/10.3390/e27070712
APA StyleKnoll, L. T., Magnoni, A. G., & Larotonda, M. A. (2025). Experimental Advances in Phase Estimation with Photonic Quantum States. Entropy, 27(7), 712. https://doi.org/10.3390/e27070712