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Quantum parameter estimation and state discrimination in continuous variable quantum optics

Mark Bradshaw

Published Jan 1, 2021• 0 citations
Publisher

Abstract

Exemplified by the Heisenberg uncertainty principle, quantum mechanics imposes a fundamental limit to the precision we can measure complementary properties of an object, for example position and momentum. Similarly, quantum mechanics tells us there is a limit to which we can discriminate between two quantum states that are nonorthogonal. Therefore, it is an interesting question to ask, given the laws of quantum mechanics, what exactly is the best measurement outcome we can achieve? This thesis examines various problems related to quantum state discrimination and quantum parameter estimation of continuous variable quantum states. We first examine quantum illumination, which can be regarded as a quantum state discrimination problem, where a quantum probe state is used to determine the presence or absence of an object in a noisy environment. If we use a probe that is quantum entangled with another probe, we can obtain a better performance outcome than is possible by using a probe without entanglement. We show that the source of enhancement is due to quantum discord. We also investigate what is the best probe state when entanglement is not used (conventional illumination) and when entanglement is used (quantum illumination). We analytically prove the optimal probe for various cases, and also provide some numerical results. We also investigate the optimal estimation of parameters that parametrise a quantum state. We provide methods of calculating the Holevo Cramer Rao bound, which is a useful quantity for quantum parameter estimation. The Holevo Cramer Rao bound not only is a good lower bound to the sum of the mean squared error, in some cases it can provide a measurement that obtains it. We consider the example of the estimation of conjugate quadrature parameters of a displacement experienced by one mode of a two-mode EPR state. We not only prove that the near-optimal measurement scheme proposed by previous authors is optimal for high squeezing, we find a different measurement that is optimal for low squeezing. Finally, we investigate using a probabilistic amplifier for quantum state discrimination and parameter estimation for coherent states. We find that the probabilistic amplifier results in improvement in discrimination and parameter estimation of coherent states under certain conditions such as when there is a high cost associated with measuring a sample.

DOI
10.25911/ffsz-9232

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Exemplified by the Heisenberg uncertainty principle, quantum mechanics imposes a fundamental limit to the precision we can measure complementary properties of an object, for example position and momentum. Similarly, quantum mechanics tells us there is a limit to which we can discriminate between two quantum states that are nonorthogonal. Therefore, it is an interesting question to ask, given the laws of quantum mechanics, what exactly is the best measurement outcome we can achieve? This thesis examines various problems related to quantum state discrimination and quantum parameter estimation of continuous variable quantum states. We first examine quantum illumination, which can be regarded as a quantum state discrimination problem, where a quantum probe state is used to determine the presence or absence of an object in a noisy environment. If we use a probe that is quantum entangled with another probe, we can obtain a better performance outcome than is possible by using a probe without entanglement. We show that the source of enhancement is due to quantum discord. We also investigate what is the best probe state when entanglement is not used (conventional illumination) and when entanglement is used (quantum illumination). We analytically prove the optimal probe for various cases, and also provide some numerical results. We also investigate the optimal estimation of parameters that parametrise a quantum state. We provide methods of calculating the Holevo Cramer Rao bound, which is a useful quantity for quantum parameter estimation. The Holevo Cramer Rao bound not only is a good lower bound to the sum of the mean squared error, in some cases it can provide a measurement that obtains it. We consider the example of the estimation of conjugate quadrature parameters of a displacement experienced by one mode of a two-mode EPR state. We not only prove that the near-optimal measurement scheme proposed by previous authors is optimal for high squeezing, we find a different measurement that is optimal for low squeezing. Finally, we investigate using a probabilistic amplifier for quantum state discrimination and parameter estimation for coherent states. We find that the probabilistic amplifier results in improvement in discrimination and parameter estimation of coherent states under certain conditions such as when there is a high cost associated with measuring a sample.
In this PhD thesis we address the problem of characterizing quantum states and parameters of systems that are of particular interest for quantum technologies. In the first part we consider continuous variable systems and in particular Gaussian states; we address the estimation of quantities characterizing single-mode Gaussian states as the displacement and squeezing parameter and we study the improvement in the parameter estimation by introducing a Kerr nonlinearity. Moreover, we address the discrimination of noisy channels by means of Gaussian states as probe states considering two problems: the detection of a lossy channel against the alternative hypothesis of an ideal lossless channel and the discrimination of two Gaussian noisy channels. In the last part of the thesis, we consider the one dimensional quantum Ising model in a transverse magnetic field. We exploit the recent results about the geometric approach to quantum phase transitions to derive the optimal estimator of the coupling constant of the model at zero and finite temperature in both cases of few spins and in the thermodynamic limit. We also analyze the effects of temperature and the scaling properties of the estimator of the coupling constant. Finally, we consider the discrimination problem for two ground states or two thermal states of the model.
Quantum illumination is a technique for detecting the presence of a target in a noisy environment by means of a quantum probe. We prove that the two-mode squeezed vacuum state is the optimal probe for quantum illumination in the scenario of asymmetric discrimination, where the goal is to minimize the decay rate of the probability of a false positive with a given probability of a false negative. Quantum illumination with two-mode squeezed vacuum states offers a 6 dB advantage in the error probability exponent compared to illumination with coherent states. Whether more advanced quantum illumination strategies may offer further improvements had been a longstanding open question. Our fundamental result proves that nothing can be gained by considering more exotic quantum states, such as, e.g., multimode entangled states. Our proof is based on a fundamental entropic inequality for the noisy quantum Gaussian attenuators. We also prove that without access to a quantum memory, the optimal probes for quantum illumination are the coherent states.
This thesis presents three studies in quantum-enhanced sensing and target detection. The first study explores covert target detection using optical or microwave probes, establishing quantum-mechanical limits on the error probabilities of entanglement-assisted detection methods while maintaining the sender's covertness. It identifies the minimal energy required to preserve covertness and reduce error probabilities, compares two-mode squeezed vacuum probes and coherent states against these limits, and extends the analysis to discriminating thermal loss channels and non-adversarial quantum illumination. The second study focuses on phase-insensitive optical amplifiers, determining the quantum limit on the precision of gain estimation using multimode probes possibly entangled with ancillary systems. It finds that the average photon number and the number of input modes are interchangeable resources for achieving optimal gain sensing precision, contrasting classical probes with quantum probes and highlighting the advantages of the latter, even with single-photon inputs and inefficient photodetection. It also provides a closed-form expression for the energy-constrained Bures distance between two amplifier channels. The third study compares three probe states -- coherent state, two-mode squeezed vacuum (TMSV), and single-photon entangled state (SPES) -- in quantum-enhanced target detection, assessing their performance under signal energy constraints relevant to covert radar sensing. SPES is uniquely positioned as a practical probe due to its non-classical properties after thermal loss and ease of generation. Numerical analysis shows that at low signal energies, the error exponent of TMSV aligns with SPES, indicating comparable detection capabilities and that SPES surpasses the best classical state-the coherent state-in accuracy for certain signal strengths.
Abstract We discuss the ultimate precision bounds on the multiparameter estimation of single- and two-mode pure Gaussian states. By leveraging on previous approaches that focused on the estimation of a complex displacement only, we derive the Holevo Cramér–Rao bound (HCRB) for both displacement and squeezing parameter characterizing single and two-mode squeezed states. In the single-mode scenario, we obtain an analytical bound and find that it degrades monotonically as the squeezing increases. Furthermore, we prove that heterodyne detection is nearly optimal in the large squeezing limit, but in general the optimal measurement must include non-Gaussian resources. On the other hand, in the two-mode setting, the HCRB improves as the squeezing parameter grows and we show that it can be attained using double-homodyne detection.
Quantum illumination is the task of determining the presence of an object in a noisy environment. We determine the optimal continuous-variable states for quantum illumination in the limit of zero object reflectivity. We prove that the optimal single-mode state is a coherent state, while the optimal two-mode state is the two-mode squeezed-vacuum state. We find that these probes are not optimal at nonzero reflectivity, but remain near optimal. This demonstrates the viability of the continuous-variable platform for an experimentally accessible, near optimal quantum illumination implementation.
The ability to perform high precision measurements underpins a plethora of applications. Several techniques for force sensing, phase estimation and discrimination, as well as surface reconstruction for complex features of three-dimensional samples, have been developed in recent years. The main aim of this thesis is to investigate metrology enhancements due to quantum resources (probes and measurements), by using quantum parameter estimation and channel discrimination techniques. The thesis focuses on two main scenarios. In the first one, we deal with three-dimensional superlocalisation. By using tools from multiparameter quantum metrology, we show that a simultaneous estimation of all three components of the separation between two incoherent point sources in the paraxial approximation is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the sub-wavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the point spread function of the imaging system, and quantitatively illustrated in detail for Gaussian beams. In this case, we show that a method of measuring the position of each photon at the imaging plane based on discrimination in terms of Hermite-Gaussian spatial modes reaches the quantum precision bound in the limit of infinitesimal separation. In the second part of the thesis, we investigate the role of quantum coherence as a resource for channel discrimination tasks. We consider a probe state of arbitrary dimension entering a black box, in which a phase shift is implemented, with the unknown phase randomly sampled from a finite set of predetermined possibilities. At the output, an optimal measurement is performed in order to guess which specific phase was applied in the process. We show that the presence of quantum coherence (superposition with respect to the eigenbasis of the generator of the phase shift) in the input probe directly determines an enhancement in the probability of success for this task, compared to the use of incoherent probes. We prove that such a quantum advantage is exactly quantified by the robustness of coherence, a full monotone with respect to the recently formulated resource theories of quantum coherence, whose properties and applications are developed and explored in detail.