Quantum tomography

We find that the Measurement Based Quantum Computing (MBQC) search algorithm on an unsorted list is not the same as Grover's search algorithm (GSA). More importantly, the MBQC search algorithm is exponentially faster than both GSA and its classical counterpart. We develop and describe the numerical formalism we utilize that confirms the MBQC exponential speedup over the GSA and classical search algorithms.

The development of the advanced Radio Frequency Timer of electrons is described. It is based on a helical deflector, which performs circular or elliptical sweeps of keV electrons, by means of 500 MHz radio frequency field. By converting a time distribution of incident electrons to a hit position distribution on a circle or ellipse, this device achieves extremely precise timing. Streak Cameras, based on similar principles, routinely operate in the ps and sub-ps time domain, but have substantial slow readout system. Here, we report a device, where the position sensor, consisting of microchannel plates and a delay-line anode, produces ~ns duration pulses which can be processed by using regular fast electronics. A photon sensor based on this technique, the Radio Frequency Photo-Multiplier Tube (RFPMT), has demonstrated a timing resolution of ~10 ps and a time stability of ~0.5 ps, FWHM. This makes the apparatus highly suited for Time Correlated Single Photon Counting which is widely used in optical microscopy and tomography of biological samples. The first application in lifetime measurements of quantum states of graphene, under construction at the A. I. Alikhanyan National Science Laboratory (AANL), is outlined. This is followed by a description of potential RFPMT applications in time-correlated Diffuse Optical Tomography, time-correlated Stimulated Emission Depletion microscopy, hybrid FRET/STED nanoscopy and Time-of-Flight Positron Emission Tomography.

Only a few classes of quantum algorithms are known which provide a speed-up over classical algorithms. However, these and any new quantum algorithms provide important motivation for the development of quantum computers. In this article new quantum algorithms are given which are based on quantum state tomography. These include an algorithm for the calculation of several quantum mechanical expectation values and an algorithm for the determination of polynomial factors. These quantum algorithms are important in their own right. However, it is remarkable that these quantum algorithms are immune to a large class of errors. We describe these algorithms and provide conditions for immunity.

We present two measures of distance between quantum processes based on the superfidelity, introduced recently to provide an upper bound for quantum fidelity. We show that the introduced measures partially fulfill the requirements for distance measure between quantum processes. We also argue that they can be especially useful as diagnostic measures to get preliminary knowledge about imperfections in an experimental setup. In particular we provide quantum circuit which can be used to measure the superfidelity between quantum processes. As the behavior of the superfidelity between quantum processes is crucial for the properties of the introduced measures, we study its behavior for several families of quantum channels. We calculate superfidelity between arbitrary one-qubit channels using affine parametrization and superfidelity between generalized Pauli channels in arbitrary dimensions. Statistical behavior of the proposed quantities for the ensembles of quantum operations in low dimensions indicates that the proposed measures can be indeed used to distinguish quantum processes.

We present two measures of distance between quantum processes based on the superfidelity, introduced recently to provide an upper bound for quantum fidelity. We show that the introduced measures partially fulfill the requirements for distance measure between quantum processes. We also argue that they can be especially useful as diagnostic measures to get preliminary knowledge about imperfections in an experimental setup. In particular we provide quantum circuit which can be used to measure the superfidelity between quantum processes. As the behavior of the superfidelity between quantum processes is crucial for the properties of the introduced measures, we study its behavior for several families of quantum channels. We calculate superfidelity between arbitrary one-qubit channels using affine parametrization and superfidelity between generalized Pauli channels in arbitrary dimensions. Statistical behavior of the proposed quantities for the ensembles of quantum operations in low dimensions indicates that the proposed measures can be indeed used to distinguish quantum processes.

State tomography on qubit pairs is routinely carried out by measuring the two qubits separately, while one expects a higher efficiency from tomography with highly symmetric joint measurements of both qubits. Our numerical study of simulated experiments does not support such expectations.

High-speed-photon detectors are some of the most important tools for observations of high energy cosmic rays. As technologies of photon detectors and their read-out electronics improved rapidly, the time resolution of some cosmic ray detectors became better than one nanosecond. To utilize such devices effectively, calibrations using a short-pulse light source are necessary. We have developed a pulsed laser of 80 picosecond width and adjustable peak intensity up to 100 mW. This pulsed laser is composed of a simple electric circuit and a laser diode. Details of this pulsed laser and its application for quality controls of photon detectors are reported in this contribution.

We present a foundational theoretical framework for the optimization and control of semantic fields through the introduction of Quantum Noetics (QN). This framework establishes a rigorous mathematical formalism operating in infinite-dimensional Hilbert spaces, leveraging principles from quantum mechanics, semantic field theory, and topological optimization.

The framework provides formal proofs for:
(1) The completeness of the semantic state space representation
(2) The convergence of semantic optimization protocols
(3) The stability of evolved semantic states
(4) The optimality of quantum-inspired semantic transformations
(5) The universality of the meta-semantic approach

This work establishes the mathematical foundations for a new paradigm in semantic field optimization, bridging quantum mechanical principles with classical semantic frameworks through rigorous mathematical formalism. The theoretical results suggest fundamental advantages over classical semantic optimization approaches while maintaining mathematical precision and formal rigor throughout the development.

When an informationally complete measurement is not available, the reconstruction of the density operator that describes the state of a quantum system can be accomplish, in a reliable way, by adopting the maximum entropy principle (MaxEnt principle), as an additional criterion, to obtain the least biased estimation. In this paper, we study the performance of the MaxEnt method for quantum state estimation when there is prior information about symmetries of the unknown state. We explicitly describe how to work with this method in the most general case, and present an algorithm that allows to improve the estimation of quantum states with arbitrary symmetries. Furthermore, we implement this algorithm to carry out numerical simulations estimating the density matrix of several three-qubit states of particular interest for quantum information tasks. We observed that, for most states, our approach allows to considerably reduce the number of independent measurements needed to obtain a sufficiently high fidelity in the reconstruction of the density matrix. Moreover, we analyze the performance of the method in realistic scenarios, showing that it is robust even when considering the effect of finite statistics, and under the presence of typical experimental noise.

This review serves as a concise introductory survey of modern compressive tomography developed since 2019. These are schemes meant for characterizing arbitrary low-rank quantum objects, be it an unknown state, a process or detector, using minimal measuring resources (hence compressive) without any a priori assumptions (rank, sparsity, eigenbasis, etc.) about the quantum object. This article contains a reasonable amount of technical details for the quantum-information community to start applying the methods discussed here. To facilitate the understanding of formulation logic and physics of compressive tomography, the theoretical concepts and important numerical results (both new and cross-referenced) shall be presented in a pedagogical manner.

Standard Bayesian credible-region theory for constructing an error region on the unique estimator of an unknown state in general quantum-state tomography to calculate its size and credibility relies on heavy Monte Carlo sampling of the state space followed by filtering to obtain the correct region sample. This conventional methodology typically gives negligible yield for very small error regions originating from large datasets. In this article, we discuss at length the in-region sampling theory for computing both size and credibility from region-average quantities that avoids this general problem altogether. Among the many possible numerical choices, we study the performance and properties of accelerated hit-and-run Monte Carlo algorithm for inregion sampling and provide its complexity estimates for quantum states. Finally with our in-region concept, by alternatively quantifying the region capacity with the region-average distance between two states in the region (measured for instance with either the Hilbert-Schmidt, trace-class or Bures distance), we derive approximation formulas to analytically estimate both region capacity and credibility without Monte Carlo computation.

We perform several numerical studies for our recently published adaptive compressive tomography scheme [D. Ahn et al. Phys. Rev. Lett. 122, 100404 (2019)], which significantly reduces the number of measurement settings to unambiguously reconstruct any rank-deficient state without any a priori knowledge besides its dimension. We show that both entangled and product bases chosen by our adaptive scheme perform comparably well with recently-known compressed-sensing element-probing measurements, and also beat random measurement bases for low-rank quantum states. We also numerically conjecture asymptotic scaling behaviors for this number as a function of the state rank for our adaptive schemes. These scaling formulas appear to be independent of the Hilbert space dimension. As a natural development, we establish a faster hybrid compressive scheme that first chooses random bases, and later adaptive bases as the scheme progresses. As an epilogue, we reiterate important elements of informational completeness for our adaptive scheme.

We present a compressive quantum process tomography scheme that fully characterizes any rank-deficient completely-positive process with no a priori information about the process apart from the dimension of the system on which the process acts. It uses randomly-chosen input states and adaptive output von Neumann measurements. Both entangled and tensor-product configurations are flexibly employable in our scheme, the latter which naturally makes it especially compatible with many-body quantum computing. Two main features of this scheme are the certification protocol that verifies whether the accumulated data uniquely characterize the quantum process, and a compressive reconstruction method for the output states. We emulate multipartite scenarios with high-order electromagnetic transverse modes and optical fibers to positively demonstrate that, in terms of measurement resources, our assumption-free compressive strategy can reconstruct quantum processes almost equally efficiently using all types of input states and basis measurement operations, operations, independent of whether or not they are factorizable into tensor-product states.

In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantum-state and quantum-process tomography are arranged together in the form of a tutorial review article that is catered to students and researchers who are new to the field of quantum tomography, with focus on maximum-likelihood related techniques as instructive examples to illustrate these ideas.

Pulsed homodyne quantum tomography usually requires a high detection efficiency limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency (< 50%) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian type. This result is obtained by applying the so-called "minimax" adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our finding paves the way to a more extensive use of quantum tomographic methods, even in physical situations in which high detection efficiency is unattainable.

We investigate generalized measurements, based on positive-operator-valued measures, and von Neumann measurements for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. In particular, we derive the conditions under which the failure probability of the measurement can reach its absolute lower bound, proportional to the fidelity of the states. The optimum measurement strategy yielding the fidelity bound of the failure probability is explicitly determined for a number of cases. One example involves two density operators of rank d that jointly span a 2d-dimensional Hilbert space and are related in a special way. We also present an application of the results to the problem of unambiguous quantum state comparison, generalizing the optimum strategy for arbitrary prior probabilities of the states.

Quantum State Tomography (QST) is a fundamental technique in Quantum Information Processing (QIP) for reconstructing unknown quantum states. However, the conventional QST methods are limited by the number of measurements required, which makes them impractical for large-scale quantum systems. To overcome this challenge, we propose the integration of Quantum Machine Learning (QML) techniques to enhance the efficiency of QST. In this paper, we conduct a comprehensive investigation into various approaches for QST, encompassing both classical and quantum methodologies; We also implement different QML approaches for QST and demonstrate their effectiveness on various simulated and experimental quantum systems, including multi-qubit networks. Our results show that our QML-based QST approach can achieve high fidelity (98%) with significantly fewer measurements than conventional methods, making it a promising tool for practical QIP applications.

The matrix product state has been demonstrated to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions for one-dimensional interacting quantum many-body systems. Inspired by this success, here we propose a clustering algorithm based on the matrix product state, which first maps the classical data into quantum states represented as matrix product states, and then minimizes the loss function using a variational matrix product states algorithm in the enlarged space. We demonstrate this algorithm by applying it to several commonly used machine learning data sets, showing that this algorithm could reach higher learning precision and that it is less likely to be trapped in local minima compared to the standard K-means algorithm. We also show that this algorithm can achieve state-of-the-art learning precision on popular computer vision data sets when used in combination with better initialization schemes.

Matrix product state has become the algorithm of choice when studying one-dimensional interacting quantum many-body systems, which demonstrates to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions. Here we propose a quantum inspired K-means clustering algorithm which first maps the classical data into quantum states represented as matrix product states, and then minimize the loss function using the variational matrix product states method in the enlarged space. We demonstrate the performance of this algorithm by applying it to several commonly used machine learning datasets and show that this algorithm could reach higher prediction accuracies and that it is less likely to be trapped in local minima compared to the classical K-means algorithm.

We present a tomographic method which requires only 4d − 3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme we have experimentally reconstructed a large number of pure states of dimension d = 7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, what represents an improvement with respect to other similar methods and contributes to answer the question of how many observables are needed to uniquely determine any pure state.

In this article we report that the entanglement produced by single photon subtraction is maximum, by studying the entanglement of multi photon subtraction from two mode squeezed states. We argue that the single photon subtraction produces maximum entanglement as it is more non-classical in nature than many photon subtractions.

We report the first measurement of the joint photon-number probability distribution for a two-mode quantum state created by a nondegenerate optical parametric amplifier. The measured distributions exhibit up to 1.9 dB of quantum correlation between the signal and idler photon numbers, whereas the marginal distributions are thermal as expected for parametric fluorescence.

Complete and precise characterization of a quantum dynamical process can be achieved via the method of quantum process tomography. Using a source of correlated photons, we have implemented several methods, each investigating a wide range of processes, e.g., unitary, decohering, and polarizing. One of these methods, ancilla-assisted process tomography (AAPT), makes use of an additional "ancilla system," and we have theoretically determined the conditions when AAPT is possible. Surprisingly, entanglement is not required. We present data obtained using both separable and entangled input states. The use of entanglement yields superior results, however.

Recently introduced shadow tomography protocols use "classical shadows" of quantum states to predict many target functions of an unknown quantum state. Unlike full quantum state tomography, shadow tomography does not insist on accurate recovery of the density matrix for high rank mixed states. Yet, such a protocol makes multiple accurate predictions with high confidence, based on a moderate number of quantum measurements. One particular influential algorithm, proposed by Huang et al. [Huang, Kueng, and Preskill, Nat. Phys. 16, 1050 (2020)], requires additional circuits for performing certain random unitary transformations. In this paper, we avoid these transformations but employ an arbitrary informationally complete positive operator-valued measure and show that such a procedure can compute k-bit correlation functions for quantum states reliably. We also show that, for this application, we do not need the median of means procedure of Huang et al. Finally, we discuss the contrast between the computation of correlation functions and fidelity of reconstruction of low rank density matrices.

We report on a new class of dimension witnesses, based on quantum random access codes, which are a function of the recorded statistics and that have different bounds for all possible decompositions of a high-dimensional physical system. Thus, it certifies the dimension of the system and has the new distinct feature of identifying whether the high-dimensional system is decomposable in terms of lower dimensional subsystems. To demonstrate the practicability of this technique, we used it to experimentally certify the generation of an irreducible 1024-dimensional photonic quantum state. Therefore, certifying that the state is not multipartite or encoded using noncoupled different degrees of freedom of a single photon. Our protocol should find applications in a broad class of modern quantum information experiments addressing the generation of high-dimensional quantum systems, where quantum tomography may become intractable.

The potential of achieving computational hardware with quantum advantage depends heavily on the quality of quantum gate operations. However, the presence of imperfect two-qubit gates poses a significant challenge and acts as a major obstacle in developing scalable quantum information processors. Google's Quantum AI and collaborators claimed to have conducted a supremacy regime experiment. In this experiment, a new two-qubit universal gate called the Sycamore gate is constructed and employed to generate random quantum circuits (RQCs), using a programmable quantum processor with 53 qubits. These computations were carried out in a computational state space of size 9 × 10 15. Nevertheless, even in strictly-controlled laboratory settings, quantum information on quantum processors is susceptible to various disturbances, including undesired interaction with the surroundings and imperfections in the quantum state. To address this issue, we conduct both quantum state tomography (QST) and quantum process tomography (QPT) experiments on Google's Sycamore gate using different artificial architectural superconducting quantum computer. Furthermore, to demonstrate how errors affect gate fidelity at the level of quantum circuits, we design and conduct full QST experiments for the five-qubit eight-cycle circuit, which was introduced as an example of the programability of Google's Sycamore quantum processor. These quantum tomography experiments are conducted in three distinct environments: noise-free, noisy simulation, and on IBM Quantum's genuine quantum computer. Our results offer valuable insights into the performance of IBM Quantum's hardware and the robustness of Sycamore gates within this experimental setup. These findings contribute to our understanding of quantum hardware performance and provide valuable information for optimizing quantum algorithms for practical applications.

Quantum state tomography, an important task in quantum information processing, aims at reconstructing a state from prepared measurement data. Bayesian methods are recognized to be one of the good and reliable choices in estimating quantum states [9]. Several numerical works showed that Bayesian estimations are comparable to, and even better than other methods in the problem of 1-qubit state recovery. However, the problem of choosing prior distribution in the general case of n qubits is not straightforward. More importantly, the statistical performance of Bayesian type estimators have not been studied from a theoretical perspective yet. In this paper, we propose a novel prior for quantum states (density matrices), and we define pseudo-Bayesian estimators of the density matrix. Then, using PAC-Bayesian theorems [16], we derive rates of convergence for the posterior mean. The numerical performance of these estimators are tested on simulated and real datasets.

Information metrics give lower bounds for the estimation of parameters. The Cencov-Morozova-Petz Theorem classifies the monotone quantum Fisher metrics. The optimum bound for the quantum estimation problem is offered by the metric which is obtained from the symmetric logarithmic derivative. To get a better bound, it means to go outside this family of metrics, and thus inevitably, to relax some general conditions. In the paper we defined logarithmic derivatives through a phase-space correspondence. This introduces a function which quantifies the deviation from the symmetric derivative. Using this function we have proved that there exist POVMs for which the new metric gives a higher bound from that of the symmetric derivative. The analysis was performed for the one qubit case.

From the experimental measurement of probability distributions of quadrature-field amplitudes, fol- lowed by numerical inversion (optical homodyne tomography), we have determined distributions and/or moments of the optical phase of small-photon-number fields for several definitions of the phase variable, including those based on Hermitian operators and on quasiprobability distributions. These measurements were performed on a vacuum field, and a weakly squeezed field. It is found that each definition of phase yields di6'erent distributions and/or moments for the experimental data. In addition, all of the ex- perimentally determined quantities agree with the corresponding theoretical predictions.

We have performed experimental quantum state tomography of NOON states with up to four photons. The measured states are generated by mixing a classical coherent state with spontaneous parametric down-conversion. We show that this method produces states which exhibit a high fidelity with ideal NOON states. The fidelity is limited by the overlap of the two-photon down-conversion state with any two photons originating from the coherent state, for which we introduce and measure a figure of merit. A second limitation on the fidelity set by the total setup transmission is discussed. We also apply the same tomography procedure for characterizing correlated photon hole states.

We establish a general principle for the tomographic approach to quantum state reconstruction, till now based on a simple rotation transformation in the phase space, which allows us to consider other types of transformations. Then, we will present different realizations of the principle in specific examples.

Entanglement is not only fundamental for understanding multipartite quantum systems but also generally useful for quantum information applications. Despite much effort devoted so far, little is known about minimal resources for detecting entanglement and also comparisons to tomography which reveals the full characterization. Here, we show that all entangled states can in general be detected in an experimental scheme that estimates the fidelity of two pure quantum states. An experimental proposal is presented with a single Hong-Ou-Mandel interferometry in which only two detectors are applied regardless of the dimensions or the number of modes of quantum systems. This shows measurement settings for entanglement detection are in general inequivalent to tomography: the number of detectors in quantum tomography increases with the dimensions and the modes whereas it is not the case in estimation of fidelity which detects entangled states.

This review covers latest developments in continuous-variable quantum-state tomography of optical fields and photons, placing a special accent on its practical aspects and applications in quantum information technology. Optical homodyne tomography is reviewed as a method of reconstructing the state of light in a given optical mode. A range of relevant practical topics are discussed, such as state-reconstruction algorithms (with emphasis on the maximum-likelihood technique), the technology of time-domain homodyne detection, mode matching issues, and engineering of complex quantum states of light. The paper also surveys quantum-state tomography for the transverse spatial state (spatial mode) of the field in the special case of fields containing precisely one photon. Contents References 29

Does chaos in the dynamics enable information gain in quantum tomography or impedes it? We address this question by considering continuous measurement tomography in which the measurement record is obtained as a sequence of expectation values of a Hermitian observable evolving under the repeated application of the Floquet map of the quantum kicked top. For a given dynamics and Hermitian observables, we observe completely opposite behavior in the tomography of well-localized spin coherent states compared to random states. As the chaos in the dynamics increases, the reconstruction fidelity of spin coherent states decreases. This contrasts with the previous results connecting information gain in tomography of random states with the degree of chaos in the dynamics that drives the system. The rate of information gain and hence the fidelity obtained in tomography depends not only on the degree of chaos in the dynamics and to what extent it causes the initial observable to spread in various...

We introduce and experimentally demonstrate a technique for performing quantum state tomography on multiple-qubit states despite incomplete knowledge about the unitary operations used to change the measurement basis. Given unitary operations with unknown rotation angles, our method can be used to reconstruct the density matrix of the state up to localσ z rotations as well as recover the magnitude of the unknown rotation angle. We demonstrate high-fidelity self-calibrating tomography on polarization-encoded one-and two-photon states. The unknown unitary operations are realized in two ways: using a birefringent polymer sheet-an inexpensive smartphone screen protector-or alternatively a liquid crystal wave plate with a tuneable retardance. We explore how our technique may be adapted for quantum state tomography of systems such as biological molecules where the magnitude and orientation of the transition dipole moment is not known with high accuracy.

We examine a variety of strategies for numerical quantum-state estimation from data of the sort commonly measured in experiments involving quantum-state tomography. We find that, in some important circumstances, an elaborate and time-consuming numerical optimization to obtain the optimum density matrix corresponding to a given data set is not necessary and that cruder, faster numerical techniques may well be sufficient; in other words, "the best" is the enemy of "good enough."

We propose a technique for performing quantum state tomography of photonic polarizationencoded multi-qubit states. Our method uses a single rotating wave plate, a polarizing beam splitter and two photon-counting detectors per photon mode. As the wave plate rotates, the photon counters measure a pseudo-continuous signal which is then Fourier transformed. The density matrix of the state is reconstructed using the relationship between the Fourier coefficients of the signal and the Stokes' parameters that represent the state. The experimental complexity, i.e. different wave plate rotation frequencies, scales linearly with the number of qubits.

Tomography is an indispensable part of quantum computation as it enables diagnosis of a quantum process through state reconstruction. Existing tomographic protocols are based on determining expectation values of various Pauli operators which typically require single-qubit rotations. However, in realistic systems, qubits often develop some form of unavoidable stray coupling making it difficult to manipulate one qubit independent of its partners. Consequently, standard protocols applied to those systems result in unfaithful reproduction of the true quantum state. We have developed a protocol, called coupling compensated tomography, that can correct for errors due to parasitic couplings completely in software and accurately determine the quantum state. We demonstrate the performance of our scheme on a system of two transmon qubits with always-on ZZ coupling. Our technique is a generic tomography tool that can be applied to large systems with different types of stray inter-qubit couplings and facilitates the use of arbitrary tomography pulses and even non-orthogonal axes of rotation.

We develop a technique for single qubit quantum state tomography using the mathematical setup of generalized quantization scheme for games. In this technique Alice sends an unknown pure quantum state to Bob who appends it with |0 0| and then applies the unitary operators on the appended quantum state and finds the payoffs for Alice and himself. Is is shown that for a particular set of unitary operators these payoffs are equal to Stokes parameters for an unknown quantum state. In this way an unknown quantum state can be measured and reconstructed. Strictly speaking this technique is not a game as no strategic competitions are involved.

When an informationally complete measurement is not available, the reconstruction of the density operator that describes the state of a quantum system can be accomplish, in a reliable way, by adopting the maximum entropy principle (MaxEnt principle), as an additional criterion, to obtain the least biased estimation. In this paper, we study the performance of the MaxEnt method for quantum state estimation when there is prior information about symmetries of the unknown state. We explicitly describe how to work with this method in the most general case, and present an algorithm that allows to improve the estimation of quantum states with arbitrary symmetries. Furthermore, we implement this algorithm to carry out numerical simulations estimating the density matrix of several three-qubit states of particular interest for quantum information tasks. We observed that, for most states, our approach allows to considerably reduce the number of independent measurements needed to obtain a sufficiently high fidelity in the reconstruction of the density matrix. Moreover, we analyze the performance of the method in realistic scenarios, showing that it is robust even when considering the effect of finite statistics, and under the presence of typical experimental noise.

The photon statistics and, moreover, the density matrix (quantum state) of a single light mode can be sampled using homodyne detection. That is, the density matrix is computed by averaging a set of sampling functions with respect to the measured quadrature value:;. We develop a practical procedure for evaluating these functions. The algorithm is simple, stable, has small computer-memory requirements, and is fast enough for real-time data processing. (Interestingly, our method involves unnormalizable solutions of the stationary Schrijdinger equation. We develop the annihilation-and-creation formalism for these solutions ar d derive their semicIassica1 approximations.) We quantify the bin width required to determine the density matrix up to a maximal quantum number. FinalIy, we show how the statistical errors of the reconstructed density matrix can be determined from the measured data.

We develop a technique for single qubit quantum state tomography using the mathematical setup of generalized quantization scheme for games. In this technique Alice sends an unknown pure quantum state to Bob who appends it with |0 0| and then applies the unitary operators on the appended quantum state and finds the payoffs for Alice and himself. Is is shown that for a particular set of unitary operators these payoffs are equal to Stokes parameters for an unknown quantum state. In this way an unknown quantum state can be measured and reconstructed. Strictly speaking this technique is not a game as no strategic competitions are involved.

Non-classical states are of practical interest in quantum computing and quantum metrology. These states can be detected through their Wigner function negativity in some regions. In this paper, we calculate the ground state of the threelevel generalised Dicke model for a single atom and determine the structure of its phase diagram using a fidelity criterion. We also calculate the Wigner function of the electromagnetic modes of the ground state through the corresponding reduced density matrix, and show in the phase diagram the regions where entanglement is present. A finer classification for the continuous phase transitions is obtained through the computation of the surface of maximum Bures distance.

Non-classical states are of practical interest in quantum computing and quantum metrology. These states can be detected through their Wigner function negativity in some regions. We show that the surfaces of minimum fidelity or maximum Bures distance constitute a signature of quantum phase transitions. Additionally the behaviour of the Wigner function associated to the field modes carry the information of both, the entanglement properties between matter and field sectors, and the regions of the parameter space where the quantum phase transitions take place. A finer classification for the continuous phase transitions is obtained through the computation of the surface of maximum Bures distance.

The development of the advanced Radio Frequency Timer of electrons is described. It is based on a helical deflector, which performs circular or elliptical sweeps of keV electrons, by means of 500 MHz radio frequency field. By converting a time distribution of incident electrons to a hit position distribution on a circle or ellipse, this device achieves extremely precise timing. Streak Cameras, based on similar principles, routinely operate in the ps and sub-ps time domain, but have substantial slow readout system. Here, we report a device, where the position sensor, consisting of microchannel plates and a delay-line anode, produces ~ns duration pulses which can be processed by using regular fast electronics. A photon sensor based on this technique, the Radio Frequency Photo-Multiplier Tube (RFPMT), has demonstrated a timing resolution of ~10 ps and a time stability of ~0.5 ps, FWHM. This makes the apparatus highly suited for Time Correlated Single Photon Counting which is widely used in optical microscopy and tomography of biological samples. The first application in lifetime measurements of quantum states of graphene, under construction at the A. I. Alikhanyan National Science Laboratory (AANL), is outlined. This is followed by a description of potential RFPMT applications in time-correlated Diffuse Optical Tomography, time-correlated Stimulated Emission Depletion microscopy, hybrid FRET/STED nanoscopy and Time-of-Flight Positron Emission Tomography.

We propose an interferometric scheme for the estimation of a linear combination with non-negative weights of an arbitrary number M > 1 of unknown phase delays, distributed across an M-channel linear optical network, with Heisenberg-limited sensitivity. This is achieved without the need of any sources of photon-number or entangled states, photon-number resolving detectors or auxiliary interferometric channels. Indeed, the proposed protocol remarkably relys upon a single squeezedstate source, an antisqueezing operation at the interferometer output, and on-off photodetectors.

The polarization properties of macroscopic Bell states are characterized using three-dimensional quantum polarization tomography. This method utilizes three-dimensional inverse Radon transform to reconstruct the polarization quasiprobability distribution function of a state from the probability distributions measured for various Stokes observables. The reconstructed 3D distributions obtained for the macroscopic Bell states are compared with those obtained for a coherent state with the same mean photon number. The results demonstrate squeezing in one or more Stokes observables.

Polarization quantum tomography is performed on 4 mode squeezed vacuum states. Three dimensional polarization quasiprobability functions are obtained and compared to that of an equal intensity coherent state. These distributions clearly demonstrate the difference in the polarization properties of the considered states. The reconstruction quality of the coherent state distribution is also analyzed by comparing the theo retically and experimentally obtained shapes for this state.

The development of the advanced Radio Frequency Timer of electrons is described. It is based on a helical deflector, which performs circular or elliptical sweeps of keV electrons, by means of 500 MHz radio frequency field. By converting a time distribution of incident electrons to a hit position distribution on a circle or ellipse, this device achieves extremely precise timing. Streak Cameras, based on similar principles, routinely operate in the ps and sub-ps time domain, but have substantial slow readout system. Here, we report a device, where the position sensor, consisting of microchannel plates and a delay-line anode, produces ~ns duration pulses which can be processed by using regular fast electronics. A photon sensor based on this technique, the Radio Frequency Photo-Multiplier Tube (RFPMT), has demonstrated a timing resolution of ~10 ps and a time stability of ~0.5 ps, FWHM. This makes the apparatus highly suited for Time Correlated Single Photon Counting which is widely used in optical microscopy and tomography of biological samples. The first application in lifetime measurements of quantum states of graphene, under construction at the A. I. Alikhanyan National Science Laboratory (AANL), is outlined. This is followed by a description of potential RFPMT applications in time-correlated Diffuse Optical Tomography, time-correlated Stimulated Emission Depletion microscopy, hybrid FRET/STED nanoscopy and Time-of-Flight Positron Emission Tomography.