We propose a practical strategy for choosing sets of input coherent states that are near-optimal for reconstructing single-mode Gaussian quantum processes with output-state heterodyne measurements. We first derive analytical expressions for the mean squared-error that quantifies the reconstruction accuracy for general process tomography and large data. Using such expressions, upon relaxing the trace-preserving constraint, we introduce an error-reducing set of input coherent states that is independent of the measurement data or the unknown true process -- the geometrical set. We numerically show that process reconstruction from such input coherent states is nearly as accurate as that from the best possible set of coherent states chosen with the complete knowledge about the process. This allows us to efficiently characterize Gaussian processes even with reasonably low-energy coherent states. We numerically observe that the geometrical strategy without trace preservation beats all nonadaptive strategies for arbitrary trace-preserving Gaussian processes of typical parameter ranges so long as the displacement components are not too large.
Explicit expressions for optical tomograms of the photon-added coherent states, even/odd photon-added coherent states and photon-added thermal states are given in terms of Hermite polynomials. Suggestions for experimental homodyne detection of the considered photon states are presented.
Squeezed and rotated quadrature of an ion in a Paul trap is discussed in connection with reconstructing its quantum state using symplectic-tomography method. Marginal distributions of the quadrature for squeezed and correlated states and for nonlinear coherent states of a trapped ion are obtained and the density matrices in the Fock basis are expressed explicitly in terms of these marginal distributions.
Using coherent states in optical quantum process tomography is a practically-relevant approach. Here, we develop a framework for complete characterization of quantum-optical processes in terms of normally-ordered moments by using coherent states as probes. We derive the associated superoperator tensors for several optical processes. We also show that our technique can be used to determine nonclassicality features of quantum-optical states and processes. Furthermore, we investigate identification of multi-mode Gaussian processes and show that the number of necessary probe coherent states scales…
An arbitrary quantum-optical process (channel) can be completely characterized by probing it with coherent states using the recently developed coherent-state quantum process tomography (QPT) [Lobino et al., Science 322, 563 (2008)]. In general, precise QPT is possible if an infinite set of probes is available. Thus, realistic QPT of infinite-dimensional systems is approximate due to a finite experimentally-feasible set of coherent states and its related energy-cut-off approximation. We show with explicit formulas that one can completely identify a quantum-optical Gaussian process just with a f…
Coherent-state quantum process tomography (csQPT) is a method of completely characterizing a quantum-optical "black box" by probing it with coherent states and performing homodyne measurements on the output [M. Lobino et al, Science 322, 563 (2008)]. We present a technique for csQPT that is fully based on statistical inference, specifically, quantum expectation-maximization. The method relies on the Jamiolkowski isomorphism and iteratively reconstructs the process tensor in the Fock basis directly from the experimental data. This approach permits incorporation of a priori constraints into the …
Single-photon subtraction plays important roles in optical quantum information processing as it provides a non-Gaussian characteristic in continuous-variable quantum information. While the conventional way of implementing single-photon subtraction based on a low-reflectance beam splitter works properly for a single-mode quantum state, it is unsuitable for a multimode quantum state because a single photon is subtracted from all multiple modes without maintaining their mode coherence. Here we experimentally implement and characterize a mode-tunable coherent single-photon subtractor based on sum …
A powerful tool for studying geometrical problems in Hilbert space is developed. In particular, we study the quantum pure state tomography problem in finite dimensions from the point of view of dynamical systems and bifurcations theory. First, we introduce a generalization of the Hellinger metric for probability distributions which allows us to find a geometrical interpretation of the quantum state tomography problem. Thereafter, we prove that every solution to the state tomography problem is an attractive fixed point of the so-called physical imposition operator. Additionally, we demonstrate …
Entangled measurement is a crucial tool in quantum technology. We propose a new entanglement measure of multi-mode detection, which estimates the amount of entanglement that can be created in a measurement. To illustrate the proposed measure, we perform quantum tomography of a two-mode detector that is comprised of two superconducting nanowire single photon detectors. Our method utilizes coherent states as probe states, which can be easily prepared with accuracy. Our work shows that a separable state such as a coherent state is enough to characterize a potentially entangled detector. We invest…
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 th…
We develop an enhanced technique for characterizing quantum optical processes based on probing unknown quantum processes only with coherent states. Our method substantially improves the original proposal [M. Lobino et al., Science 322, 563 (2008)], which uses a filtered Glauber-Sudarshan decomposition to determine the effect of the process on an arbitrary state. We introduce a new relation between the action of a general quantum process on coherent state inputs and its action on an arbitrary quantum state. This relation eliminates the need to invoke the Glauber-Sudarshan representation for sta…
Quantum processes involving single-photon states are of broad interest in particular for quantum communication. Extending to continuous values a recent proposal by Yuan et al \cite{YUAN16}, we show that single-photon quantum processes can be characterized using phase randomized coherent states (PRCS) as inputs. As a proof of principle, we present the experimental investigation of single-photon tomography using PRCS. The probability distribution of field quadratures measurements for single-photon states can be accurately derived from the PRCS data. As a consequence, the Wigner function and the …
In this paper, we will construct the quantum states of target space coordinates from world-sheet states, using quantum state tomography. To perform quantum state tomography of an open string, we will construct suitable quadrature operators. We do this by first defining the quadrature operators in world-sheet, and then using them to construct the quantum target space quadrature operators for an open string. We will connect the quantum target space to classical geometry using coherent string states. We will be using a novel construction based on a string displacement operator to construct these …
In this work we develop an experimental procedure to interrogate the single- and multiphoton scattering matrices of an unknown quantum system interacting with propagating photons. Our proposal requires coherent state laser or microwave inputs and homodyne detection at the scatterer's output, and provides simultaneous information about multiple ---elastic and inelastic--- segments of the scattering matrix. The method is resilient to detector noise and its errors can be made arbitrarily small by combining experiments at various laser powers. Finally, we show that the tomography of scattering has…
We show with explicit formulas that one can completely identify an unknown quantum process with only one weakly entangled state; and identify a quantum optical Gaussian process with either one two-mode squeezed state or a few different coherent states. In tomography of a multi-mode process, our method reduces the number of different test states exponentially compared with existing methods.
We report a technique for experimental characterization of an $M$-mode quantum optical process, generalizing the single-mode coherent-state quantum-process tomography method [M. Lobino et al., Science 322, 563 (2008); A. Anis and A.I. Lvovsky, New J. Phys. 14, 105021 (2012)]. By measuring effect of the process on multi-mode coherent states via balanced homodyne tomography, we obtain the process tensor in the Fock basis. This rank-$4M$ tensor, which predicts the effect of the process on an arbitrary density matrix, is iteratively reconstructed directly from the experimental data via the maximum…
We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detai…
Conditional quantum optical processes enable a wide range of technologies from generation of highly non-classical states to implementation of quantum logic operations. The process fidelity that can be achieved in a realistic implementation depends on a number of system parameters. Here we experimentally examine Fock-state filtration, a canonical example of a broad class of conditional quantum operations acting on a single optical field mode. This operation is based upon interference of the mode to be manipulated with an auxiliary single-photon state at a beam splitter, resulting in the entangl…
Conventional tomographic techniques are becoming increasingly infeasible for reconstructing the operators of quantum devices of growing sophistication. We describe a novel tomographic procedure using coherent states which begins by reconstructing the diagonals of the operator, and then each successive off-diagonal in a recursive manner. Each recursion is considerably more efficient than reconstructing the operator in its entirety, and each successive recursion involves fewer parameters. We apply our technique to reconstruct the positive-operator-valued measure (POVM) corresponding to a recentl…
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
An analysis of the homodyne tomography process that is often used to determine the Wigner functions of quantum optical states is performed to consider the effects of the spatiotemporal degrees of freedom. The homodyne tomography process removes those parts of the input state that are not associated with the mode of the local oscillator by tracing out those degrees of freedom. Using a functional approach to incorporate all the spatiotemporal degrees of freedom, we find that this reduction in the degrees of freedom introduces distortions in the observed Wigner function. The analysis also shows h…
Long-lived exciton coherences have been recently observed in photosynthetic complexes via ultrafast spectroscopy, opening exciting possibilities for the study and design of coherent exciton transport. Yet, ambiguity in the spectroscopic signals has led to arguments for interpreting them in terms of the exciton dynamics, demanding more stringent tests. We propose a novel strategy, Quantum Process Tomography (QPT) for ultrafast spectroscopy, to reconstruct the evolving quantum state of excitons in double-walled supramolecular light-harvesting nanotubes at room temperature. The protocol calls for…
A simple, and elegant geometrical representation is developed to describe the concept of coherence and squeezing for angular momentum operators. Angular momentum squeezed states were obtained by applying Bogoliubov transformation on the angular momentum coherent states in Schwinger representation [Phys. Rev. {\bf{A 51}} (1995)]. We present the geometrical phase space description of angular momentum coherent and squeezed states and relate with the harmonic oscillator. The unique feature of our geometric representation is the portraying of the expectation values of the angular momentum component…
Flexible characterization techniques that identify and quantify experimental imperfections under realistic assumptions are crucial for the development of quantum computers. Gate set tomography is a characterization approach that simultaneously and self-consistently extracts a tomographic description of the implementation of an entire set of quantum gates, as well as the initial state and measurement, from experimental data. Obtaining such a detailed picture of the experimental implementation is associated with high requirements on the number of sequences and their design, making gate set tomog…
Single-shot coherent process tomography and mid-infrared polarimetry with undetected photons
Humboldt-Universität zu Berlin（柏林洪堡大学）；University of Science and Technology of China（中国科学技术大学）；Ferdinand-Braun-Institut, Leibniz-Institut für Höchstfrequenztechnik（费迪南德·布劳恩研究所，莱布尼茨高频技术研究所）
We remind the properties of the intelligent (and quasi-intelligent) spin states introduced by Aragone et al. We use these states to construct families of coherent wave packets on the sphere and we sketch the time evolution of these wave packets for a rigid body molecule.
Quantum gate set tomography (GST) has emerged as a promising method for the full characterization of quantum logic gates. In contrast to quantum process tomography (QPT), GST self-consistently and correctly accounts for state preparation and measurement (SPAM) errors. It therefore provides significantly more accurate estimates than QPT as gate fidelities increase into the fault-tolerant regime. We give a detailed review of GST and provide a self-contained guide to its implementation. The method is presented in a step-by-step fashion and relevant mathematical background material is included. Ou…
The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented, and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently-developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically de…
A central task in quantum information processing is to characterize quantum processes. In the realm of optical quantum information processing, this amounts to characterizing the transformations of the mode creation and annihilation operators. This transformation is unitary for linear optical systems, whereas these yield the well-known Bogoliubov transformations for systems with Hamiltonians that are quadratic in the mode operators. In this paper, we show that a modified Mach-Zehnder interferometer can characterize both these kinds of evolutions for multimode systems. While it suffices to use c…
Efficient Characterization of Coherent and Correlated Low-Degree Noise in Layers of Gates
Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Dipartimento di Fisica e Astronomia “G. Galilei” & Padua Quantum Technologies Research Center, Università degli Studi di Padova（帕多瓦大学伽利略物理与天文学系及帕多瓦量子技术研究中心）
Any optical quantum information processing machine would be comprised of fully-characterized constituent devices for both single state manipulations and tasks involving the interaction between multiple quantum optical states. Ideally for the latter, would be an apparatus capable of deterministic optical phase shifts that operate on input quantum states with the action mediated solely by auxiliary signal fields. Here we present the complete experimental characterization of a system designed for optically controlled phase shifts acting on single-photon level probe coherent states. Our setup is b…
I derive simple non-local dynamic boundary conditions, suitable for modelling of radiation emission from stacked Josephson junctions, and employ them for analysis of flux-flow emission from intrinsic Josephson junctions in high-$T_c$ superconductors. It is shown that due to the lack of Lorenz contraction of fluxons in stacked junctions, high quality geometrical resonances are prerequisite for high power emission from the stack. This leads to a dual role of the radiative impedance: on the one hand, small impedance increases the efficiency of emission from the stack, on the other hand, enhanced …
It is generally assumed that every process in quantum physics can be described mathematically by a completely positive map. However, experimentally reconstructed processes are not necessarily completely positive due to statistical or systematic errors. In this paper, we introduce a test for discriminating statistical from systematic errors which is necessary to interpret experimentally reconstructed, non-completely positive maps.Wedemonstrate the significance of the test using several examples given by experiments and simulations. In particular, we demonstrate experimentally how an initial cor…
In the past decades, quantum plasmonics has become an active area due to its potential applications in on-chip plasmonic devices for quantum information processing. However, the fundamental physical process, i.e., how a quantum state of light evolves in the photon-plasmon conversion process, has not been clearly understood. Here, we report a complete characterization of the plasmon-assisted extraordinary optical transmission process through quantum process tomography. By inputting various coherent states to interact with the plasmonic structure and detecting the output states with a homodyne d…
We consider a system of static spin qubits embedded in a one-dimensional spin coherent channel and develop a scheme to readout the state of one and two qubits separately. We use unpolarized flying qubits for this purpose that scatter off from the static qubits due to the Heisenberg exchange interaction. Analysing the transmission coefficient as a function of density matrix elements along with additional unitary gates we reconstruct the state of static qubits.
The exponential growth of photonic quantum technologies is driving the demand of tools for measuring the quality of their information carriers. One of the most prominent is Stimulated Emission Tomography (SET), which uses classical coherent fields to measure the Joint Spectral Amplitude (JSA) of photon pairs with high speed and resolution. While the modulus of the JSA can be directly addressed from a single intensity measurement, the retrieval of the Joint Spectral Phase (JSP) is far more challenging and received minor attentions. However, a wide class of spontaneous sources of technological r…
All the $n(2n+3)$ mean and covariance parameters of an $n$-mode Gaussian states are expressed in terms of the expectation values of the same number of conjugates of the total number observable. This permits a complete tomography of the state. The same is applied to outputs of a Gaussian channel corresponding to selected coherent states to perform the complete tomography of the channel. This leads to some interesting problems concerning the distribution of the number operator and also tomographic complexity.
We employ quantum state and process tomography with time-bin qubits to benchmark a city-wide metropolitan quantum communication system. Over this network, we implement real-time feedback control systems for stabilizing the phase of the time-bin qubits, and obtain a 99.3% quantum process fidelity to the ideal channel, indicating the high quality of the whole quantum communication system. This allows us to implement field trial of high performance quantum key distribution using coherent one way protocol with average quantum bit error rate and visibility of 0.25% and 99.2% during 12 hours over 61…
Quantum memory advantage for quantum process tomography
Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（达勒姆复杂量子系统中心，柏林自由大学）；School of Engineering and Applied Sciences, Harvard University（哈佛工程学院与应用科学学院）
The description of excited state dynamics in multichromophoric systems constitutes both a theoretical and experimental challenge in modern physical chemistry. An experimental protocol which can systematically characterize both coherent and dissipative processes at the level of the evolving quantum state of the chromophores is desired. In this article, we show that a carefully chosen set of polarization controlled two-color heterodyned photon-echo experiments can be used to reconstruct the time-evolving density matrix of the one-exciton manifold of a heterodimer. This possibility in turn allows…
