I propose an approach to adaptive homodyne detection of digitally modulated quantum optical pulses in which the phase of the local oscillator is chosen to maximize the average information gain, i.e., the mutual information, at each step of the measurement. I study the properties of this adaptive detection scheme by considering the problem of classical information content of ensembles of coherent states. Using simulations of quantum trajectories and visualizations of corresponding measurement operators, I show that the proposed measurement scheme adapts itself to the features of each ensemble. For all considered ensembles of coherent states, it consistently outperforms heterodyne detection and Wiseman's adaptive scheme for phase measurements [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)].
An adaptive optimization technique to improve precision of quantum homodyne tomography is presented. The method is based on the existence of so-called null functions, which have zero average for arbitrary state of radiation. Addition of null functions to the tomographic kernels does not affect their mean values, but changes statistical errors, which can then be reduced by an optimization method that "adapts" kernels to homodyne data. Applications to tomography of the density matrix and other relevant field-observables are studied in detail.
We present an experimental demonstration of the power of real-time feedback in quantum metrology, confirming a theoretical prediction by Wiseman regarding the superior performance of an adaptive homodyne technique for single-shot measurement of optical phase. For phase measurements performed on weak coherent states with no prior knowledge of the signal phase, we show that the variance of adaptive homodyne estimation approaches closer to the fundamental quantum uncertainty limit than any previously demonstrated technique. Our results underscore the importance of real-time feedback for reaching …
We show a protocol achieving the ultimate Heisenberg-scaling sensitivity in the estimation of a parameter encoded in a generic linear network, without employing any auxiliary networks, and without the need of any prior information on the parameter nor on the network structure. As a result, this protocol does not require a prior coarse estimation of the parameter, nor an adaptation of the network. The scheme we analyse consists of a single-mode squeezed state and homodyne detectors in each of the $M$ output channels of the network encoding the parameter, making it feasible for experimental appl…
Fast and accurate measurement is a highly desirable, if not vital, feature of quantum computing architectures. In this work we investigate the usefulness of adaptive measurements in improving the speed and accuracy of qubit measurement. We examine a particular class of quantum computing architectures, ones based on qubits coupled to well controlled harmonic oscillator modes (reminiscent of cavity-QED), where adaptive schemes for measurement are particularly appropriate. In such architectures, qubit measurement is equivalent to phase discrimination for a mode of the electromagnetic field, and w…
We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schrödinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitativel…
The preparation stage of optical qubits is an essential task in all the experimental setups employed for the test and demonstration of Quantum Optics principles. We consider a deterministic protocol for the preparation of qubits as a superposition of vacuum and one photon number states, which has the advantage to reduce the amount of resources required via phase-sensitive measurements using a local oscillator ('dyne detection'). We investigate the performances of the protocol using different phase measurement schemes: homodyne, heterodyne, and adaptive dyne detection (involving a feedback loop…
We address phase-shift estimation by means of squeezed vacuum probe and homodyne detection. We analyze Bayesian estimator, which is known to asymptotically saturate the classical Cramer-Rao bound to the variance, and discuss convergence looking at the a posteriori distribution as the number of measurements increases. We also suggest two feasible adaptive methods, acting on the squeezing parameter and/or the homodyne local oscillator phase, which allow to optimize homodyne detection and approach the ultimate bound to precision imposed by the quantum Cramer-Rao theorem. The performances of our t…
Homodyne detection is considered as a way to improve the efficiency of communication near the single-photon level. The current lack of commercially available {\it infrared} photon-number detectors significantly reduces the mutual information accessible in such a communication channel. We consider simulating direct detection via homodyne detection. We find that our particular simulated direct detection strategy could provide limited improvement in the classical information transfer. However, we argue that homodyne detectors (and a polynomial number of linear optical elements) cannot simulate ph…
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on $\mathbb R^2$ which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < η\leq 1$, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the $\mathbb L_\infty$-risk over a class of infinitely differentiable. We compute also the lower …
We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems. The state is represented through the Wigner function, a generalized probability density on $\mathbb{R}^2$ which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The effect of the losses due to detection inefficiencies, which are always present in a real experiment, is the addition to the tomographic data of independent Gaussian noise. We construct a kernel estimator for the Wigner function, prove tha…
Asymmetry effects in homodyne and heterodyne measurements: Positive operator-valued measures and asymptotic security of Gaussian-continuous-variable quantum key distribution
ITMO University（伊托夫大学）；Laboratory for Quantum Communications, ITMO University（伊托夫大学量子通信实验室）；Laboratory of Quantum Processes and Measurements, ITMO University（伊托夫大学量子过程与测量实验室）
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…
Quantum transducers play a crucial role in hybrid quantum networks. A good quantum transducer can faithfully convert quantum signals from one mode to another with minimum decoherence. Most investigations of quantum transduction are based on the protocol of direct mode conversion. However, the direct protocol requires the matching condition, which in practice is not always feasible. Here we propose an adaptive protocol for quantum transducers, which can convert quantum signals without requiring the matching condition. The adaptive protocol only consists of Gaussian operations, feasible in vario…
We propose an experimental design for universal continuous-variable quantum computation that incorporates recent innovations in linear-optics-based continuous-variable cluster state generation and cubic-phase gate teleportation. The first ingredient is a protocol for generating the bilayer-square-lattice cluster state (a universal resource state) with temporal modes of light. With this state, measurement-based implementation of Gaussian unitary gates requires only homodyne detection. Second, we describe a measurement device that implements an adaptive cubic-phase gate, up to a random phase-spa…
We present a strategy for estimation of d-level quantum states and for the simple adaption of corresponding measurements. The adaption method is inspired by mutually unbiased measurements, but it is also applicable in cases for which no complete set of mutually unbiased bases is known. We present results of Monte Carlo simulations, that demonstrate the fidelity gain of the adaptive strategy compared to a non-adaptive estimation.
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated codewords with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations…
Beyond their use as numerical tools, quantum trajectories can be ascribed a degree of reality in terms of quantum measurement theory. In fact, they arise naturally from considering continuous observation of a damped quantum system. A particularly useful form of quantum trajectories is as linear (but non-unitary) stochastic Schrodinger equations. In the limit where a strong local oscillator is used in the detection, and where the system is not driven, these quantum trajectories can be solved. This gives an alternate derivation of the probability distributions for completed homodyne and heterody…
We consider the effect of loss on quantum-optical communication channels. The channel based on direct detection of number states, which for a lossless transmission line would achieve the ultimate quantum channel capacity, is easily degraded by loss. The same holds true for the channel based on homodyne detection of squeezed states, which also is very fragile to loss. On the contrary, the ``classical'' channel based on heterodyne detection of coherent states is loss-invariant. We optimize the a priori probability for the squeezed-state and the number-state channels, taking the effect of loss in…
We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisen…
The problem of measuring a time-varying phase, even when the statistics of the variation is known, is considerably harder than that of measuring a constant phase. In particular, the usual bounds on accuracy - such as the $1/(4\bar{n})$ standard quantum limit with coherent states - do not apply. Here, restricting to coherent states, we are able to analytically obtain the achievable accuracy - the equivalent of the standard quantum limit - for a wide class of phase variation. In particular, we consider the case where the phase has Gaussian statistics and a power-law spectrum equal to $κ^{p-1}/|ω…
In this letter we revisit the problem of optimal design of quantum tomographic experiments. In contrast to previous approaches where an optimal set of measurements is decided in advance of the experiment, we allow for measurements to be adaptively and efficiently re-optimised depending on data collected so far. We develop an adaptive statistical framework based on Bayesian inference and Shannon's information, and demonstrate a ten-fold reduction in the total number of measurements required as compared to non-adaptive methods, including mutually unbiased bases.
We study the problem of calibrating a quantum receiver for optical coherent states when transmitted on a quantum optical channel with variable transmissivity, a common model for long-distance optical-fiber and free/deep-space optical communication. We optimize the error probability of legacy adaptive receivers, such as Kennedy's and Dolinar's, on average with respect to the channel transmissivity distribution. We then compare our results with the ultimate error probability attainable by a general quantum device, computing the Helstrom bound for mixtures of coherent-state hypotheses, for the fi…
Quantum filtering is a signal processing technique that estimates the posterior state of a quantum system under continuous measurements and has become a standard tool in quantum information processing, with applications in quantum state preparation, quantum metrology, and quantum control. If the filter assumes a nominal model that differs from reality, however, the estimation accuracy is bound to suffer. Here I derive identities that relate the excess error caused by quantum filter mismatch to the relative entropy between the true and nominal observation probability measures, with one identity…
