We address the estimation of the loss parameter of a bosonic channel probed by arbitrary signals. Unlike the optimal Gaussian probes, which can attain the ultimate bound on precision asymptotically either for very small or very large losses, we prove that Fock states at any fixed photon number saturate the bound unconditionally for any value of the loss. In the relevant regime of low-energy probes, we demonstrate that superpositions of the first low-lying Fock states yield an absolute improvement over any Gaussian probe. Such few-photon states can be recast quite generally as truncations of de-Gaussified photon-subtracted states.
Optical channels, such as fibers or free-space links, are ubiquitous in today's telecommunication networks. They rely on the electromagnetic field associated with photons to carry information from one point to another in space. As a result, a complete physical model of these channels must necessarily take quantum effects into account in order to determine their ultimate performances. Specifically, Gaussian photonic (or bosonic) quantum channels have been extensively studied over the past decades given their importance for practical purposes. In spite of this, a longstanding conjecture on the o…
The quantum Fisher information (QFI) in SU(2) and SU(1,1) interferometers was considered, and the QFI-only calculation was overestimated. In general, the phase estimation as a two-parameter estimation problem, and the quantum Fisher information matrix (QFIM) is necessary. In this paper, we theoretically generalize the model developed by Escher et al [Nature Physics 7, 406 (2011)] to the QFIM case with noise and study the ultimate precision limits of SU(2) and SU(1,1) interferometers with photon losses because photon losses as a very usual noise may happen to the phase measurement process. Usin…
We establish the fundamental limit of communication capacity within Gaussian schemes under phase-insensitive Gaussian channels, which employ multimode Gaussian states for encoding and collective Gaussian operations and measurements for decoding. We prove that this Gaussian capacity is additive, i.e., its upper bound occurs with separable encoding and separable receivers so that a single-mode communication suffices to achieve the largest capacity under Gaussian schemes. This rigorously characterizes the gap between the ultimate Holevo capacity and the capacity within Gaussian communication, sho…
The major problem of multiparameter quantum estimation theory is to find an ultimate measurement scheme to go beyond the standard quantum limits that each quasi-classical estimation measurement is limited by. Although, in some specifics quantum protocols without environmental noise, the ultimate sensitivity of a multiparameter quantum estimation can beat the standard quantum limit. However, the presence of noise imposes limitations on the enhancement of precision due to the inevitable existence of environmental fluctuations. Here, we address the motivation behind the usage of Gaussian quantum …
Multimode Gaussian quantum light, which includes multimode squeezed and multipartite quadrature entangled light, is a very general and powerful quantum resource with promising applications in quantum information processing and metrology. In this paper, we determine the ultimate sensitivity in the estimation of any parameter when the information about this parameter is encoded in such light, irrespective of the information extraction protocol used in the estimation and of the measured observable. In addition we show that an appropriate homodyne detection scheme allows us to reach this ultimate …
The Holevo Cramér Rao bound is a lower bound on the sum of the mean square error of estimates for parameters of a state. We provide a method for calculating the Holevo Cramér-Rao bound for estimation of quadrature mean parameters of a Gaussian state by formulating the problem as a semidefinite program. In this case, the bound is tight; it is attained by purely Guassian measurements. We consider the example of a symmetric two-mode squeezed thermal state undergoing an unknown displacement on one mode. We calculate the Holevo Cramér-Rao bound for joint estimation of the conjugate parameters for t…
Relativistic quantum metrology provides an optimal strategy for the estimation of parameters encoded in quantum fields in flat and curved spacetime. These parameters usually correspond to physical quantities of interest such as proper times, accelerations, gravitational field strengths, among other spacetime parameters. The precise estimation of these parameters can lead to novel applications in gravimeters, spacetime probes and gravitational wave detectors. Previous work in this direction only considered pure probe states. In realistic situations, however, probe states are mixed. In this pape…
The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measure…
We investigate the performance of entangled coherent state for quantum enhanced phase estimation. An exact analytical expression of quantum Fisher information is derived to show the role of photon losses on the ultimate phase sensitivity. We find a transition of the sensitivity from the Heisenberg scaling to the classical scaling due to quantum decoherence of the photon state. This quantum-classical transition is uniquely determined by the number of photons being lost, instead of the number of incident photons or the photon loss rate alone. Our results also reveal that a crossover of the sensi…
Multimode photon-subtraction provides an experimentally feasible option to construct large non-Gaussian quantum states in continuous-variable quantum optics. The non-Gaussian features of the state can lead towards the more exotic aspects of quantum theory, such as negativity of the Wigner function. However, the pay-off for states with such delicate quantum properties is their sensitivity to decoherence. In this paper, we present a general model that treats the most important source of decoherence in a purely optical setting: losses. We use the framework of open quantum systems and master equat…
Quantum attributes of light have been related to non-classicality so far, i. e. to incompatibility with mixtures of coherent states. The progress in quantum optics indicates that this feature does not suffice to witness exotic behavior of light. Contrary, quantum non-Gaussianity is starting to appear as a promising and applicable property reflecting interesting states of light that are suitable for quantum protocols. We investigate a newly introduced hierarchy of criteria of quantum non-Gaussianity and predict this attribute can be observed on light emitted from many single photon emitters, ev…
In this paper, we compare single-photon qubits and hybrid qubits as information carriers through quantum teleportation using a Gaussian continuous-variable channel. A hybrid qubit in our study is in the form of entanglement between a coherent state and a single photon. We find that hybrid qubits outperform photonic qubits when coherent amplitudes of the hybrid qubits are as low as $α\lesssim 1$, while photonic qubits yield better results for larger amplitudes. We analyze effects of photon losses, and observe that the overall character of teleportation for different qubits remains the same alth…
When standard light sources are employed, the precision of the phase determination is limited by the shot noise. Quantum entanglement provides means to exceed this limit with the celebrated example of N00N states that saturate the ultimate Heisenberg limit on precision, but at the same time are extremely fragile to losses. In contrast, we provide experimental evidence that appropriately engineered quantum states outperform both standard and N00N states in the precision of phase estimation when losses are present.
Loss measurements are at the base of spectroscopy and imaging, thus perme- ating all the branches of science, from chemistry and biology to physics and material science. However, quantum mechanics laws set the ultimate limit to the sensitivity, constrained by the probe mean energy. This can be the main source of uncertainty, for example when dealing with delicate system such as biological samples or photosensitive chemicals. It turns out that ordinary (clas- sical) probe beams, namely with Poissonian photon number distribution, are fundamentally inadequate to measure small losses with the high…
Bounds on the ultimate precision attainable in the estimation of a parameter in Gaussian quantum metrology are obtained when the average number of bosonic probes is fixed. We identify the optimal input probe state among generic (mixed in general) Gaussian states with a fixed average number of probe photons for the estimation of a parameter contained in a generic multimode interferometric optical circuit, namely, a passive linear circuit preserving the total number of photons. The optimal Gaussian input state is essentially a single-mode squeezed vacuum, and the ultimate precision is achieved b…
The ability to confine light into tiny spatial dimensions is important for applications such as microscopy, sensing and nanoscale lasers. While plasmons offer an appealing avenue to confine light, Landau damping in metals imposes a trade-off between optical field confinement and losses. We show that a graphene-insulator-metal heterostructure can overcome that trade-off, and demonstrate plasmon confinement down to the ultimate limit of the lengthscale of one atom. This is achieved by far-field excitation of plasmon modes squeezed into an atomically thin hexagonal boron nitride dielectric h-BN s…
We find and investigate the optimal scheme of quantum distributed Gaussian sensing for estimation of the average of independent phase shifts. We show that the ultimate sensitivity is achievable by using an entangled symmetric Gaussian state, which can be generated using a single-mode squeezed vacuum state, a beam-splitter network, and homodyne detection on each output mode in the absence of photon loss. Interestingly, the maximal entanglement of a symmetric Gaussian state is not optimal although the presence of entanglement is advantageous as compared to the case using a product symmetric Gaus…
We address the estimation of the magnetic field B acting on an ensemble of atoms with total spin J subjected to collective transverse noise. By preparing an initial spin coherent state, for any measurement performed after the evolution, the mean-square error of the estimate is known to scale as $1/J$, i.e. no quantum enhancement is obtained. Here, we consider the possibility of continuously monitoring the atomic environment, and conclusively show that strategies based on time-continuous non-demolition measurements followed by a final strong measurement may achieve Heisenberg-limited scaling $1…
We have studied theoretical un-symmetric multi-photon subtracted twin beam state and demonstrated a method for generating states that resembles to high photon number states with the increase in the number of subtracted photons through Wigner distribution function, which can be reconstructed experimentally by Homodyne measurement. A crucial point is high non-classicality is obtained by photon subtraction when mean photons per mode of twin beam state is low. We have calculated photon statistics from the phase space distribution function and found sub-poissonian behaviour in the same low mean pho…
We address the estimation of the loss parameter of a bosonic channel probed by Gaussian signals. We derive the ultimate quantum bound on precision and show that no improvement may be obtained by having access to the environment degrees of freedom. We found that, for small losses, the variance of the optimal estimator is proportional to the loss parameter itself, a result that represents a qualitative improvement over the shot noise limit. An observable based on the symmetric logarithmic derivative is derived, which attains the ultimate bound and may be implemented using Gaussian operations and…
Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér-Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A 52, 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in t…
Many different quantum information communication protocols such as teleportation, dense coding and entanglement based quantum key distribution are based on the faithful transmission of entanglement between distant location in an optical network. The distribution of entanglement in such a network is however hampered by loss and noise that is inherent in all practical quantum channels. Thus, to enable faithful transmission one must resort to the protocol of entanglement distillation. In this paper we present a detailed theoretical analysis and an experimental realization of continuous variable e…
Subtle quantum properties offer exciting new prospects in optical communications. Quantum entanglement enables the secure exchange of cryptographic keys and the distribution of quantum information by teleportation. Entangled bright beams of light attract increasing interest for such tasks, since they enable the employment of well-established classical communications techniques. However, quantum resources are fragile and undergo decoherence by interaction with the environment. The unavoidable losses in the communication channel can lead to a complete destruction of useful quantum properties -- …
I show that classical capacity per unit cost of noisy bosonic Gaussian channels can be attained by employing generalized on-off keying modulation format and a projective measurement of individual output states. This means that neither complicated collective measurements nor phase-sensitive detection is required to communicate over optical channels at the ultimate limit imposed by laws of quantum mechanics in the limit of low average cost.
A long-standing problem on the classical capacity of bosonic Gaussian channels has recently been resolved by proving the minimum output entropy conjecture. It is also known that the ultimate capacity quantified by the Holevo bound can be achieved asymptotically by using an infinite number of channels. However, it is less understood to what extent the communication capacity can be reached if one uses a finite number of channels, which is a topic of practical importance. In this paper, we study the capacity of Gaussian communication, i.e., employing Gaussian states and Gaussian measurements to e…
Optical communication channels are ultimately quantum-mechanical in nature, and we must therefore look beyond classical information theory to determine their communication capacity as well as to find efficient encoding and decoding schemes of the highest rates. Thermal channels, which arise from linear coupling of the field to a thermal environment, are of particular practical relevance; their classical capacity has been recently established, but their quantum capacity remains unknown. While the capacity sets the ultimate limit on reliable communication rates, it does not promise that such rat…
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cramér-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optima…
Quantum continuous variables are being explored as an alternative means to implement quantum key distribution, which is usually based on single photon counting. The former approach is potentially advantageous because it should enable higher key distribution rates. Here we propose and experimentally demonstrate a quantum key distribution protocol based on the transmission of gaussian-modulated coherent states (consisting of laser pulses containing a few hundred photons) and shot-noise-limited homodyne detection; squeezed or entangled beams are not required. Complete secret key extraction is ach…
We consider an instance of black-box quantum metrology in the Gaussian framework, where we aim to estimate the amount of squeezing applied on an input probe, without previous knowledge on the phase of the applied squeezing. By taking the quantum Fisher information (QFI) as the figure of merit, we evaluate its average and variance with respect to this phase in order to identify probe states that yield good precision for many different squeezing directions. We first consider the case of single-mode Gaussian probes with the same energy, and find that pure squeezed states maximize the average quan…
We consider the estimation of noise parameters in a quantum channel, assuming the most general strategy allowed by quantum mechanics. This is based on the exploitation of unlimited entanglement and arbitrary quantum operations, so that the channel inputs may be interactively updated. In this general scenario we draw a novel connection between quantum metrology and teleportation. In fact, for any teleportation-covariant channel (e.g., Pauli, erasure, or Gaussian channel), we find that adaptive noise estimation cannot beat the standard quantum limit, with the quantum Fisher information being det…
We address the experimental estimation of Gaussian quantum discord for two-mode squeezed thermal state, and demonstrate a measurement scheme based on a pair of homodyne detectors assisted by Bayesian analysis which provides nearly optimal estimation for small value of discord. Besides, though homodyne detection is not optimal for Gaussian discord, the noise ratio to the ultimate quantum limit, as dictacted by the quantum Cramer-Rao bound, is limited to about 10 dB.
We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We find that while for displaced thermal states an increase in temperature reduces the estimation fidelity, for squeezed thermal states a larger temperature can enhance the estimation fidelity. The many-copy optimal bounds are compared with the minimum variance achieved by three important single-shot measurement strategies. We show that the single-copy canonical phase measurement does not always…
We explore the advantages offered by twin light beams produced in parametric down-conversion for precision measurement. The symmetry of these bipartite quantum states, even under losses, suggests that monitoring correlations between the divergent beams permits a high-precision inference of any symmetry-breaking effect, e.g. fiber birefringence. We show that the quantity of entanglement is not the key feature for such an instrument. In a lossless setting, scaling of precision at the ultimate `Heisenberg' limit is possible with photon counting alone. Even as photon losses approach 100% the preci…
We introduce and experimentally explore the concept of quantum non-Gaussian depth of single-photon states with a positive Wigner function. The depth measures the robustness of a single-photon state against optical losses. The directly witnessed quantum non-Gaussianity withstands significant attenuation, exhibiting a depth of 18 dB, while the nonclassicality remains unchanged. Quantum non-Gaussian depth is an experimentally approachable quantity that is much more robust than the negativity of the Wigner function. Furthermore, we use it to reveal significant differences between otherwise strongl…
Highly quantum non-linear interactions between different bosonic modes lead to the generation of quantum non-Gaussian states, i.e. states that cannot be written as mixtures of Gaussian states. A paradigmatic example is given by Schroedinger's cat states, that is coherent superpositions of coherent states with opposite amplitude. We here consider a novel quantum non-Gaussianity criterion recently proposed in the literature and prove its effectiveness on Schroedinger cat states evolving in a lossy bosonic channel. We prove that quantum non-Gaussianity can be effectively detected for high values …
We investigate the capacity of bosonic quantum channels for the transmission of quantum information. Achievable rates are determined from measurable moments of the channel by showing that every channel can asymptotically simulate a Gaussian channel which is characterized by second moments of the initial channel. We calculate the quantum capacity for a class of Gaussian channels, including channels describing optical fibers with photon losses, by proving that Gaussian encodings are optimal. Along the way we provide a complete characterization of degradable Gaussian channels and those arising fr…
We theoretically present the quantum Cramér-Rao bounds (QCRB) of an SU(1,1) interferometer for Gaussian states input with and without the internal photonic losses. The phase shifts in the single arm and in the double arms are studied and the corresponding analytical expressions of quantum Fisher information with Gaussian input states are presented. Different from the traditional Mach-Zehnder interferometer, the QCRB of single arm case is slightly higher or lower than that of double arms case depending on the input states. With a fixed mean photon number and for pure Gaussian state input, the o…
The main challenge in deterministic quantum state transfer between remote mechanical resonators is the local decoherence and the transmission losses in the communication channel. In the path of overcoming this limitation, here we employ a shortcut to adiabatic passage protocol to devise a fast and reliable evolution path between two remote mechanical modes in separate optomechanical systems. A quantum state transfer between the two nodes is conceived by engineering their coupling to an intermediate fiber optical channel. The coupling pulses are operated such that the dark eigenmode of the syst…
We propose a quantum optics experiment where a single two-mode Gaussian entangled state is used for realizing the paradigm of an amendable Gaussian channel recently presented in Phys. Rev. A, \textbf{87}, 062307 (2013). Depending on the choice of the experimental parameters the entanglement of the probe state is preserved or not and the relative map belongs or not to the class of entanglement breaking channels. The scheme has been optimized to be as simple as possible: it requires only a single active non-linear operation followed by four passive beam-splitters. The effects of losses, detectio…
The entanglement negativity is a versatile measure of entanglement that has numerous applications in quantum information and in condensed matter theory. It can not only efficiently be computed in the Hilbert space dimension, but for non-interacting bosonic systems, one can compute the negativity efficiently in the number of modes. However, such an efficient computation does not carry over to the fermionic realm, the ultimate reason for this being that the partial transpose of a fermionic Gaussian state is no longer Gaussian. To provide a remedy for this state of affairs, in this work we introd…
The influence of losses in the transmission of continuous-variable entangled light through linear devices such as optical fibers is studied, with special emphasis on Gaussian states. Upper bounds on entanglement and the distance to the set of separable Gaussian states are calculated. Compared with the distance measure, the bounds can substantially overestimate the entanglement and thus do not show the drastic decrease of entanglement with increasing mean photon number, as does the distance measure. In particular, it shows that losses give rise to entanglement saturation, which principally limi…
We investigate the continuous-variable entanglement swapping protocol in a non-Gaussian setting, with non- Gaussian states employed either as entangled inputs and/or as swapping resources. The quality of the swapping protocol is assessed in terms of the teleportation fidelity achievable when using the swapped states as shared entangled resources in a teleportation protocol. We thus introduce a two-step cascaded quantum communication scheme that includes a swapping protocol followed by a teleportation protocol. The swapping protocol is fed by a general class of tunable non-Gaussian states, the …
A fundamental requirement for enabling fault-tolerant quantum information processing is an efficient quantum error-correcting code (QECC) that robustly protects the involved fragile quantum states from their environment. Just as classical error-correcting codes are indispensible in today's information technologies, it is believed that QECC will play a similarly crucial role in tomorrow's quantum information systems. Here, we report on the first experimental demonstration of a quantum erasure-correcting code that overcomes the devastating effect of photon losses. Whereas {\it errors} translate,…
We address local quantum estimation of bilinear Hamiltonians probed by Gaussian states. We evaluate the relevant quantum Fisher information (QFI) and derive the ultimate bound on precision. Upon maximizing the QFI we found that single- and two-mode squeezed vacuum represent an optimal and universal class of probe states, achieving the so-called Heisenberg limit to precision in terms of the overall energy of the probe. We explicitly obtain the optimal observable based on the symmetric logarithmic derivative and also found that homodyne detection assisted by Bayesian analysis may achieve estimat…
We demonstrate quantum correlations in the transverse plane of continuous wave light beams by producing -4.0 dB, -2.6 dB and -1.5 dB of squeezing in the TEM00, TEM10 and TEM20 Hermite- Gauss modes with an optical parametric amplifier, respectively. This has potential applications in quantum information networking, enabling parallel quantum information processing. We describe the setup for the generation of squeezing and analyze the effects of various experimental issues such as mode overlap between pump and seed and nonlinear losses.
We study interferometry with Gaussian states and show that an ancilla-assisted scheme outperforms coherent state interferometry for all levels of loss. We also compare the ancilla-assisted scheme to other interferometric schemes involving squeezing, and show that it is the most advantage in the high-loss, high photon-number regime. In fact, in the presence of high loss, it out-performs many other strategies proposed to date. We find the optimal measurement observable for each scheme discussed. We also find that with the appropriate measurement, the achievable precision of the proposal by Caves…
University of York（约克大学）；University of Florence（佛罗伦萨大学）；INFN Sezione di Firenze（意大利国家核物理研究所佛罗伦萨分所）
Quantum Channel Discrimination (QCD) presents a fundamental task in quantum information theory, with critical applications in quantum reading, illumination, data-readout and more. The extension to multiple quantum channel discrimination has seen a recent focus to characterise potential quantum advantage associated with quantum enhanced discriminatory protocols. In this paper, we study thermal imaging as an environment localisation task, in which thermal images are modelled as ensembles of Gaussian phase insensitive channels with identical transmissivity, and pixels possess properties according…
