Gaussian Operations Research Articles

Gaussian quantum resource theories

We develop a general framework to assess capabilities and limitations of the Gaussian toolbox in continuous variable quantum information theory. Our framework allows us to characterize the structure and properties of quantum resource theories specialized to Gaussian states and Gaussian operations, establishing rigorous methods for their description and yielding a unified approach to their quantification. We show in particular that, under a few intuitive and physically motivated assumptions on the set of free states, no Gaussian quantum resource can be distilled with free Gaussian operations, even when an unlimited supply of the resource state is available. This places fundamental constraints on state manipulations in all such Gaussian resource theories. We discuss in particular the applications to quantum entanglement, where we extend previously known results by showing that Gaussian entanglement cannot be distilled even with Gaussian operations preserving the positivity of the partial transpose, as well as to other Gaussian resources such as steering and optical nonclassicality. A comprehensive semidefinite programming representation of all these resources is explicitly provided.

Convex resource theory of non-Gaussianity

Continuous-variable systems realized in quantum optics play a major role in quantum information processing, and it is also one of the promising candidates for a scalable quantum computer. We introduce a resource theory for continuous-variable systems relevant to universal quantum computation. In our theory, easily implementable operations---Gaussian operations combined with feed-forward---are chosen to be the free operations, making the convex hull of the Gaussian states the natural free states. Since our free operations and free states cannot perform universal quantum computation, genuine non-Gaussian states---states not in the convex hull of Gaussian states---are the necessary resource states for universal quantum computation together with free operations. We introduce a monotone to quantify the genuine non-Gaussianity of resource states, in analogy to the stabilizer theory. A direct application of our resource theory is to bound the conversion rate between genuine non-Gaussian states. Finally, we give a protocol that probabilistically distills genuine non-Gaussianity---increases the genuine non-Gaussianity of resource states---only using free operations and postselection on Gaussian measurements, where our theory gives an upper bound for the distillation rate. In particular, the same protocol allows the distillation of cubic phase states, which enable universal quantum computation when combined with free operations.

Resource theory of non-Gaussian operations

Non-Gaussian states and operations are crucial for various continuous-variable quantum information processing tasks. To quantitatively understand non-Gaussianity beyond states, we establish a resource theory for non-Gaussian operations. In our framework, we consider Gaussian operations as free operations, and non-Gaussian operations as resources. We define entanglement-assisted non-Gaussianity generating power and show that it is a monotone that is non-increasing under the set of free super-operations, i.e., concatenation and tensoring with Gaussian channels. For conditional unitary maps, this monotone can be analytically calculated. As examples, we show that the non-Gaussianity of ideal photon-number subtraction and photon-number addition equal the non-Gaussianity of the single-photon Fock state. Based on our non-Gaussianity monotone, we divide non-Gaussian operations into two classes: (1) the finite non-Gaussianity class, e.g., photon-number subtraction, photon-number addition and all Gaussian-dilatable non-Gaussian channels; and (2) the diverging non-Gaussianity class, e.g., the binary phase-shift channel and the Kerr nonlinearity. This classification also implies that not all non-Gaussian channels are exactly Gaussian-dilatable. Our resource theory enables a quantitative characterization and a first classification of non-Gaussian operations, paving the way towards the full understanding of non-Gaussianity.

Nonclassicality by Local Gaussian Unitary Operations for Gaussian States.

A measure of nonclassicality in terms of local Gaussian unitary operations for bipartite Gaussian states is introduced. is a faithful quantum correlation measure for Gaussian states as product states have no such correlation and every non product Gaussian state contains it. For any bipartite Gaussian state , we always have , where the upper bound 1 is sharp. An explicit formula of for -mode Gaussian states and an estimate of for -mode Gaussian states are presented. A criterion of entanglement is established in terms of this correlation. The quantum correlation is also compared with entanglement, Gaussian discord and Gaussian geometric discord.

Quantum Transduction with Adaptive Control

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 various physical platforms. Moreover, we show that the adaptive protocol can be robust against imperfections associated with finite squeezing, thermal noise, and homodyne detection, and it can be implemented to realize quantum state transfer between microwave and optical modes.

Arbitrary multimode Gaussian operations on mechanical cluster states

We consider opto- and electro-mechanical quantum systems composed of a driven cavity mode interacting with a set of mechanical resonators. It has been proposed that the latter can be initialized in arbitrary cluster states, including universal resource states for Measurement Based Quantum Computation (MBQC). We show that, despite the unavailability in this set-up of direct measurements over the mechanical resonators, computation can still be performed to a high degree of accuracy. In particular, it is possible to indirectly implement the measurements necessary for arbitrary Gaussian MBQC by properly coupling the mechanical resonators to the cavity field and continuously monitoring the leakage of the latter. We provide a thorough theoretical analysis of the performances obtained via indirect measurements, comparing them with what is achievable when direct measurements are instead available. We show that high levels of fidelity are attainable in parameter regimes within reach of present experimental capabilities.

Encoding qubits into oscillators with atomic ensembles and squeezed light

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator provides a number of advantages when used in a fault-tolerant architecture for quantum computing, most notably that Gaussian operations suffice to implement all single- and two-qubit Clifford gates. The main drawback of the encoding is that the logical states themselves are challenging to produce. Here we present a method for generating optical GKP-encoded qubits by coupling an atomic ensemble to a squeezed state of light. Particular outcomes of a subsequent spin measurement of the ensemble herald successful generation of the resource state in the optical mode. We analyze the method in terms of the resources required (total spin and amount of squeezing) and the probability of success. We propose a physical implementation using a Faraday-based quantum non-demolition interaction.

Robust Gaussian teleportation with attenuation and nonunity gain

The average fidelity of the teleportation of a coherent state is calculated for general Gaussian bipartite systems shared by the partners of the protocol, Alice and Bob. It is considered that the shared Gaussian bipartite modes suffer independent attenuations before the processing of Alice and Bob. Moreover the classical communication between the partners can be controlled by a gain not necessarily unitary. Comparing with the classical fidelity threshold of measure-and-prepare methods, we establish several genuinely quantum teleportation conditions which depend on the gain and the local attenuations. Considering that the gain can be tuned to maximize the bipartite state set able to genuinely quantum teleportation, a condition for teleportation robust to local attenuations is found. This condition is demonstrated to be essentially equivalent to the condition of robust Gaussian bipartite entanglement, obtained in previous articles, showing that the attenuation robustness is an entanglement property relevant for characterization and application of bipartite systems. For the derivation of the robust teleportation conditions, the Gaussian operations onto the bipartite system are thoroughly studied, so that the transformations that maintain the fidelity invariant are found. Some scenarios for different Gaussian bipartite states are presented and discussed.

Gaussian Intrinsic Entanglement.

We introduce a cryptographically motivated quantifier of entanglement in bipartite Gaussian systems called Gaussian intrinsic entanglement (GIE). The GIE is defined as the optimized mutual information of a Gaussian distribution of outcomes of measurements on parts of a system, conditioned on the outcomes of a measurement on a purifying subsystem. We show that GIE vanishes only on separable states and exhibits monotonicity under Gaussian local trace-preserving operations and classical communication. In the two-mode case, we compute GIE for all pure states as well as for several important classes of symmetric and asymmetric mixed states. Surprisingly, in all of these cases, GIE is equal to Gaussian Rényi-2 entanglement. As GIE is operationally associated with the secret-key agreement protocol and can be computed for several important classes of states, it offers a compromise between computable and physically meaningful entanglement quantifiers.

Passivity and practical work extraction using Gaussian operations

Quantum states that can yield work in a cyclical Hamiltonian process form one of the primary resources in the context of quantum thermodynamics. Conversely, states whose average energy cannot be lowered by unitary transformations are called passive. However, while work may be extracted from non-passive states using arbitrary unitaries, the latter may be hard to realize in practice. It is therefore pertinent to consider the passivity of states under restricted classes of operations that can be feasibly implemented. Here, we ask how restrictive the class of Gaussian unitaries is for the task of work extraction. We investigate the notion of Gaussian passivity, that is, we present necessary and sufficient criteria identifying all states whose energy cannot be lowered by Gaussian unitaries. For all other states we give a prescription for the Gaussian operations that extract the maximal amount of energy. Finally, we show that the gap between passivity and Gaussian passivity is maximal, i.e., Gaussian-passive states may still have a maximal amount of energy that is extractable by arbitrary unitaries, even under entropy constraints.

Blind Image Quality Assessment with Local Contrast Features

The aim of this research is to create a tool to evaluate distortion in images without the information about original image. Work is to extract the statistical information of the edges and boundaries in the image and to study the correlation between the extracted features. Change in the structural information like shape and amount of edges of the image derives quality prediction of the image. Local contrast features are effectively detected from the responses of Gradient Magnitude (G) and Laplacian of Gaussian (L) operations. Using the joint adaptive normalisation, G and L are normalised. Normalised values are quantized into M and N levels respectively. For these quantised M levels of G and N levels of L, Probability (P) and conditional probability(C) are calculated. Four sets of values namely marginal distributions of gradient magnitude Pg, marginal distributions of Laplacian of Gaussian Pl, conditional probability of gradient magnitude Cg and probability of Laplacian of Gaussian Cl are formed. These four segments or models are Pg, Pl, Cg and Cl. The assumption is that the dependencies between features of gradient magnitude and Laplacian of Gaussian can formulate the level of distortion in the image. To find out them, Spearman and Pearson correlations between Pg, Pl and Cg, Cl are calculated. Four different correlation values of each image are the area of interest. Results are also compared with classical tool Structural Similarity Index Measure (SSIM)

Gaussian benchmark for optical communication aiming towards ultimate capacity

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, showing that Gaussian regime is not sufficient to achieve the Holevo bound particularly in the low-photon regime. Furthermore the Gaussian benchmark established here can be used to critically assess the performance of non-Gaussian protocols for optical communication. We move on to identify non-Gaussian schemes to beat the Gaussian capacity and show that a non-Gaussian receiver recently implemented by Becerra et al. [Nat. Photon. 7, 147 (2013)] can achieve this aim with an appropriately chosen encoding strategy.

Entropy generation in Gaussian quantum transformations: applying the replica method to continuous-variable quantum information theory

AbstractIn spite of their simple description in terms of rotations or symplectic transformations in phase space, quadratic Hamiltonians such as those modelling the most common Gaussian operations on bosonic modes remain poorly understood in terms of entropy production. For instance, determining the quantum entropy generated by a Bogoliubov transformation is notably a hard problem, with generally no known analytical solution, while it is vital to the characterisation of quantum communication via bosonic channels. Here we overcome this difficulty by adapting the replica method, a tool borrowed from statistical physics and quantum field theory. We exhibit a first application of this method to continuous-variable quantum information theory, where it enables accessing entropies in an optical parametric amplifier. As an illustration, we determine the entropy generated by amplifying a binary superposition of the vacuum and a Fock state, which yields a surprisingly simple, yet unknown analytical expression.

Channel purification via continuous-variable quantum teleportation with Gaussian postselection

We present a protocol based on continuous-variable quantum teleportation and Gaussian post- selection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian operations.

Amplification uncertainty relation for probabilistic amplifiers

Traditionally, quantum amplification limit refers to the property of inevitable noise addition on canonical variables when the field amplitude of an unknown state is linearly transformed through a quantum channel. Recent theoretical studies have determined amplification limits for cases of probabilistic quantum channels or general quantum operations by specifying a set of input states or a state ensemble. However, it remains open how much excess noise on canonical variables is unavoidable and whether there exists a fundamental trade-off relation between the canonical pair in a general amplification process. In this paper we present an uncertainty-product form of amplification limits for general quantum operations by assuming an input ensemble of Gaussian distributed coherent states. It can be derived as a straightforward consequence of canonical uncertainty relations and retrieves basic properties of the traditional amplification limit. In addition, our amplification limit turns out to give a physical limitation on probabilistic reduction of an Einstein-Podolsky-Rosen uncertainty. In this regard, we find a condition that probabilistic amplifiers can be regarded as local filtering operations to distill entanglement. This condition establishes a clear benchmark to verify an advantage of non-Gaussian operations beyond Gaussian operations with a feasible input set of coherent states and standard homodyne measurements.

Detection of copy–move forgery using discrete analytical Fourier–Mellin transform

In recent years, digital image processing has become commonplace with growing powerful and available image editing software. People without any professional technique can also manipulate and forge digital images easily. One of the most popular manners of digital image forgeries is the copy–move image forgery. Extensive researches in detecting copy–move forgery have made a deal of achievements, but most presented methods based on these researches have been only focus on some simple composite forgeries and not able to detect different types of post-processed forgeries. In this paper, we aim to deal with the post-processed forgery operations and scenarios, mainly geometric distortion. We introduce analytical Fourier–Mellin transform (AFMT) and focus on its discretization. We propose discrete analytical Fourier–Mellin transform (DAFMT). We also pay attention to high performance of DAFMT in detecting the copy–move image forgeries. Due to the AFMT described in polar coordinate, so we need to convert coordinate system from polar to Cartesian coordinates. To be computed conveniently, we define an auxiliary disk template to accomplish this conversion. We devote to the use of our proposed DAFMT in detection of image forgeries. A great deal of researches and experiments show that the proposed DAFMT can effectively resist translation, rotation, scaling, and added Gaussian noise operations. Compared with other relevant up-to-date methods, experiments also prove that DAFMT has made a progress in detecting and identifying the forgery images which are suffered from geometric distortion operations.

Bounds to precision for quantum interferometry with Gaussian states and operations

We address high-precision measurements by active and passive interferometric schemes based on Gaussian states and operations. In particular, we look for the best states to be injected into their ports according to the quantum Cramer–Rao bound, i.e., maximizing the quantum Fisher information over all the involved parameters, given a constraint on the overall mean number of photons entering into the interferometer. We found that for passive interferometers involving only beam splitters, the optimal input leading to Heisenberg scaling is a pair of identical squeezed-coherent states with at most one-third of the total energy employed in squeezing. For active interferometers involving optical amplifiers, input coherent signals are enough to achieve Heisenberg scaling, given an optimal value of the amplification gain. For passive schemes our results clarify the role of squeezing in improving both the reference phase and the signal phase of an interferometer.

Testing nonclassicality and non-gaussianity in phase space.

We theoretically propose and experimentally demonstrate a nonclassicality test of a single-mode field in phase space, which has an analogy with the nonlocality test proposed by Banaszek and Wódkiewicz [Phys. Rev. Lett. 82, 2009 (1999)]. Our approach to deriving the classical bound draws on the fact that the Wigner function of a coherent state is a product of two independent distributions as if the orthogonal quadratures (position and momentum) in phase space behave as local realistic variables. Our method detects every pure nonclassical Gaussian state, which can also be extended to mixed states. Furthermore, it sets a bound for all Gaussian states and their mixtures, thereby providing a criterion to detect a genuine quantum non-Gaussian state. Remarkably, our phase-space approach with invariance under Gaussian unitary operations leads to an optimized test for a given non-Gaussian state. We experimentally show how this enhanced method can manifest quantum non-Gaussianity of a state by simply choosing phase-space points appropriately, which is essentially equivalent to implementing a squeezing operation on a given state.

Photon sorting, efficient bell measurements, and a deterministic controlled-Z gate using a passive two-level nonlinearity.

Although the strengths of optical nonlinearities available experimentally have been rapidly increasing in recent years, significant challenges remain to using such nonlinearities to produce useful quantum devices such as efficient optical Bell state analyzers or universal quantum optical gates. Here we describe a new approach that avoids the current limitations by combining strong nonlinearities with active Gaussian operations in efficient protocols for Bell state analyzers and controlled-sign gates.

Repeat-until-success cubic phase gate for universal continuous-variable quantum computation

In order to achieve universal quantum computation using continuous variables, one needs to jump out of the set of Gaussian operations and have a non-Gaussian element, such as the cubic phase gate. However, such a gate is currently very difficult to implement in practice. Here we introduce an experimentally viable 'repeat-until-success' approach to generating the cubic phase gate, which is achieved using sequential photon subtractions and Gaussian operations. We find that our scheme offers benefits in terms of the expected time until success, although we require a primitive quantum memory.