Properties Of Quantum Systems Research Articles

Exploring Nonlocal Correlations in Arbitrarily High-Dimensional Systems

Abstract The utilization of higher-dimensional systems holds promise for unveiling novel phenomena and enhancing the efficacy of practical tasks, such as entanglement-based quantum cryptography. However, detecting quantum correlations within qudit systems poses a formidable challenge. In this study, we delve into the intricacies of Bell’s nonlocality within higher-dimensional systems. We introduce a novel correlation function and leverage it to construct a family of Bell inequalities adapted for quantum systems of arbitrary dimensionality. By gauging the probability of local realism violation under random measurements as our primary metric, we explore the relation between pure state entanglement and nonlocality. Our findings align closely with predictions derived from the comprehensive polytope analysis via linear programming methods. While our focus predominantly centers on the two-qudit scenario, our methodology readily extends to the N -partite case, accommodating an arbitrary number of measurements per party.

Fermi’s golden rule in tunneling models with quantum waveguides perturbed by Kato class measures

Abstract In this paper we consider two dimensional quantum system with an infinite waveguide of the width d and a transversally invariant profile. Furthermore, we assume that at a distant ρ there is a perturbation defined by the Kato measure. We show that, under certain conditions, the resolvent of the Hamiltonian has the second sheet pole which reproduces the resonance at z ( ρ ) with the asymptotics z ( ρ ) = E β ; n + O ( exp ⁡ ( − 2 | E β ; n | ρ ) ρ ) for ρ large and with the resonant energy E β ; n . Moreover, we show that the imaginary component of z ( ρ ) satisfies Fermi’s golden rule which we explicitly derive.

Quantum subspace controllability implying full controllability

In the analysis of controllability of finite dimensional quantum systems, subspace controllability refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics.We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state (Pure State Controllability) is verified.

Learning quantum properties from short-range correlations using multi-task networks

Characterizing multipartite quantum systems is crucial for quantum computing and many-body physics. The problem, however, becomes challenging when the system size is large and the properties of interest involve correlations among a large number of particles. Here we introduce a neural network model that can predict various quantum properties of many-body quantum states with constant correlation length, using only measurement data from a small number of neighboring sites. The model is based on the technique of multi-task learning, which we show to offer several advantages over traditional single-task approaches. Through numerical experiments, we show that multi-task learning can be applied to sufficiently regular states to predict global properties, like string order parameters, from the observation of short-range correlations, and to distinguish between quantum phases that cannot be distinguished by single-task networks. Remarkably, our model appears to be able to transfer information learnt from lower dimensional quantum systems to higher dimensional ones, and to make accurate predictions for Hamiltonians that were not seen in the training.

Compressing continuous variable quantum measurements

We generalize the notion of joint measurability to continuous variable systems by extending a recently introduced compression algorithm of quantum measurements to this realm. The extension results in a property that asks for the minimal dimensional quantum system required for representing a given set of quantum measurements. To illustrate the concept, we show that the canonical pair of position and momentum is completely incompressible. We translate the concept of measurement compression to the realm of quantum correlations, where it results in a generalization of continuous variable quantum steering. In contrast to the steering scenario, which detects entanglement, the generalization detects the dimensionality of entanglement. We illustrate the bridge between the concepts by showing that an analogue of the original EPR argument is genuinely infinite-dimensional with respect to our figure of merit, and that a fundamental discrete variable result on preparability of unsteerable state assemblages with separable states does not directly carry over to the continuous variable setting. We further prove a representation result for partially entanglement breaking channels that can be of independent interest.

Parameterized Multipartite Entanglement and Genuine Entanglement Measures Based on q-Concurrence.

We study genuine multipartite entanglement (GME) and multipartite k-entanglement based on q-concurrence. Well-defined parameterized GME measures and measures of multipartite k-entanglement are presented for arbitrary dimensional n-partite quantum systems. Our GME measures show that the GHZ state is more entangled than the W state. Moreover, our measures are shown to be inequivalent to the existing measures according to entanglement ordering. Detailed examples show that our measures characterize the multipartite entanglement finer than some existing measures, in the sense that our measures identify the difference of two different states while the latter fail.

Path integrals, complex probabilities and the discrete Weyl representation

A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex probabilities is motivated by a recent reinterpretation of the real-time path integral as the expectation value of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The discrete formulation in this work is based on a discrete version of the Weyl algebra that can be applied to any observable with a finite number of outcomes. The origin of the complex probability in this work is the completeness relation. In the discrete formulation the complex probability exactly factors into products of conditional probabilities and exact unitarity is maintained at each level of approximation. The approximation of infinite dimensional quantum systems by discrete systems is discussed. The method is illustrated by applying it to scattering theory and quantum field theory. The implications of these applications for quantum computing is discussed.

Polarized ZZ pairs in gluon fusion and vector boson fusion at the LHC

Pair production of helicity-polarized weak bosons (Vλ=Wλ±,Zλ) from gluon fusion (gg→VλVλ′′) and weak boson fusion (V1V2→VλVλ′′) are powerful probes of the Standard Model, new physics, and properties of quantum systems. Measuring cross sections of polarized processes is a chief objective of the Large Hadron Collider's (LHC) Run 3 and high luminosity programs, but progress is limited by the simulation tools that are presently available. We propose a method for computing polarized cross sections that works by directly modifying Feynman rules instead of (squared) amplitudes. The method is applicable to loop-induced processes, and can capture the interference between arbitrary polarization configurations, interference with non-resonant diagrams, as well as off-shell/finite-width effects. By construction, previous results that work at the (squared) amplitude level are recoverable. As a demonstration, we report the prospect of observing and studying polarized ZλZλ′ pairs when produced via gluon fusion and electroweak processes in final-states with four charged leptons at the LHC, using the new method to simulate the gluon fusion process. Our Feynman rules are publicly available as a set of Universal FeynRules Object libraries called SM_Loop_VPolar.

Quantum Measurement Classification Using Statistical Learning

Interpreting the results of a quantum computer can pose a significant challenge due to inherent noise in these mesoscopic quantum systems. Quantum measurement, a critical component of quantum computing, involves determining the probabilities linked with quantum states post-multiple circuit computations based on quantum readout values provided by hardware. While there are promising classification-based solutions, they can either misclassify or necessitate excessive measurements, thereby proving to be costly. This article puts forth an efficient method to discern the quantum state by analyzing the probability distributions of data post-measurement. Specifically, we employ cumulative distribution functions to juxtapose the measured distribution of a sample against the distributions of basis states. The efficacy of our approach is demonstrated through experimental results on a superconducting transmon qubit architecture, which shows a substantial decrease (88%) in single qubit readout error compared to state-of-the-art measurement techniques. Moreover, we report additional error reduction (12%) compared to state-of-the-art measurement techniques when our technique is applied to enhance existing multi-qubit classification techniques. We also demonstrate the applicability of our proposed method for higher dimensional quantum systems, including classification of single qutrits as well as multiple qutrits.

On complementarity and distribution of imaginarity in finite dimensions

We investigate the complementarity relation among the l1 norm of imaginarity, linear entropy and the Brukner–Zeilinger invariant information in the presence of conjugate symmetry. By using an equality derived from the complementarity relation, we define a quantity named index of imaginarity distribution with respect to the imaginarity, linear entropy and invariant information in finite dimensional quantum systems. We show that the index of imaginarity distribution is nonnegative and upper bounded by the square of the l1 norm of imaginarity, and serves as an invaluable tool in characterizing effectively the imaginarity distribution. Detailed examples are presented to illustrate our theoretical analysis.

Multipartite entanglement detection via generalized Wigner–Yanase skew information

The detection of multipartite entanglement in multipartite quantum systems is a fundamental and key issue in quantum information theory. In this paper, we investigate k-nonseparability and k-partite entanglement of N-partite quantum systems from the perspective of the generalized Wigner–Yanase skew information. We deduce the upper bounds on the generalized Wigner–Yanase skew information of k-separable states and k-producible states in arbitrary dimensional multipartite quantum systems. And based on the upper bounds, we develop two different approaches in form of simple inequalities to construct entanglement criteria, which are expressed in terms of the generalized Wigner–Yanase skew information. Any violation of these inequalities by a quantum state reveals its k-nonseparability or k-partite entanglement, so these inequalities present the hierarchic classifications of k-nonseparability or k-partite entanglement for all N-partite quantum states from N-nonseparability to 2-nonseparability or from 2-partite entanglement to N-partite entanglement. Some well-known entanglement criteria are the special cases of our criteria. We illustrate the significance of our criteria by showing that they can reveal k-nonseparability and k-partite entanglement undetectable by well-known criteria through several typical examples, and their advantages over well-known criteria.

Quantum Bisimilarity via Barbs and Contexts: Curbing the Power of Non-deterministic Observers

Past years have seen the development of a few proposals for quantum extensions of process calculi. The rationale is clear: with the development of quantum communication protocols, there is a need to abstract and focus on the basic features of quantum concurrent systems, like CCS and CSP have done for their classical counterparts. So far, though, no accepted standard has emerged, neither for the syntax nor for the behavioural semantics. Indeed, the various proposals do not agree on what should be the observational properties of quantum values, and as a matter of fact, the soundness of such properties has never been validated against the prescriptions of quantum theory. To this aim, we introduce a new calculus, Linear Quantum CCS (lqCCS), and investigate the features of behavioural equivalences based on barbs and contexts. Our calculus can be thought of as an asynchronous, linear version of qCCS, which is in turn based on value-passing CCS. The combination of linearity and asynchronous communication fits well with the properties of quantum systems (e.g. the no-cloning theorem), since it ensures that each qubit is sent exactly once, precisely specifying which qubits of a process interact with the context. We exploit contexts to examine how bisimilarities relate to quantum theory. We show that the observational power of general contexts is incompatible with quantum theory: roughly, they can perform non-deterministic moves depending on quantum values without measuring (hence perturbing) them. Therefore, we refine the operational semantics in order to prevent contexts from performing unfeasible non-deterministic choices. This induces a coarser bisimilarity that better fits the quantum setting: (i) it lifts the indistinguishability of quantum states to the distributions of processes and, despite the additional constraints, (ii) it preserves the expressiveness of non-deterministic choices based on classical information. To the best of our knowledge, our semantics is the first one that satisfies the two properties above.

Typical bipartite steerability and generalized local quantum measurements

Recently proposed correlation-matrix-based sufficient conditions for bipartite steerability from Alice to Bob for arbitrary measurements are applied to local informationally complete positive operator valued measures (POVMs) of the (N,M)-type. These POVMs include a large class of local generalized measurements of current interest [Siudzińska K 2022 Phys. Rev. A 105 042209]. It is shown that the trace norm of correlation matrices with local (N,M)-POVMs is proportional to that of local orthonormal hermitian operator bases (LOOs). This implies that all types of informationally complete (N,M)-POVMs are equally powerful in detecting bipartite steerability from Alice to Bob and, in addition, they are as powerful as LOOs. In order to explore the typicality of steering numerical calculations of lower bounds on Euclidean volume ratios between steerable bipartite quantum states from Alice to Bob and all quantum states are determined with the help of a hit-and-run Monte-Carlo algorithm. These results demonstrate that with the single exception of two qubits this correlation-matrix-based sufficient condition significantly underestimates these volume ratios. These results are also compared to a recently proposed method that determines bipartite steerability from Alice’s qubit to Bob’s arbitrary dimensional quantum system by bipartite entanglement detection. It is demonstrated that in general this method is significantly more effective in detecting typical steerability provided non local entanglement detection methods are used which transcend local measurements.

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.

Closed-form analytic expressions for shadow estimation with brickwork circuits

Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haar-random (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables.We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.

A hybrid-qudit representation of digital RGB images

Quantum image processing is an emerging topic in the field of quantum information and technology. In this paper, we propose a new quantum image representation of RGB images with deterministic image retrieval, which is an improvement over all the similar existing representations in terms of using minimum resource. We use two entangled quantum registers constituting of total 7 qutrits to encode the color channels and their intensities. Additionally, we generalize the existing encoding methods by using both qubits and qutrits to encode the pixel positions of a rectangular image. This hybrid-qudit approach aligns well with the current progress of NISQ devices in incorporating higher dimensional quantum systems than qubits. We then describe the image encoding method using higher-order qubit-qutrit gates, and demonstrate the decomposition of these gates in terms of simpler elementary gates. We use the Google Cirq’s quantum simulator to verify the image preparation in both the ideal noise-free scenario and in presence of realistic noise modelling. We show that the complexity of the image encoding process is linear in the number of pixels. Lastly, we discuss the image compression and some basic RGB image processing protocols using our representation.

Data re-uploading with a single qudit

Quantum two-level systems, i.e., qubits, form the basis for most quantum machine learning approaches that have been proposed throughout the years. However, higher dimensional quantum systems constitute a promising alternative and are increasingly explored in theory and practice. Here, we explore the capabilities of multi-level quantum systems, so-called qudits, for their use in a quantum machine learning context. We formulate classification and regression problems with the data re-uploading approach and demonstrate that a quantum circuit operating on a single qudit is able to successfully learn highly non-linear decision boundaries of classification problems such as the MNIST digit recognition problem. We demonstrate that the performance strongly depends on the relation between the qudit states representing the labels and the structure of labels in the training data set. Such a bias can lead to substantial performance improvement over qubit-based circuits in cases where the labels, the qudit states, and the operators employed to encode the data are well-aligned. Furthermore, we elucidate the influence of the choice of the elementary operators and show that a squeezing operator is necessary to achieve good performances. We also show that there exists a trade-off for qudit systems between the number of circuit-generating operators in each processing layer and the total number of layers needed to achieve a given accuracy. Finally, we compare classification results from numerically exact simulations and their equivalent implementation on actual IBM quantum hardware. The findings of our work support the notion that qudit-based algorithms exhibit attractive traits and constitute a promising route to increasing the computational capabilities of quantum machine learning approaches.

Counting stabiliser codes for arbitrary dimension

In this work, we compute the number of [[n,k]]d stabilizer codes made up of d-dimensional qudits, for arbitrary positive integers d. In a seminal work by Gross \cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm, but when the qudits are Galois-qudits). The proof in \cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n,k]]d codes. Our work overlaps with \cite{Gross2006} when d is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether d is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.

Disorder effect on intersubband optical absorption of n-type δ-doped quantum well in GaAs

The inevitable structural disorder associated with the fluctuation of the applied external electric field, laser intensity, and bidimensional density in the low dimensional quantum system can affect noticeably optical absorption properties and the related phenomena. In this work, we study the effect of structural disorder on the optical absorption properties in delta-doped quantum wells (DDQWs). Starting from effective mass approximation and the Thomas-Fermi approach as well as using the matrix density, the electronic structure and the optical absorption coefficients of DDQWs are determined. It is found that the optical absorption properties depend on the strength and the type of structural disorder. Particularly, the bidimensional density disorder suppresses strongly the optical properties. Whilst, the disordered external applied electric field fluctuates moderately in the properties. In contrast, the disordered laser holds absorption properties unalterable. So, our results specify that to have and preserve good optical absorption properties in DDQWs, requires precise control of the bidimensional. Besides, the finding may improve the understanding of the impact of the disorder on the optoelectronic properties based on DDQWs.

Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor.

Finding the transient and steady state properties of open quantum systems is a central problem in various fields of quantum technologies. Here, we present a quantum-assisted algorithm to determine the steady states of open system dynamics. By reformulating the problem of finding the fixed point of Lindblad dynamics as a feasibility semidefinite program, we bypass several well-known issues with variational quantum approaches to solving for steady states. We demonstrate that our hybrid approach allows us to estimate the steady states of higher dimensional open quantum systems and discuss how our method can find multiple steady states for systems with symmetries.