Properties Of Quantum Systems Research Articles

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.

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.

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.

Universal Out-of-Equilibrium Dynamics of 1D Critical Quantum Systems Perturbed by Noise Coupled to Energy

We consider critical one dimensional quantum systems initially prepared in their groundstate and perturbed by a smooth noise coupled to the energy density. By using conformal field theory, we deduce a universal description of the out-of-equilibrium dynamics. In particular, the full time-dependent distribution of any $2$--pt chiral correlation function can be obtained from solving two coupled ordinary stochastic differential equations. In contrast with the general expectation of heating, we demonstrate that the system reaches a non-trivial and universal stationary state characterized by broad distributions. As an example, we analyse the local energy density: while its first moment diverges exponentially fast in time, the stationary distribution, which we derive analytically, is symmetric around a negative median and exhibits a fat tail with $3/2$ decay exponent. We obtain a similar result for the entanglement entropy production associated to a given interval of size $\ell$. The corresponding stationary distribution has a $3/2$ right tail for all $\ell$, and converges to a one-sided Levy stable for large $\ell$. Our results are benchmarked via analytical and numerical calculations for a chain of non-interacting spinless fermions with excellent agreement.

Gaussian Quantum Systems and Kahler Geometrical Structure

In this article, we study the phase-space distribution of the quantum state as a framework to describe the different properties of quantum systems in continuous-variable systems. The natural approach to quantum systems is given the Gaussian Wigner representation, to unify the description of bosonic and fermionic quantum states, we study the structure of the Kahler space geometry as the geometry generated by three forms under the agreement conditions depended on the nature of the state bosonic or fermionic. Multimode light is studied, and we established that the Fock space vacuum corresponds to a certain homogeneous Gaussian state.

Possibilities are quantum

Creative possibilities require imagination, intuition, vision, creativity, comprehension, and inventiveness, properties that are associated with human consciousness and free will. New developments in quantum information theory have allowed to recently postulate with high confidence that consciousness and free will are properties of quantum systems in pure quantum states. This new development allows saying that possibilities emerge from the quantum level of reality since a classical world is deterministic, that is, algorithmic and predictable, and thus incapable of real creativity. Only a creative agent like a human being could imagine real possibilities in a near-infinite combinatorial sea of patterns, and then figure out some intelligent way to realize them. Even if nature were aimlessly creating random patterns someone should have the comprehension to recognize a creative possibility among them. But comprehension is a non-algorithmic property that requires consciousness.

Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach

Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine, and induced) and hidden exceptional points, which are not recognized directly in amplitude spectra, are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in infinitely dimensional open quantum systems.

Detecting quantum phase transition via magic resource in the XY spin model

Quantum phase transition in the $XY$ spin model with three-spin interaction is investigated using magic resource (non-stabilizerness), which is crucial in universal fault-tolerant quantum computation. The magic quantifier we employ here is defined straightforwardly via characteristic functions of quantum states, which are well defined for all dimensional quantum systems (in sharp contrast to those defined by discrete Wigner functions) and can be easily calculated. We show that the magic quantifier of both the reduced single-site spins and two-site spins of the system ground state increase to their maximum around the critical points for quantum phase transition. This indicates that the magic resource can be used to detect the critical phenomena in the $XY$ spin model and reveals a connection between quantum phase transition in many-body systems and quantum resource in stabilizer quantum computation.

Quantum algorithms from fluctuation theorems: Thermal-state preparation

Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with HamiltoniansH0andH1=H0+V. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state ofH1at inverse temperatureβ≥0starting from a purification of the thermal state ofH0. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, isO~(eβ(ΔA−wl)/2), whereΔAis the free-energy difference betweenH1andH0,andwlis a work cutoff that depends on the properties of the work distribution and the approximation errorϵ>0. If the non-equilibrium process is trivial, this complexity is exponential inβ‖V‖, where‖V‖is the spectral norm ofV. This represents a significant improvement of prior quantum algorithms that have complexity exponential inβ‖H1‖in the regime where‖V‖≪‖H1‖. The dependence of the complexity inϵvaries according to the structure of the quantum systems. It can be exponential in1/ϵin general, but we show it to be sublinear in1/ϵifH0andH1commute, or polynomial in1/ϵifH0andH1are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value ofwland improve the complexity even further. To this end, we analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes and see significant complexity improvements.

Topologically-imposed vacancies and mobile solid 3He on carbon nanotube

Low dimensional fermionic quantum systems are exceptionally interesting because they reveal distinctive physical phenomena, including among others, topologically protected excitations, edge states, frustration, and fractionalization. Our aim was to confine 3He on a suspended carbon nanotube to form 2-dimensional Fermi-system. Here we report our measurements of the mechanical resonance of the nanotube with adsorbed sub-monolayer down to 10 mK. At intermediate coverages we have observed the famous 1/3 commensurate solid. However, at larger monolayer densities we have observed a quantum phase transition from 1/3 solid to an unknown, soft, and mobile solid phase. We interpret this mobile solid phase as a bosonic commensurate crystal consisting of helium dimers with topologically-induced zero-point vacancies which are delocalized at low temperatures. We thus demonstrate that 3He on a nanotube merges both fermionic and bosonic phenomena, with a quantum phase transition between fermionic solid 1/3 phase and the observed bosonic dimer solid.

Direct determination of arbitrary dimensional entanglement monotones using statistical correlators and minimal complementary measurements

Higher dimensional quantum systems (qudits) present a potentially more efficient means, compared to qubits, for implementing various information theoretic tasks. One of the ubiquitous resources in such explorations is entanglement. Entanglement monotones (EMs) are of key importance, particularly for assessing the efficacy of a given entangled state as a resource for information theoretic tasks. Till date, investigations towards determination of EMs have focused on providing their tighter lower bounds. There is yet no general scheme available for direct determination of the EMs. Consequently, an empirical determination of any EM has not yet been achieved for entangled qudit states. The present paper fills this gap, both theoretically as well as experimentally. First, we derive analytical relations between statistical correlation measures i.e. mutual predictability (MP), mutual information (MI) and Pearson correlation coefficient (PCC) and standard EMs i.e. negativity (N) and entanglement of formation (EOF) in arbitrary dimensions. As a proof of concept, we then experimentally measure MP, MI and PCC of two-qutrit pure states and determine their N and EOF using these derived relations. This is a useful addition to the experimenter’s toolkit wherein by using a limited number of measurements (in this case 1 set of measurements), one can directly measure the EMs in a bipartite arbitrary dimensional system. We obtain the value of N for our bipartite qutrit to be 0.907 ± 0.013 and the EOF to be 1.323 ± 0.022. Since the present scheme enables determination of more than one EM by the same limited number of measurements, we argue that it can serve as a unique experimental platform for quantitatively comparing and contrasting the operational implications of EMs as well as showing their non-monotonicity for a given bipartite pure qudit state.