Arbitrary Pure State Research Articles

Classical simulation of non-Gaussian fermionic circuits

We propose efficient algorithms for classically simulating fermionic linear optics operations applied to non-Gaussian initial states. By gadget constructions, this provides algorithms for fermionic linear optics with non-Gaussian operations. We argue that this problem is analogous to that of simulating Clifford circuits with non-stabilizer initial states: Algorithms for the latter problem immediately translate to the fermionic setting. Our construction is based on an extension of the covariance matrix formalism which permits to efficiently track relative phases in superpositions of Gaussian states. It yields simulation algorithms with polynomial complexity in the number of fermions, the desired accuracy, and certain quantities capturing the degree of non-Gaussianity of the initial state. We study one such quantity, the fermionic Gaussian extent, and show that it is multiplicative on tensor products when the so-called fermionic Gaussian fidelity is. We establish this property for the tensor product of two arbitrary pure states of four fermions with positive parity.

Quantum G-coherence factorization law under fully and strictly incoherent operations

Quantum coherence plays an important role in quantum resource theory, which is strongly related with entanglement. In order to quantify the full coherence of qudit states, we define G-coherence and convex roof of G-coherence, and prove that the G-coherence is a strong coherence monotone and the convex roof of G-coherence is a coherence measure under fully and strictly incoherent operation (FSIO), respectively. Similar to the entanglement factorization law, we prove a coherence factorization law for arbitrary d-dimensional quantum pure and mixed states under FSIO channels, which generalizes the entanglement factorization law for bipartite pure states. Our results will play an important role in the simplification of dynamical coherence measure.

Quantifying Bell nonlocality of a pure two-qudit state via its entanglement

For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.

Unifying nonclassical correlation via Tsallis [formula omitted]entropy and its application in bilocal scenario

This article introduces a new variant of a family of measurement-induced nonlocality (MIN) measures using Tsallis α−relative entropy (TαE). We show that the proposed MIN measure satisfies all the axioms of a faithful measure of quantum correlation and resolves the local ancilla issue of Hilbert–Schmidt MIN. Analytically, we evaluate the MIN for an arbitrary pure state and 2×n-dimensional mixed state. We also establish the simple relations between Tsallis MIN and the other MINs available in the literature. Extending the notion of the Tsallis entropy–based MIN, we characterize the nonbilocal correlation in a bilocal scenario and identify the closer connection between the nonlocal and nonbilocal correlation measures.

Real quantum operations and state transformations

Resource theory of imaginarity provides a useful framework to understand the role of complex numbers, which are essential in the formulation of quantum mechanics, in a mathematically rigorous way. In the first part of this article, we study the properties of ‘real’ (quantum) operations both in single-party and bipartite settings. As a consequence, we provide necessary and sufficient conditions for state transformations under real operations and show the existence of ‘real entanglement’ monotones. In the second part of this article, we focus on the problem of single copy state transformation via real quantum operations. When starting from pure initial states, we completely solve this problem by finding an analytical expression for the optimal fidelity of transformation, for a given probability of transformation and vice versa. Moreover, for state transformations involving arbitrary initial states and pure final states, we provide a semidefinite program to compute the optimal achievable fidelity, for a given probability of transformation.

Enhancing nonclassical properties of quantum states of light using linear optics.

In this Letter, we present a simple and versatile scheme for enhancing the nonclassical properties of light states using only linear optics and photodetectors. By combining a coherent state |α〉 and an arbitrary pure state of light |ϕ〉 (excluding coherent states) at two beam splitters, we show that the amplitude α of the coherent state can be tuned to filter out specific Fock components and generate states of light with increased nonclassical features. We provide two examples of input states and demonstrate the effectiveness of our scheme in enhancing the sub-Poissonian statistics or the quadrature squeezing of the output states.

Controlling gain with loss: Bounds on localizable entanglement in multiqubit systems

We investigate the relation between the amount of entanglement localized on a chosen subsystem of a multiqubit system via local measurements on the rest of the system, and the bipartite entanglement that is lost during this measurement process. We study a number of paradigmatic pure states, including the generalized Greenberger--Horne--Zeilinger (GHZ), the generalized $W$ (gW), Dicke, and the generalized Dicke states. For the generalized GHZ and $W$ states, we analytically derive bounds on localizable entanglement in terms of the entanglement present in the system prior to the measurement. Also, for the Dicke and the generalized Dicke states, we demonstrate that with increasing system size, localizable entanglement tends to be equal to the bipartite entanglement present in the system over a specific partition before measurement. We extend the investigation numerically in the case of arbitrary multiqubit pure states. We also analytically determine the modification of these results, including the proposed bounds, in situations where these pure states are subjected to single-qubit phase-flip noise on all qubits. Additionally, we study one-dimensional paradigmatic quantum spin models, namely, the transverse-field XY model and the XXZ model in an external field, and numerically demonstrate a cubic dependence of the localized entanglement on the lost entanglement. We show that this relation is robust even in the presence of disorder in the strength of the external field.

Engineering superpositions of N00N states using an asymmetric non-linear Mach–Zehnder interferometer

We revisit a method for mapping arbitrary single-mode pure states into superpositions of N00N states using an asymmetric non-linear Mach–Zehnder interferometer (ANLMZI). This method would allow one to tailor-make superpositions of N00N states where each axis of the two-mode joint-photon number distribution is weighted by the statistics of any single-mode pure state. The non-linearity of the ANLMZI comes in the form of a χ(3) self-Kerr interaction occurring on one of the intermediary modes of the interferometer. Motivated by the non-classical interference effects that occur at a beam splitter, we introduce inverse-engineering techniques aimed toward extrapolating optimal transformations for generating N00N state superpositions. These techniques are general enough so as to be employed to probe the means of generating states of any desired quantum properties.

Bounds on positive operator-valued measure based coherence of superposition

Quantum coherence is a fundamental feature of quantum physics and plays a significant role in quantum information processing. By generalizing the resource theory of coherence from von Neumann measurements to positive operator-valued measures (POVMs), POVM-based coherence measures have been proposed with respect to the relative entropy of coherence, the l 1 norm of coherence, the robustness of coherence and the Tsallis relative entropy of coherence. We derive analytically the lower and upper bounds on these POVM-based coherence of an arbitrary given superposed pure state in terms of the POVM-based coherence of the states in superposition. Our results can be used to estimate range of quantum coherence of superposed states. Detailed examples are presented to verify our analytical bounds.

Characterizing nonclassical correlations of tensorizing states in a bilocal scenario

In the present paper, we attempt to address the question of “can tensorizing states $$(\rho \otimes \rho \; \text {or}\;\rho \otimes \rho ')$$ have quantum advantages?”. To answer this question, we exploit the notion of measurement-induced nonlocality (MIN) and advocate a fidelity-based nonbilocal measure to capture the nonlocal effects of tensorizing states due to locally invariant von Neumann projective measurements. We show that the properties of the fidelity-based nonbilocal measures are retrieved from that of MIN. Analytically, we evaluate the nonbilocal measure for any arbitrary pure state. The upper bounds of the nonbilocal measure based on fidelity are also obtained in terms of eigenvalues of correlation matrix. As an illustration, we have computed the nonbilocality for some popular input states.

Unification of coherence and quantum correlations in tripartite systems

In quantum resource theories (QRTs), there exists evidences of intrinsic connections among different measures of quantum resources, including entanglement, coherence, quantum steering, and so on. However, building the relations among different quantum resources is a vital yet challenging task in multipartite quantum systems. Here, we focus on a unified framework of interpreting the interconversions among different quantum resources in tripartite systems. In particular, an exact relation between the generalized geometric measure and the genuinely multipartite concurrence are derived for tripartite entanglement states. Then we obtain the tradeoff relation between the first-order coherence and the genuine tripartite entanglement by the genuinely multipartite concurrence and concurrence fill. Furthermore, the tradeoff relation between the maximum steering inequality violation and concurrence fill for an arbitrary three-qubit pure state is found. In addition, we investigate the close relation between the maximum steering inequality violation and the first-order coherence. The results show that these quantum resources are intrinsic related and can be converted to each other in the framework of QRTs, although they are still regarded to be different.

SWAP test for an arbitrary number of quantum states

We develop a recursive algorithm to generalize the quantum SWAP test for an arbitrary number m of quantum states requiring O(m) controlled-swap (CSWAP) gates and $$O(\log m)$$ ancillary qubits. We construct a quantum circuit able to simultaneously measure overlaps $$|\langle \phi _i, \phi _j\rangle |^2$$ of m arbitrary pure states $${|{\phi _1\ldots \phi _m}\rangle }$$ . Our construction relies on a pairing unitary that generates a superposition state where every pair of input states is labeled by a basis state formed by the ancillaries. By implementing a simple genetic algorithm, we give numerical evidence indicating that our method of labeling each pair of inputs using CSWAP gates is optimal up to $$m=8$$ . Potential applications of the new circuits in the context of quantum machine learning are discussed.

Polarization conversion using customized subwavelength laser-induced periodic surface structures on stainless steel

Stainless steel is a basic raw material used in many industries. It can be customized by generating laser-induced periodic surface structure (LIPSS) as subwavelength gratings. Here, we present the capabilities of an LIPSS on stainless steel to modify the polarization state of the reflected radiation at the IR band. These structures have been modeled using the finite element method and fabricated by femtosecond laser processing. The Stokes parameters have been obtained experimentally and a model for the shape has been used to fit the simulated Stokes values to the experimental data. The birefringence of the LIPSS is analyzed to explain how they modify the polarization state of the incoming light. We find the geometry of the subwavelength grating that makes it work as an optical retarder that transforms a linearly polarized light into a circularly polarized wave. In addition, the geometrical parameters of the LIPSS are tuned to selectively absorb one of the components of the incoming light, becoming a linear axial polarizer. Appropriately selecting the geometrical parameters and orientation of the fabricated LIPSS makes it possible to obtain an arbitrary pure polarization state when illuminated by a pure linearly polarized state oriented at an azimuth of 45°. The overall reflectance of these transformations reaches values close to 60% with respect to the incident intensity, which is the same reflectivity obtained for non-nanostructured stainless steel flat surfaces.

Entangled subspaces and generic local state discrimination with pre-shared entanglement

Walgate and Scott have determined the maximum number of generic pure quantum states that can be unambiguously discriminated by an LOCC measurement [Journal of Physics A: Mathematical and Theoretical, 41:375305, 08 2008]. In this work, we determine this number in a more general setting in which the local parties have access to pre-shared entanglement in the form of a resource state. We find that, for an arbitrary pure resource state, this number is equal to the Krull dimension of (the closure of) the set of pure states obtainable from the resource state by SLOCC. Surprisingly, a generic resource state maximizes this number.Local state discrimination is closely related to the topic of entangled subspaces, which we study in its own right. We introduce r-entangled subspaces, which naturally generalize previously studied spaces to higher multipartite entanglement. We use algebraic-geometric methods to determine the maximum dimension of an r-entangled subspace, and present novel explicit constructions of such spaces. We obtain similar results for symmetric and antisymmetric r-entangled subspaces, which correspond to entangled subspaces of bosonic and fermionic systems, respectively.

Adaptive procedures for discriminating between arbitrary tensor-product quantum states

Discriminating between quantum states is a fundamental task in quantum information theory. Given two quantum states ${\ensuremath{\rho}}_{+}$ and ${\ensuremath{\rho}}_{\ensuremath{-}}$, the Helstrom measurement distinguishes between them with minimal probability of error. However, finding and experimentally implementing the Helstrom measurement can be challenging for quantum states on many qubits. Due to this difficulty, there is great interest in identifying local measurement schemes which are close to optimal. In the first part of this work, we generalize previous work by Acin et al. [Phys. Rev. A 71, 032338 (2005)] and show that a locally greedy scheme using Bayesian updating can optimally distinguish between any two states that can be written as a tensor product of arbitrary pure states. We then show that the same algorithm cannot distinguish tensor products of mixed states with vanishing error probability (even in a large subsystem limit), and introduce a modified locally greedy scheme with strictly better performance. In the second part of this work, we compare these simple local schemes with a general dynamic programming approach which finds both the optimal series of local measurements as well as the optimal order in which subsystems are measured.

Characterizing nonbilocal correlation: a geometric perspective

Exploiting the notion of measurement-induced nonlocality [Phys.Rev. Lett. 106, 120401 (2011)], we introduce a new measure to quantify the nonbilocal correlation. We establish a simple relation between the nonlocal and nonbilocal measures for the arbitrary pure input states. Considering the mixed states as inputs, we derive two upper bounds of affinity-based nonbilocal measure. Finally, we have studied the nonbilocality of a different combinations of input states.

Measure of Tripartite Quantum Correlation in Multiqubit Symmetric Pure State

We propose a direct measure of tripartite quantum correlation in an arbitrary symmetric pure state of N correlated two-level atoms (qubits). We compute the third-order moments of the collective pseudo-spin operators in terms of the individual atomic operators of the atoms in the assembly and find that all the bipartite quantum correlation terms among the atoms cancel out leaving only the all possible tripartite quantum correlation terms among them. We observe that the third-order moments are made up of solely the tripartite quantum correlations among the atoms. This helps to extract out only the tripartite quantum correlations among the N atoms and propose a measure and quantification of these correlations. We also propose the necessary and sufficient condition for the presence of tripartite quantum correlation in such multi-atomic systems. We conjecture the way of determining the third-order moments of the pseudo-spin operators of the two-level atoms in the assembly, experimentally.

Complementary relation between tripartite entanglement and the maximum steering inequality violation

We investigate the relation between tripartite entanglement of the three-qubit system and the maximum steering inequality violation of the reduced two-qubit states. Firstly, it is found that a single parameter family of entangled three-qubit pure states have the maximum steering inequality violation among all of the three-qubit pure states for a fixed amount of tripartite entanglement. The tripartite entanglement is quantified by the genuinely multipartite concurrence, generalized geometric measure, and tangle. Subsequently, the complementary relation between tripartite entanglement and the maximum steering inequality violation for an arbitrary three-qubit pure state can be established. Particularly, the result also holds for three-qubit mixed states if the entanglement measure is tangle. The complementary relations indicate that the maximum steering inequality violation of the reduced two-qubit system is at the expense of tripartite entanglement.

Quantum Fisher information maximization in an unbalanced interferometer

In this paper we provide the answer to the following question: given an arbitrary pure input state and a general, unbalanced, Mach-Zehnder interferometer, what transmission coefficient of the first beam splitter maximizes the quantum Fisher information (QFI)? We consider this question for both single- and two-parameter QFI, or, in other words, with or without having access to an external phase reference. We give analytical results for all involved scenarios. It turns out that, for a large class of input states, the balanced (50:50) scenario yields the optimal two-parameter QFI; however, this is far from being a universal truth. When it comes to the single-parameter QFI, the balanced scenario is rarely the optimal one and an unbalanced interferometer can bring a significant advantage over the balanced case. We also state the condition imposed upon the input state so that no metrological advantage can be exploited via an external phase reference. Finally, we illustrate and discuss our assertions through a number of examples, including both Gaussian and non-Gaussian input states.

Asymmetry-induced nonclassical correlation

In quantum resource theory (QRT), asymmetry recognized as a valid resource for the advantage of various quantum information processing. In this paper, we establish the resource theory of asymmetry using quantum Fisher information (QFI). By defining the average Fisher information as a measure of asymmetry, it is shown that the discrepancy of bipartite global and local asymmetries naturally induces the nonclassical correlation between the subsystems. This measure satisfies all the necessary axioms of a faithful measure of bipartite quantum correlation. As an illustration, we have studied the proposed measure for an arbitrary pure state and Bell diagonal state.