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Majorana representation for states that maximize the Wehrl entropy and maximize M 1 , which are Kings of Quantumness and Queens of Quantumness. In each row, we indicate the value of S. In the top-left corner of each cell, we indicate (in red) the corresponding degree of unpolarization. The degeneracies at some points of the constellations are indicated in blue.
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The striking differences between quantum and classical systems predicate disruptive quantum technologies. We peruse quantumness from a variety of viewpoints, concentrating on phase-space formulations because they can be applied beyond particular symmetry groups. The symmetry-transcending properties of the Husimi Q function make it our basic tool. I...
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... on the positions of the points. The distribution of points that correspond to an optimized value of the function may not be unique, even though the criterion that the points are as far as possible from one another remains. Numerical results for some specific values of the dimension S can be found in Ref. 124, and we present some of them in Fig. 2. ...
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... [133][134][135] Finding states that minimize the maximum in Eq. (134) is achieved by restricting the search to pure states with specific rotational symmetries or to directions n pointing in a subset of all directions, as these restrictions are guaranteed to contain the optimal results. 79 We present the results for some dimensions in Fig. ...
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... unit sphere, . c ¼ P N i¼1 i jn i ihn i j, with large N. Then, M 1 becomes a quadratic function of the coefficients i , which has to be minimized under the constraints i ! 0 and P i i ¼ 1. This is a quadratic program that can be solved by a variety of algorithms, and the solutions seem to be unique. We compare the results for some dimensions in Fig. ...
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... configurations and the degrees of unpolarization of the corresponding states are also shown in Fig. 2. We see that for small S, the AVS Quantum Science REVIEW scitation.org/journal/aqs configurations are almost identical for all the extremal principles. For larger S, they differ in general, and the degree of unpolarization is also markedly different. This highlights the need for a Royal Family of states, where each member is the most ...
Citations
... is the Euclidean norm of the Kth multipole. The quantities A M (ρ) can be be used to furnish a generalized uncertainty principle for CV [31] and they are a good indicator of quantumness [46,47]. For spin variables, it has been shown that A M (ρ) are maximized to all orders M by SU(2)-coherent states, which are the least quantum states in this context, and vanish for the most quantum states, which are called the Kings of Quantumness, the furthest in some sense from coherent states [48][49][50]. ...
... Like their SU(2) counterparts, these multipoles can be used to quantify quantumness, with the most quantum states having the lowest cumulative multipole moments and vice versa. Since in many known contexts the vacuum states and Fock states define the limits of the least and most quantum states [46], this implies that measuring the lowest-order multipoles could already be well defined for inspecting the quantumness of a quantum state. ...
Coherent-state representations are a standard tool to deal with continuous-variable systems, as they allow one to efficiently visualize quantum states in phase space. Here, we work out an alternative basis consisting of monomials on the basic observables, with the crucial property of behaving well under symplectic transformations. This basis is the analogue of the irreducible tensors widely used in the context of SU(2) symmetry. Given the density matrix of a state, the expansion coefficients in that basis constitute the multipoles, which describe the state in a canonically covariant form that is both concise and explicit. We use these quantities to assess properties such as quantumness or Gaussianity and to furnish direct connections between tomographic measurements and quasiprobability distribution reconstructions.
... In Eq. (A.9) we used the following notation. It is worth mentioning that the stellar function can also be defined in H ∞ (continuous-variable quantum systems) using the canonical coherent states [134,135]. F (z) is a holomorphic function which provides an analytic representation of a quantum state. The transformation in Eq. (A.9) defines the Bargmann-Segal transform on the spin coherent states which maps a quantum state to the space of holomorphic functions, called the Segal-Bargmann space [133][134][135][136]. ...
... F (z) is a holomorphic function which provides an analytic representation of a quantum state. The transformation in Eq. (A.9) defines the Bargmann-Segal transform on the spin coherent states which maps a quantum state to the space of holomorphic functions, called the Segal-Bargmann space [133][134][135][136]. The Bargmann representation is directly related to the Husimi Q-function, which is a quasi-probability distribution on the phase space. ...
... It can be interpreted as an information measure for a joint noisy measurement of position and momentum of the quantum system. Lieb and Solovej proved that Wehrl entropy on H S is lower bounded by 13) and the lower bound is attained only for the Bloch coherent states for which the Majorana constellations are concentrated at a single point on the Poincare sphere [135,137]. The more spread out the points are on the sphere, the larger the value of the Wehrl entropy becomes. ...
Classical polarimetry is a well-established discipline with diverse applications across different branches of science. The burgeoning interest in leveraging quantum resources to achieve highly sensitive measurements has spurred researchers to elucidate the behavior of polarized light within a quantum mechanical framework, thereby fostering the development of a quantum theory of polarimetry. In this work, drawing inspiration from polarimetric investigations in biological tissues, we investigate the precision limits of polarization rotation angle estimation about a known rotation axis, in a quantum polarimetric process, comprising three distinct quantum channels. The rotation angle to be estimated is induced by the retarder channel on the Stokes vector of the probe state. The diattenuator and depolarizer channels, acting on the probe state, can be thought of as effective noise processes. We explore the precision constraints inherent in quantum polarimetry by evaluating the quantum Fisher information (QFI) for probe states of significance in quantum metrology, namely NOON, Kings of Quantumness, and Coherent states. The effects of the noise channels as well as their ordering is analyzed on the estimation error of the rotation angle to characterize practical and optimal quantum probe states for quantum polarimetry. Furthermore, we propose an experimental framework tailored for NOON state quantum polarimetry, aiming to bridge theoretical insights with empirical validation.
... It also minimizes the cumulative multi-poles of its polarization distribution, for which it has earned the moniker "king of quantumness" [49,50], and is the optimal state for rotation sensing, explaining the sobriquet "quantum rotosensor" [51,52]. The tetrahedron state has numerous other extremal properties [53], including maximizing delocalization measures in SU(2) phase space such as the Wehrl entropy [54]. It has also been studied for its entanglement properties [55][56][57][58][59], which have been generalized to multi-dimensional systems with multiple qudits [60]. ...
... It was proven in Refs. [53,66] that the optimal states for the task at hand are pure states that exhibit the following properties: ...
It is often thought that the super-sensitivity of a quantum state to an observable comes at the cost of a decreased sensitivity to other non-commuting observables. For example, a squeezed state squeezed in position quadrature is super-sensitive to position displacements, but very insensitive to momentum displacements. This misconception was cleared with the introduction of the compass state [Nature 412, 712 (2001)10.1038/35089017], a quantum state equally super-sensitive to displacements in position and momentum. When looking at quantum states used to measure spin rotations, N00N states are known to be more advantageous than classical methods as long as they are aligned to the rotation axis. When considering the estimation of a rotation with unknown direction and amplitude, a certain class of states stands out with interesting properties. These states are equally sensitive to rotations around any axis, are second-order unpolarized, and can possess the rotational properties of Platonic solids in particular dimensions. Importantly, these states are optimal for simultaneously estimating the three parameters describing a rotation. In the asymptotic limit, estimating all d parameters describing a transformation simultaneously rather than sequentially can lead to a reduction of the appropriately weighted sum of the measured parameters’ variances by a factor of d. We report the experimental creation and characterization of the lowest-dimensional such state, which we call the “tetrahedron state” due to its tetrahedral symmetry. This tetrahedron state is created in the symmetric subspace of four optical photons’ polarization in a single spatial and temporal mode, which behaves as a spin-2 particle. While imperfections due to the hardware limited the performance of our method, ongoing technological advances will enable this method to generate states which out-perform any other existing strategy in per-photon comparisons.
... In the case of a spin- 1 2 particle, it reduces to the celebrated Bloch representation of a two-level quantum system (qubit). This representation is used in various contexts such as spinor Bose gases [8][9][10][11], entanglement classification in multiqubit systems [12][13][14][15][16][17][18][19][20], the Berry phase associated with the cyclic evolution of the state [21][22][23], investigating Lipkin-Meshkov-Glick model [24,25] and studying symmetries and properties of spin states [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. ...
... In the Majorana representation the most coherent state is represented by one 2j-degenerated point on a sphere. On the other hand, the most quantum, or the most anticoherent state |ψ⟩ should "point nowhere" and be represented as 2j points equally distributed on a sphere [39][40][41]. However, even if the polarization (coherence) disappears, the higher moments of coherence might not vanish. ...
... On the other hand, we aim to identify the quantum measurement "as quantum as possible", composed of orthogonal vectors which are the least spin-coherent and maximize the average measure of anticoherence. Several quantities can be used for this purpose, including the Wehrl entropy, which characterizes localization of the state in the phase space [43][44][45][46][48][49][50][51], measures related to the distribution of Majorana stars representing a state [18] or cumulative distribution based on multipole expansion of density matrix [39,52]. In this work we rely on the quantity introduced by Baguette and Martin [31], under the name of measure of anticoherence. ...
Spin anticoherent states acquired recently a lot of attention as the most "quantum" states. Some coherent and anticoherent spin states are known as optimal quantum rotosensors. In this work, we introduce a measure of quantumness for orthonormal bases of spin states, determined by the average anticoherence of individual vectors and the Wehrl entropy. In this way, we identify the most coherent and most quantum states, which lead to orthogonal measurements of extreme quantumness. Their symmetries can be revealed using the Majorana stellar representation, which provides an intuitive geometrical representation of a pure state by points on a sphere. Results obtained lead to maximally (minimally) entangled bases in the 2 j + 1 dimensional symmetric subspace of the 2 2 j dimensional space of states of multipartite systems composed of 2 j qubits. Some bases found are iso-coherent as they consist of all states of the same degree of spin-coherence.
... Examples include geometrical measures of entanglement [55][56][57], spherical t-designs [58,59], and the Thomson [60][61][62][63][64][65] and Tammes [66][67][68] problems. Moreover, a number of states with remarkable properties, such as queens [69] and kings of quantumness [70], maximally entangled states [71], k-uniform states [72][73][74], and states with maximal Wehrl entropy [75], can be aptly understood in terms of the properties of their corresponding constellations [76]. ...
... 2 K = 1. These coefficients 2 K have been used as measures of localization [101]; as quantifiers of quantumness [76], which is useful for applications such as rotation sensing [43]; and to quantify mode-decomposition-independent entanglement properties [14]. ...
Majorana stars, the 2S spin coherent states that are orthogonal to a spin-S state, offer an elegant method to visualize quantum states, disclosing their intrinsic symmetries. These states are naturally described by the corresponding multipoles. These quantities can be experimentally determined and allow for an SU(2)-invariant analysis. We investigate the relationship between Majorana constellations and state multipoles, thus providing insights into the underlying symmetries of the system. We illustrate our approach with some relevant and informative examples.
... where the integral is evaluated over the plane, dα = dRe(α)dIm(α), and ψ(ᾱ) = ⟨α|ψ⟩ [37]. Unlike Fock states, coherent states form an uncountable, overcomplete basis for the ...
... We compute the Fock probabilities via Eq. (37), which are enumerated on the x-axis, sorted by the probability. optical unitaryÛ, with transfer matrixû. ...
We introduce a framework for simulating quantum optics by decomposing the system into a finite rank (number of terms) superposition of coherent states. This allows us to define a resource theory, where linear optical operations are “free” (i.e., do not increase the rank), and the simulation complexity for an m-mode system scales quadratically in m, in stark contrast to the Hilbert space dimension. We outline this approach explicitly in the Fock basis, relevant in particular for Boson sampling, where the simulation time (space) complexity for computing output amplitudes, to arbitrary accuracy, scales as O(m² 2 n ) [O(m2 n )] for n photons distributed among m modes. We additionally demonstrate that linear optical simulations with the n photons initially in the same mode scales efficiently, as O(m² n). This paradigm provides a practical notion of “non-classicality,” i.e., the classical resources required for simulation. Moreover, by making connections to the stellar rank formalism, we show this comes from two independent contributions, the number of single-photon additions and the amount of squeezing.
... For concreteness, the single-photon-added coherent state enables us to continuously tune the photon system from a coherent state (representing quantum states in their 'classical' limit) to a single Fock number state (which we take as a uniquely 'quantum' state) 12,13 . Note that, indeed, the coherent state delineates the border regarding the classicality and quantumness of photon states from different perspectives 14 . We thus define a parameterized photon state as the basis for possible investigation of the fuzzy border that may separate the 'quantum' from 'classical' regimes in the above sense, utilizing the coupling with a single electron wavepacket as a measuring pointer. ...
How does the quantum-to-classical transition of measurement occur? This question is vital for both foundations and applications of quantum mechanics. Here, we develop a new measurement-based framework for characterizing the classical and quantum free electron–photon interactions and then experimentally test it. We first analyze the transition from projective to weak measurement in generic light–matter interactions and show that any classical electron-laser-beam interaction can be represented as an outcome of weak measurement. In particular, the appearance of classical point-particle acceleration is an example of an amplified weak value resulting from weak measurement. A universal factor, $$\exp \left(-{\Gamma }^{2}/2\right)$$ exp − Γ 2 / 2 , quantifies the measurement regimes and their transition from quantum to classical, where $$\Gamma$$ Γ corresponds to the ratio between the electron wavepacket size and the optical wavelength. This measurement-based formulation is experimentally verified in both limits of photon-induced near-field electron microscopy and the classical acceleration regime using a DLA. Our results shed new light on the transition from quantum to classical electrodynamics, enabling us to employ the essence of the wave-particle duality of both light and electrons in quantum measurement for exploring and applying many quantum and classical light–matter interactions.
... We pursue this idea and work out a general method for finding stars in the discrete cylinder. Since the geometry of a constellation elegantly encapsulates all the properties of the state [65], we characterize the quantumness using the multipoles associated to the stars. ...
Majorana stellar representation, which visualizes a quantum spin as points on the Bloch sphere, allows quantum mechanics to accommodate the concept of trajectory, the hallmark of classical physics. We extend this notion to the discrete cylinder, which is the phase space of the canonical pair angle and orbital angular momentum. We demonstrate that the geometrical properties of the ensuing constellations aptly encapsulate the quantumness of the state.
... Using this representation, the general task of characterizing the most nonclassical spin state is equivalent to the task of characterizing a spherical distribution of points-a historic mathematical problem with many acceptable solutions [33,34]. Several notions that quantify the nonclassicality of angular momentum have been studied in this stellar context, including anticoherence [35][36][37][38], P-representability [39,40], and the geometric measure of entanglement [41,42]; see also [43][44][45][46]. ...
... Here we discuss some observations in the context of states that maximize other measures of nonclassicality. In particular, the constellations of such alternative maximal states are in general highly symmetric, highly delocalized, or both [37,45]. And while the Wigner-maximal constellations partially display these qualities in the spins considered, they do not follow an obvious geometric guiding principle. ...
The Majorana stellar representation is used to characterize spin states that have a maximally negative Wigner quasiprobability distribution on a spherical phase space. These maximally Wigner-negative spin states generally exhibit a partial but not high degree of symmetry within their star configurations. In particular, for spin j > 2, maximal constellations do not correspond to a Platonic solid when available and do not follow an obvious geometric pattern as dimension increases. In addition, they are generally different from spin states that maximize other measures of nonclassicality such as anticoherence or geometric entanglement. Random states display on average a relatively high amount of negativity, but the extremal states and those with similar negativity are statistically rare in Hilbert space. We also prove that all spin coherent states of arbitrary dimension have non-zero Wigner negativity. This offers evidence that all pure spin states also have non-zero Wigner negativity. The results can be applied to qubit ensembles exhibiting permutation invariance.
... In the case of a spin- 1 2 particle, it reduces to the celebrated Bloch representation of a twolevel quantum system (qubit). This representation is used in various contexts such as spinor Bose gases [8][9][10][11], entanglement classification in multiqubit systems [12][13][14][15][16][17][18][19], the Berry phase associated with the cyclic evolution of the state [21][22][23], investigating Lipkin-Meshkov-Glick model [24,25] and studying symmetries and properties of spin states [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. ...
... Hence, in the Majorana representation, the most coherent state is represented by one 2j-degenerated point on a sphere. On the other hand, the anticoherent state |ψ⟩ should "point nowhere", and be represented as 2j points equally distributed on a sphere [39][40][41]. However, even if the polarization (coherence) disappears, the higher moments of coherence might not vanish. ...
Spin anticoherent states acquired recently a lot of attention as the most "quantum" states. Some coherent and anticoherent spin states are known as optimal quantum rotosensors. In this work we introduce a measure of spin coherence for orthonormal bases, determined by the average anticoherence of individual vectors, and identify the most and the least coherent bases which lead to orthogonal measurements of extreme coherence. Their symmetries can be revealed using the Majorana stellar representation, which provides an intuitive geometrical representation of a pure state by points on a sphere. Results obtained lead to maximally (minimally) entangled bases in the $2j+1$ dimensional symmetric subspace of the $2^{2j}$ dimensional space of quantum states of multipartite systems composed of $2j$ qubits.
