arXiv subjects

Chryssomalis Chryssomalakos

Publications and source records attributed to Chryssomalis Chryssomalakos.

At least 19 recordsLinked to original sources

arXiv2025

Universality in fidelity-based quantum metrology

We consider the problem of identifying the quantum spin states that are the optimal sensors of a given transformation, averaged over all possible orientations of the spin system. Our geometric approach to the problem is based on a fidelity criterion and is entirely general, encompassing any unitary transformation. This formalism leads to a universality result: For any value of the spin, there exists a zero-measure subset of states that can be the optimal sensors for certain transformations and the worst sensors for others, and this set does not depend on the transformation under consideration. In other words, some spin states are simply the best (or worst) sensors, regardless of what they detect.

quant-ph↗

arXiv2024

Entangling power of symmetric multiqubit systems: a geometrical approach

Unitary gates with high entangling capabilities are relevant for several quantum-enhanced technologies. For symmetric multiqubit systems, such as spin states or bosonic systems, the particle exchange symmetry restricts these gates and also the set of not-entangled states. In this work, we analyze the entangling power of unitary gates in these systems by reformulating it as an inner product between vectors with components given by SU$(2)$ invariants. For small number of qubits, this approach allows us to study analytically the entangling power including the detection of the unitary gate that maximizes it. We observe that extremal unitary gates exhibit entanglement distributions with high rotational symmetry, same that are linked to a convex combination of Husimi functions of certain states. Furthermore, we explore the connection between entangling power and the Schmidt numbers admissible in some quantum state subspaces. Thus, the geometrical approach presented here suggests new paths for studying entangling power linked to other concepts in quantum information theory.

quant-ph↗

arXiv2024

Quantum metrology of rotations with mixed spin states

The efficiency of a quantum metrology protocol can be significantly diminished by the interaction of the system with its environment, leading to a loss of purity and, as a result, a mixed state for the probing system. An example is the measurement of a magnetic field through the rotation of a spin that is subject to decoherence due to its coupling to a surrounding spin or bosonic bath. In this work, we define mixed optimal quantum rotosensors (OQRs) as mixed spin-$j$ states that achieve maximum sensitivity to estimate infinitesimal rotations, when the rotation axis is unknown. We study two scenarios, where the probe states saturate the averaged fidelity or the averaged quantum Cram\'er-Rao bound, the latter giving the ultimate sensitivity. We find that mixed OQRs can achieve sensitivity equal to that of pure states and are obtained by mixing states from linear subspaces of anticoherent states. We present several examples of mixed OQRs and their associated anticoherent subspaces. We also show that OQRs maximize entanglement in a specific sense, preserving the known relation between entanglement and optimal rotation sensitivity for pure states, even in the context of mixed states. Our results highlight the interconnection between quantum metrology of rotations, anticoherence and entanglement in mixed spin states.

quant-ph↗

arXiv2021

Noise Effects on the Wilczek-Zee Geometric Phase

Non-abelian geometric phases have been proposed as an essential ingredient in logical gate implementation -- their geometric nature guarantees their invariance under reparametrizations of the associated cyclic path in parameter space. However, they are still dependent on deformations of that path, due to, e.g., noise. The first question that we tackle in this work is how to quantify in a meaningful way this effect of noise, focusing, for concreteness, on the nuclear quadrupole resonance hamiltonian -- other systems of this nature can clearly be treated analogously. We consider a precessing magnetic field that drives adiabatically a degenerate doublet, and is subjected to noise, the effects of which on the Wilczek-Zee holonomy are computed analytically. A critical review of previous related works reveals a series of assumptions, like sudden jumps in the field, or the presence of white noise, that might violate adiabaticity. We propose a state-independent measure of the effect, and then consider sinusoidal noise in the field, of random amplitude and phase. We find that all integer noise frequencies $m \neq 2$ behave similarly, in a manner reminiscent of the abelian case, but that noise of frequency $m = 2$ has a very different, and, at the same time, very pronounced effect, that might well affect robustness estimations.

quant-ph↗

arXiv2021

Symmetric Multiqudit States: Stars, Entanglement, Rotosensors

A constellation of $N=d-1$ Majorana stars represents an arbitrary pure quantum state of dimension $d$ or a permutation-symmetric state of a system consisting of $n$ qubits. We generalize the latter construction to represent in a similar way an arbitrary symmetric pure state of $k$ subsystems with $d$ levels each. For $d\geq 3$, such states are equivalent, as far as rotations are concerned, to a collection of various spin states, with definite relative complex weights. Following Majorana's lead, we introduce a multiconstellation, consisting of the Majorana constellations of the above spin states, augmented by an auxiliary, "spectator" constellation, encoding the complex weights. Examples of stellar representations of symmetric states of four qutrits, and two spin-$3/2$ systems, are presented. We revisit the Hermite and Murnaghan isomorphisms, which relate multipartite states of various spins, number of parties, and even symmetries. We show how the tools introduced can be used to analyze multipartite entanglement and to identify optimal quantum rotosensors, i.e., pure states which are maximally sensitive to rotations around a specified axis, or averaged over all axes.

quant-ph↗

arXiv2019

When geometric phases turn topological

Geometric phases, accumulated when a quantum system traces a cycle in quantum state space, do not depend on the parametrization of the cyclic path, but do depend on the path itself. In the presence of noise that deforms the path, the phase gets affected, compromising the robustness of possible applications, e.g., in quantum computing. We show that for a special class of spin states, called anticoherent, and for paths that correspond to a sequence of rotations in physical space, the phase only depends on topological characteristics of the path, in particular, its homotopy class, and is therefore immune to noise.

quant-ph↗

arXiv2017

Geometry of spin coherent states

Spin states of maximal projection along some direction in space are called (spin) coherent, and are, in many aspects, the "most classical" available. For any spin $s$, the spin coherent states form a 2-sphere in the projective Hilbert space $\mathbb{P}$ of the system. We address several questions regarding that sphere, in particular its possible intersections with complex lines. We also find that, like Dali's iconic clocks, it extends in all possible directions in $\mathbb{P}$. We give a simple expression for the Majorana constellation of the linear combination of two coherent states, and use Mason's theorem to give a lower bound on the number of distinct stars of a linear combination of two arbitrary spin-$s$ states. Finally, we plot the image of the spin coherent sphere, assuming light in $\mathbb{P}$ propagates along Fubini-Study geodesics. We argue that, apart from their intrinsic geometric interest, such questions translate into statements experimentalists might find useful.

quant-ph↗

arXiv2017

Extracting Geometry from Quantum Spacetime: Obstacles down the road

Any acceptable quantum gravity theory must allow us to recover the classical spacetime in the appropriate limit. Moreover, the spacetime geometrical notions should be intrinsically tied to the behavior of the matter that probes them. We consider some difficulties that would be confronted in attempting such an enterprise. The problems we uncover seem to go beyond the technical level to the point of questioning the overall feasibility of the project. The main issue is related to the fact that, in the quantum theory, it is impossible to assign a trajectory to a physical object, and, on the other hand, according to the basic tenets of the geometrization of gravity, it is precisely the trajectories of free localized objects that define the spacetime geometry. The insights gained in this analysis should be relevant to those interested in the quest for a quantum theory of gravity and might help refocus some of its goals.

gr-qc↗

arXiv2016

Geometric Phase of a Spin-1/2 Particle Coupled to a Quantum Vector Operator

We calculate Berry's phase when the driving field, to which a spin-1/2 is coupled adiabatically, rather than the familiar classical magnetic field, is a quantum vector operator, of noncommuting, in general, components, e.g., the angular momentum of another particle, or another spin. The geometric phase of the entire system, spin plus "quantum driving field", is first computed, and is then subdivided into the two subsystems, using the Schmidt decomposition of the total wave function -the resulting expression shows a marked, purely quantum effect, involving the commutator of the field components. We also compute the corresponding mean "classical" phase, involving a precessing magnetic field in the presence of noise, up to terms quadratic in the noise amplitude -the results are shown to be in excellent agreement with numerical simulations in the literature. Subtleties in the relation between the quantum and classical case are pointed out, while three concrete examples illustrate the scope and internal consistency of our treatment.

quant-ph↗

arXiv2013

Position Operators and Center of Mass: New Perspectives

After reviewing the work of Pryce on Center-of-Mass (CoM) definitions in special relativity, and that of Jordan and Mukunda on position operators for relativistic particles with spin, we propose two new criteria for a CoM candidate: associativity, and compatibility with the Poisson bracket structure. We find that they are not satisfied by all of Pryce's definitions, and they also rule out Dixon's CoM generalization to the curved spacetime case. We also emphasize that the various components of the CoM position do not commute among themselves, in the general case, and thus provide a natural entry point to the arena of noncommutative spacetime, without the ad-hoc assumptions of the standard paradigm.

physics.gen-ph↗

arXiv2012

Operational Geometry on de Sitter Spacetime

Traditional geometry employs idealized concepts like that of a point or a curve, the operational definition of which relies on the availability of classical point particles as probes. Real, physical objects are quantum in nature though, leading us to consider the implications of using realistic probes in defining an effective spacetime geometry. As an example, we consider de Sitter spacetime and employ the centroid of various composite probes to obtain its effective sectional curvature, which is found to depend on the probe's internal energy, spatial extension, and spin. Possible refinements of our approach are pointed out and remarks are made on the relevance of our results to the quest for a quantum theory of gravity.

gr-qc↗

arXiv2007

Star Product and Invariant Integration for Lie type Noncommutative Spacetimes

We present a star product for noncommutative spaces of Lie type, including the so called ``canonical'' case by introducing a central generator, which is compatible with translations and admits a simple, manageable definition of an invariant integral. A quasi-cyclicity property for the latter is shown to hold, which reduces to exact cyclicity when the adjoint representation of the underlying Lie algebra is traceless. Several explicit examples illuminate the formalism, dealing with kappa-Minkowski spacetime and the Heisenberg algebra (``canonical'' noncommutative 2-plane).

hep-th↗

arXiv2004

Physics, Combinatorics and Hopf Algebras

A number of problems in theoretical physics share a common nucleus of combinatoric nature. It is argued here that Hopf algebraic concepts and techiques can be particularly efficient in dealing with such problems. As a first example, a brief review is given of the recent work of Connes, Kreimer and collaborators on the algebraic structure of the process of renormalization in quantum field theory. Then the concept of $k$-primitive elements is introduced -- these are particular linear combinations of products of Feynman diagrams -- and it is shown, in the context of a toy-model, that they significantly reduce the computational cost of renormalization. As a second example, Sorkin's proposal for a family of generalizations of quantum mechanics, indexed by an integer $k>2$, is reviewed (classical mechanics corresponds to $k=1$, while quantum mechanics to $k=2$). It is then shown that the quantum measures of order $k$ proposed by Sorkin can also be described as $k$-primitive elements of the Hopf algebra of functions on an appropriate infinite dimensional abelian group.

hep-th↗

arXiv2003

Higher Order Measures, Generalized Quantum Mechanics and Hopf Algebras

We study Sorkin's proposal of a generalization of quantum mechanics and find that the theories proposed derive their probabilities from $k$-th order polynomials in additive measures, in the same way that quantum mechanics uses a probability bilinear in the quantum amplitude and its complex conjugate. Two complementary approaches are presented, a $C^*$ and a Hopf-algebraic one, illuminating both algebraic and geometric aspects of the problem.

quant-ph↗

arXiv2002

Hamiltonians for curves

We examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion. To do this we tailor the theory of deformations to the Frenet-Serret frame of the curve. The Euler-Lagrange equations describing equilibrium are obtained; Noether's theorem is exploited to identify the constants of integration of these equations as the Casimirs of the euclidean group in three dimensions. While this system appears not to be integrable in general, it {\it is} in various limits of interest. Let the energy density be given as some function of the curvature and torsion, $f(κ,τ)$. If $f$ is a linear function of either of its arguments but otherwise arbitrary, we claim that the first integral associated with rotational invariance permits the torsion $τ$ to be expressed as the solution of an algebraic equation in terms of the bending curvature, $κ$. The first integral associated with translational invariance can then be cast as a quadrature for $κ$ or for $τ$.

nlin.SI↗

arXiv2001

Lie Superalgebra Stability and Branes

The algebra of the generators of translations in superspace is unstable, in the sense that infinitesimal perturbations of its structure constants lead to non-isomorphic algebras. We show how superspace extensions remedy this situation (after arguing that remedy is indeed needed) and review the benefits reaped in the description of branes of all kinds in the presence of the extra dimensions.

hep-th↗

arXiv1997

Drinfeld twist for quantum $su(2)$ in the adjoint representation

We give a detailed description of the adjoint representation of Drinfeld's twist element, as well as of its coproduct, for $su_{q}(2)$. We also discuss, as applications, the computation of the universal R-matrix in this representation and the problem of symmetrization of identical-particle states with quantum $su(2)$ symmetry.

q-alg↗

arXiv1996

Remarks on Quantum Integration

We give a general integration prescription for finite dimensional braided Hopf algebras, deriving the N-dimensional quantum superplane integral as an example. The transformation properties of the integral on the quantum plane are found. We also discuss integration on quantum group modules that lack a Hopf structure.

q-alg↗