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arXiv:2211.05692 · Journal of Physics A Mathematical and Theoretical

Exploring weak value arguments and Bargmann invariants in N-level quantum systems through the Majorana symmetric representation

Lorena Ballesteros Ferraz, Dominique Lambert, Yves Caudano

Published Apr 29, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract This work examines the argument of weak values for general observables and develops a geometric description on the Bloch sphere. We apply the Majorana symmetric representation to reach this goal. The weak value of a general observable is proportional to the weak value of an effective projector: it is constructed from the application of the observable over the initial state, after normalization by a constant of proportionality that is real. The argument of the weak value of a projector on a pure state of an N-level system corresponds to a symplectic area in the complex projective space ( CP N − 1 ) . This symplectic area cannot be visualized directly but it can be represented geometrically with a sum of N − 1 solid angles on the Bloch sphere using the Majorana stellar representation. By combining these two ideas, we show that the argument of the weak value of any observable (i.e. not just projectors) can be described with the Majorana representation, as the sum of N − 1 solid angles on the Bloch sphere. These two approaches provide two geometrical descriptions, a first one in the complex projective space CP N − 1 and a second one on the Bloch sphere, after mapping the problem from the original N-dimensional quantum state space ( CP N − 1 ) to a multi-qubit description in three-dimensional space by making use of the Majorana representation. These results can also be applied to the argument of the third-order Bargmann invariant, the most fundamental order as the argument of any higher order invariant can be expressed as a sum of the argument of third-order Bargmann invariants, as well as to the argument of the Kirkwood–Dirac quasi-probability distribution. Finally, we focus on the argument of the weak value of a general spin-1 operator when its modulus diverges towards infinity. This divergence amplifies signals with great usefulness in experiments and appears connected to the qubit entanglement in the Majorana representation.

DOI
10.1088/1751-8121/add22a

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Abstract This work examines the argument of weak values for general observables and develops a geometric description on the Bloch sphere. We apply the Majorana symmetric representation to reach this goal. The weak value of a general observable is proportional to the weak value of an effective projector: it is constructed from the application of the observable over the initial state, after normalization by a constant of proportionality that is real. The argument of the weak value of a projector on a pure state of an N-level system corresponds to a symplectic area in the complex projective space ( CP N − 1 ) . This symplectic area cannot be visualized directly but it can be represented geometrically with a sum of N − 1 solid angles on the Bloch sphere using the Majorana stellar representation. By combining these two ideas, we show that the argument of the weak value of any observable (i.e. not just projectors) can be described with the Majorana representation, as the sum of N − 1 solid angles on the Bloch sphere. These two approaches provide two geometrical descriptions, a first one in the complex projective space CP N − 1 and a second one on the Bloch sphere, after mapping the problem from the original N-dimensional quantum state space ( CP N − 1 ) to a multi-qubit description in three-dimensional space by making use of the Majorana representation. These results can also be applied to the argument of the third-order Bargmann invariant, the most fundamental order as the argument of any higher order invariant can be expressed as a sum of the argument of third-order Bargmann invariants, as well as to the argument of the Kirkwood–Dirac quasi-probability distribution. Finally, we focus on the argument of the weak value of a general spin-1 operator when its modulus diverges towards infinity. This divergence amplifies signals with great usefulness in experiments and appears connected to the qubit entanglement in the Majorana representation.
Abstract Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in terms of three real vectors on the unit sphere in N 2 − 1 dimensions, S N 2 − 2 . These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, C P N − 1 , which has a non-trivial representation as a 2N − 2 dimensional submanifold of S N 2 − 2 (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in C P N − 1 . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell–Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates. To conclude, we discuss the interpretation and usefulness of geometric phases in weak values in connection to weak measurements.
Abstract We express modular and weak values of observables of three- and higher-level quantum systems in their polar form. The Majorana representation of N -level systems in terms of symmetric states of N − 1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N -level quantum systems can be factored in N − 1 contributions. Their modulus is determined by the product of N − 1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N − 1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three-box paradox (Aharonov and Vaidman 1991 J. Phys. A: Math. Gen . 24 2315–28) from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre- and postselected.
We investigate the geometric structure of a Hermitian two-level mixing Hamiltonian motivated by neutral meson oscillations and its connections with qubit, fermionic, and topological descriptions. Mapping the Hamiltonian to an effective qubit representation, we analyze the geometric phase associated with cyclic parameter evolution on the Bloch sphere. Without identifying the complex mixing phase with a physical CP-violation observable, we interpret it as a CP-like geometric parameter controlling the azimuthal orientation of the Hamiltonian vector and its reversal under phase conjugation. Third-order Bargmann invariants constructed from the eigenstates yield an associated discrete phase whose dependence on the mixing parameters is examined alongside the continuous geometric construction. The cyclic evolution is also represented as a single-qubit RzR_zRz​ phase operation and expressed through Majorana bilinears. Extending the construction to the momentum-dependent Bogoliubov-de Gennes Hamiltonian of the Kitaev chain, we relate momentum-space winding to the topological regimes and the quantized Berry (Zak) phase in the real-parameter model, while the corresponding Bargmann construction approaches the global phase in the continuum limit. Together, these results provide a geometric perspective connecting phase structure, qubit operations, fermionic representations, and momentum-space topology within related two-level Hamiltonians.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
Abstract Quantum measurements can be generalized to include complex quantities. It is possible to relate the quantum weak values of projection operators to the third order Bargmann invariants. The argument of the weak value becomes, up to a sign, equal to the Berry’s phase associated with the three state vectors. In case of symmetric informationally complete, positive operator valued measures, this relation takes a particularly simple form. Alternating strong and weak measurements can be used to determine Berry’s phase directly, which demonstrates that not only their real and imaginary parts but also moduli and arguments of weak values have a physical significance. For an arbitrary projection operator, weak value is real when the projector, pre- and post-selected states lie on a so-called null phase curve which includes the geodesic containing the three states as a special case.
Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis, the entanglement of symmetric multipartite states is categorized, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalization of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n-qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterization of entanglement equivalence classes such as stochastic local operations and classical communication (SLOCC) and the degeneracy configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Möbius transformations of complex analysis, which allows for a clear visualization of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to five qubits are derived. Well-known entanglement classification schemes such as the four qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.