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

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 $({\text{CP}}^{N-1})$. 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 ${\text{CP}}^{N-1}$ and a second one on the Bloch sphere, after mapping the problem from the original $N$-dimensional quantum state space $({\text{CP}}^{N-1})$ 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.

References in corpus (51)

• Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification
• Direct Measurement of the Quantum Wavefunction
• Bloch vectors for qudits
• Complex weak values in quantum measurement
• Quantum non-locality in two three-level systems
• Direct measurement of general quantum states using weak measurement
• Anomalous Weak Values Are Proofs of Contextuality
• |0>|1>+|1>|0>
• Precision frequency measurements with interferometric weak values
• Weak Values, Quantum Trajectories, and the Stony-Brook Cavity QED experiment
• Prior information: how to circumvent the standard joint-measurement uncertainty relation
• Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects
• The significance of the imaginary part of the weak value
• Weak Values are Interference Phenomena
• New wine in old bottles: Quantum measurement - direct, indirect, weak - with some applications
• Approaching quantum-limited metrology with imperfect detectors by using weak-value amplification
• Anomalous weak values and contextuality: robustness, tightness, and imaginary parts
• Entanglement and symmetry in permutation symmetric states
• Charge sensing amplification via weak values measurement
• Modular values and weak values of quantum observables
• Measuring Non-Hermitian Operators via Weak Values
• Majorana representation of symmetric multiqubit states
• Quantum circuits for measuring weak values, Kirkwood--Dirac quasiprobability distributions, and state spectra
• Weak values of electron spin in a double quantum dot
• Complete characterization of post-selected quantum statistics using weak measurement tomography
• Weak-value amplification as an optimal metrological protocol
• Uncertainty limits for quantum metrology obtained from the statistics of weak measurements
• Weak values are universal in von Neumann measurements
• Properties and Applications of the Kirkwood-Dirac Distribution
• The Three-Box Paradox Revisited
• Geometric phase in weak measurements
• Geometrical aspects of weak measurements and quantum erasers
• Interferences in quantum eraser reveal geometric phases in modular and weak values
• Classical Field Approach to Quantum Weak Measurements
• Bloch sphere representation of three-vertex geometric phases
• Non-Abelian transport distinguishes three usually equivalent notions of entropy production
• Geometric description of modular and weak values in discrete quantum systems using the Majorana representation
• Bounding Quantum Advantages in Postselected Metrology
• Operational formulation of weak values without probe systems
• Experimental probe of weak value amplification and geometric phase through the complex zeroes of the response function
• The von Neumann Model of Measurement in Quantum Mechanics
• Entangling power of symmetric two-qubit quantum gates
• Entanglement of a Single Spin-1 Object: An Example of Ubiquitous Entanglement
• Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems
• Enhancing interferometry using weak value amplification with real weak values
• Efficient quantum computing with weak measurements
• Spin squeezing in symmetric multiqubit states with two distinct Majorana spinors
• On the relevance of weak measurements in dissipative quantum systems
• Gravitational sensing with weak value based optical sensors
• Revisiting weak values through non-normality
• Geometric decomposition of geodesics and null phase curves using Majorana star representation

Cited by in corpus (1)

• Wick rotation derivation for weak values of the density and time-dependent density functional theory