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arXiv:2202.11145 · Quantum Science and Technology

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Lorena Ballesteros Ferraz, Dominique L Lambert, Yves Caudano

Published Aug 23, 2022• 11 citations
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Abstract

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.

DOI
10.1088/2058-9565/ac8bf1

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We revisit the quantum weak measurement (QWM) by sketching the polarisation state dynamics on the Poincaré sphere and the associated geometric phase, which is considered the soul of QWM. Our experimental arrangement comprises a coherent laser beam as a source of pure state, two polarizers corresponding to pre- and post-selection and a tilted wave plate placed in-between them to introduce weak interaction. The pre-selected state is a linearly polarised beam that can be represented as a point on the Poincaré sphere equator and the weak interaction with the wave plate results in a small spread in S3 axis. Now, an orthogonal projection of this state leads two different geodesics on the surface of the sphere through its poles, starting from both sides of the spread to the orthogonal point. The post-selection is made by moving the projection point slightly away from the orthogonal position so that both geodesics shift to the equator which results in a rapid accumulation of geometric phase in the beam crosssection. A gradient in the linear momentum is developed as a consequence which gives amplified shift in the beam position.

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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 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 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.
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.
The geometric interpretation of (pseudo)spin 1/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings. We here theoretically characterize a general N-level system by its coherence vector described on the higher dimensional generalized Bloch (hyper)sphere, where topological properties such as the Berry curvature and the quantum metric take simple forms set by the SU(N) algebra. Focusing on two spatial dimensions, we present a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising N-1 two-spheres. We demonstrate that for the N-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up a new avenue for the interpretation of the topological classification and the dynamical illustration of multi-level systems, which in turn is anticipated to help in the design of new experimental probes.
The geometric interpretation of (pseudo)spin-1/21/21/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high-energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings, or have involved restrictions to sub manifolds within the full Hilbert space. We here employ a coherence vector description to theoretically characterize a general NNN-level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying SU(N)\mathrm{SU}(N)SU(N) algebra and construct physically intuitive geometric pictures for topological concepts. Focusing on two spatial dimensions, we reveal a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising $N\ensuremath{-}1$ two-spheres. We demonstrate that for the NNN-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up another avenue for the interpretation of the topological classification and the dynamical illustration of multilevel systems, which in turn is anticipated to help in the design of new experimental probes.
We revisit the quantum weak measurement (QWM) by sketching the polarisation state dynamics on the Poincaré sphere and the associated geometric phase, which is considered the soul of QWM. Our experimental arrangement comprises a coherent laser beam as a source of pure state, two polarizers corresponding to pre- and post-selection and a tilted wave plate placed in-between them to introduce weak interaction. The pre-selected state is a linearly polarised beam that can be represented as a point on the Poincaré sphere equator and the weak interaction with the wave plate results in a small spread in S3 axis. Now, an orthogonal projection of this state leads two different geodesics on the surface of the sphere through its poles, starting from both sides of the spread to the orthogonal point. The post-selection is made by moving the projection point slightly away from the orthogonal position so that both geodesics shift to the equator which results in a rapid accumulation of geometric phase in the beam crosssection. A gradient in the linear momentum is developed as a consequence which gives amplified shift in the beam position.