Uncertainty relation of Anandan-Aharonov and Intelligent states

The quantum states which satisfy the equality in the generalised uncertainty relation are called intelligent states. We prove the existence of intelligent states for the Anandan-Aharonov uncertainty relation based on the geometry of the quantum state space for arbitrary parametric evolutions of quantum states when the initial and final states are non-orthogonal. email:[email protected] In recent years the study of geometry of the quantum state space and its implications have gained much importance. The introduction of Riemannian metric structure by Provost and Valle [1] and Fubini-Study metric by Anandan and Aharonov [2,3] into the projective Hilbert space of the quantum system has attracted a lots of attention. The relation between geometric distance function and geometric phase was studied and the equivalence of the above two metric strctures (up to a scale factor) was pointed out [4]. The introduction of the length of the curve [4,5] has provided us a new way of understanding geometric phases in quantum systems. Subsequently, the metric structures were generalised to mixed states by Anandan [6] and the statistical distinguishability was used to define a metric structre by Braunstein and Caves [7]. Later, the Fubini-Study metric was generalised to non-unitary and non-linear quantum systems, and a metric approach to generalised geometric phase was proposed [8]. One of the outcome of the geometric approach is the parameter-based uncertainty relation (PBUR) in quantum theory. This is often useful when we do not have a Hermitian operator canonical conjugate to another operator which represent a physical quantity of our interest. The vivid example is the quest for time-energy uncertanity relation, when we do not have a Hermitian time operator canonical conjugate to energy. In this letter we study the intelligent states (to be defined soon) for the AharonovAnandan uncertainty relation and prove the existence of such states when the initial and final states are non-orthogonal during an arbitrary parametric evolution of a quantum system. To briefly recall the essential geometric ideas, let us consider a quantum system S whose state vector |ψ(t)〉 ∈ H = CN evolves in time from time t1 to t2. Geometrically the state is represented by a point in the projective Hilbert space P = H − {0}/C∗, where C∗ is a group of non-zero complex numbers. The time evolution of the system gives us a curve C in H, i.e. C : t→ |ψ(t)〉, t1 ≤ t ≤ t2. Since H is Riemannian this curve has a length which