Restricted numerical shadow and the geometry of quantum entanglement

Abstract

The restricted numerical range W R (A) of an operator A acting on a Ddimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the of set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A -a normalized probability distribution on the complex plane supported in W R (A). Its value at point z ∈ C is equal to the probability that the expectation value ψ|A|ψ is equal to z, where |ψ represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini-Study measure. Studying restricted shadows of operators of a fixed size D = N A N B we analyse the geometry of sets of separable and maximally entangled states of the N A × N B composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on D = 2 3 dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three-qubit system.

Symbolic integration with respect to the Haar measure on the unitary groups

“…The first two moments of dµ E X are calculated in [11], and are given by The corresponding probability measure is denoted by dμ E X . For definition and basic facts concerning numerical shadows see [11,13,14].…”

“…The first two moments of dμ E X are calculated in [11], and are given by learn that the integral in Eq. …”

“…The first two moments of dµ E X are calculated in [11], and are given by where tr A and tr B denote partial traces over a specified sub-system and · HS is a Hilbert-Schmidt norm given by X HS = √ trXX † . In order to calculate (39) and (39) define symbolic matrices In this example we take d = 3.…”

Probability density of quantum expectation values

“…The authors would like to thank Karol Życzkowski for bringing to their attention references [11,12,13,14].…”

“…The probability density P A (a) has been considered also in a series of recent works [11,12,13] (see also [14] for an entry into the mathematical literature) where P A (a) is sometimes referred to as numerical shadow. In the present article we give extra care to the physical case where the total Hilbert space is that of a many-body systems and observables are extensive operators.…”