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Limit on Time-Energy Uncertainty with Multipartite Entanglement

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Limit on Time-Energy Uncertainty with Multipartite Entanglement Limit on Time-Energy Uncertainty with Multipartite Entanglement Manabendra Nath Bera, R. Prabhu, Arun Kumar Pati, Aditi Sen(De), Ujjwal Sen Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India We establish a relation between the geometric time-energy uncertainty and multipartite entan- glement. In particular, we show that the time-energy uncertainty relation is bounded below by the geometric measure of multipartite entanglement for an arbitrary quantum evolution of any multi- partite system. The product of the time-averaged speed of the quantum evolution and the time interval of the evolution is bounded below by the multipartite entanglement of the target state. This relation holds for pure as well as for mixed states. We provide examples of physical systems for which the bound reaches close to saturation. I. INTRODUCTION of quantum evolution? Or, more generally, can the geo- metric quantum uncertainty relation be connected, quan- titatively, with the multipartite entanglement present in In the beginning of the last century, the geometry of the system? This question is important not only due space-time has played an important role in the reformu- to its fundamental nature, but also because of its prac- lation of classical mechanics. Similarly, it is hoped that tical relevance in quantum information. We establish a a geometric formulation of quantum theory, and an un- relationship between the multipartite entanglement in a derstanding of the geometry of quantum state space can many-body quantum system and the total distance trav- provide new insights into the nature of quantum world eled by the state (pure or mixed) during its evolution. and, in particular, into the field of quantum information This leads us naturally to consider the geometric time- science. In quantum theory, the geometric properties of energy uncertainty relation for multiparty quantum sys- the quantum state space are characterized by the Rie- tems, and to provide a connection of the same to mul- mannian metrics defined on the corresponding projective tipartite entanglement. In particular, we find that the Hilbert space [1–6]. The discovery of geometric phase, minimum time taken by an initial quantum state (pure by Pancharatnam [7] and then by Berry [8], provided a or mixed) to reach a target entangled state is bounded new fillip to the importance of geometric properties in below by the geometric measure of multipartite entangle- the quantum domain. The connection of this phase to ment [14, 15] (cf. [16] ) upto a factor depending upon the the Riemannian metrics lead to a new understandings in corresponding path-integrated energy fluctuation. In the terms of the geometrical ideas on the projective Hilbert case of mixed states, we show that the relation is generic, space [9]. One of the significant outcomes of this geo- in that its form is independent of the specific metric used. metric approach is the geometric quantum uncertainty We exemplify this by using various metrics such as, the relation (GQUR) [2, 4, 10] which constraints the motion Fubini-Study [2, 5], the Hilbert-Schmidt [4], and the Bu- of a system in its projective Hilbert space. Importantly, res metrics [6, 10]. We then investigate the relation of it was found that for any such evolution, there exists a genuine multipartite entanglement with the GQUR by lower bound on the product of the time separation be- considering a physically realizable quantum many-body tween initial and final state and its time-averaged energy model, viz., the Heisenberg spin chain. fluctuation [4]. The structure of the article is as follows. In Sec. II, Entanglement has played a pivotal role in the recent we introduce the geometric quantum uncertainty rela- developments of quantum information [11]. On the prac- tion (GQUR) in connection to quantum evolution for tical side, the major challenge is to create, store, and pure states. We derive the relation between the GQUR arXiv:1303.0706v1 [quant-ph] 4 Mar 2013 process multiparty entanglement in a controlled manner. and the multipartite entanglement measures for the cor- This has created an immense interest to understand the responding pure states, and illustrate the relation with properties of entangled quantum states and the dynam- examples, in Sec. III. In Sec. IV, we show that a simi- ics of entanglement in a multiparticle scenario [12]. An lar connection can be obtained for mixed quantum states unentangled multiparty quantum system, if allowed to using different metrics. Finally, we conclude in Sec. V. interact, may evolve to an entangled state, depending on the initial state and the driving Hamiltonian. Hence it is important to identify the class of initial states and the class of non-local Hamiltonians that can create a de- II. GEOMETRIC QUANTUM UNCERTAINTY sired entangled quantum state. The entangling capac- RELATION ity of Hamiltonians for two qubits and the relation of the “speed” of the quantum evolution with the entan- The Hilbert space formalism of quantum mechanics glements of the initial and final states were addressed in possesses interesting geometric properties, and their in- Ref. [13]. herent features are used to build the geometric quantum In this article, we pose the question: does multipar- uncertainty relation [1–4, 6, 17]. Let Ψ be a set of {| i} N tite entanglement have any connection to the geometry vectors in a multiparticle Hilbert space = i. H ⊗i=1H 2 Since two unit vectors differing only in phase, represent Anandan is that the energy fluctuation of the state drives the same physically (i.e., quantum mechanically) equiv- the quantum evolution [2]. In the projective Hilbert space alent states, one can construct a ray space from these , an isolated system can move if and only if it is not in a equivalent classes of states. The set of rays of , via a stationaryP state, i.e., it has a non-zero energy fluctuation, projection map, forms the projective Hilbert spaceH ( ). which for the state ψ(t) is denoted as ∆H(t). dN P H | i If dim i = d i, then = C . Correspondingly, for the Suppose that a quantum state Ψ(t) is transported ∼ N | i H ∀ H Cd to a state Ψ(t + dt) after an infinitesimal time inter- projective Hilbert space = ( ) ∼= ( 0 )/U(1), | i which is a complex manifoldP ofP dimensionH (d−{N }1). This val dt, following the Schr¨odinger evolution governed by can also be considered as a real manifold of− dimension the Hamiltonian H(t). By using the Fubini-Study met- 2(dN 1). Any quantum state at a given instant of time ric in Eq. (2), the infinitesimal distance that the system can be− represented as a point in via the projection traverses during this evolution is given by [2] map Π : Ψ Ψ Ψ . The evolutionP of the state vector 2 represents| ai → curve | ihC :| t Ψ(t) in whose projection dS = ~∆H(t)dt, (5) Π(C) lies in [1–6, 9, 17→]. | i H P For a given curve, one can ask how much distance the where ∆H(t) is the energy fluctuation, defined by system has traveled during its time evolution. As the state evolves under the Schr¨odinger evolution, it traces a ∆H(t)2 = Ψ(t) H(t)2 Ψ(t) Ψ(t) H(t) Ψ(t) 2. (6) curve in the Hilbert space. Different Hamiltonians may h | | i − h | | i give different curves, but all of them may be projected The above relation implies that a higher energy fluctua- to a single curve in . For pure states, the distance, S, tion leads to a higher speed in the evolution. If the initial between any two pointsP in the projective Hilbert space , state Ψ(0) is transported to the state Ψ(τ) after a time ′P | i | i corresponding to the quantum states Ψ(λ) and Ψ(λ ) τ, it follows a path whose total distance is can be defined with the Fubini-Study| distancei [2],| as i 2 τ S = ∆H(t)dt. (7) 2 ′ 2 ~ S =4 1 Ψ(λ ) Ψ(λ) , (1) 0 − |h | i| Z ′ For a time independent Hamiltonian, the energy uncer- where λ and λ are parameters on which these states de- tainty, ∆H, is a constant throughout the evolution. In pend. The distance is in fact between rays corresponding this case, the total distance that the system traverses to the vectors of , and is hence defined in . If two vec- during its evolution, from Ψ(0) to Ψ(τ) , is given by tors differ from eachH other infinitesimallyP then we have | i | i the infinitesimal Fubini-Study metric as given by 2 S = ~τ ∆H. (8) dS2 =4 1 Ψ(λ + dλ) Ψ(λ) 2 − |h | i| i j (2) Note here that there are infinitely many Hamiltonians = 4 ( ∂iΨ ∂j Ψ ∂iΨ Ψ Ψ ∂j Ψ ) dλ dλ , h | i − h | ih | i can be used to transport the same initial state Ψ(0) to the same final state Ψ(τ) . The distance traversed| i where we have retained terms upto second order only. | i for any such path must be lower than S0, where S0 is Another distance, denoted as can be defined via the the shortest geodesic connecting Ψ(0) and Ψ(τ) in , Bargmann angle [2] S according to the same metric that| isi used| to calculatei P the actual distance traveled in the quantum evolution. 2 2 Ψ1 Ψ2 = cos S . (3) In other words, we have |h | i| 2 τ∆H > ~ cos−1( Ψ(0) Ψ(τ) ). (9) For infinitesimally close vectors, the Fubini-Study dis- |h | i| tance, and the distance obtained via the Bargmann angle where we have used the metric generated via the are the same and is given by Eq.
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