arXiv:1303.0706v1 [quant-ph] 4 Mar 2013 eeomnso unu nomto [ information quantum of developments lmnso h nta n nlsae eeadesdin addressed entan- were the states of final [ with relation and Ref. evolution initial the quantum the and of the glements of two capac- “speed” entangling for the The Hamiltonians de- of and a state. ity states create quantum can initial that entangled of Hamiltonians sired class non-local Hence the of identify class Hamiltonian. the to driving depending important the state, is and entangled it state initial an to the to on allowed evolve if may [ system, interact, scenario quantum multiparticle multiparty a dynam- unentangled in the entanglement and the of states understand ics quantum and to entangled interest store, of immense properties create, an created to manner. has controlled is This a challenge in entanglement multiparty major process the side, tical be- energy separation time-averaged time its a [ and the exists fluctuation state of final there and product evolution, initial the tween such on any Importantly, for bound space. that lower Hilbert found projective was its it in system uncertainty a of quantum [ geometric (GQUR) the relation is Hilbert approach projective metric the on [ ideas to in space geometrical understandings phase the new this a of in of to terms lead connection properties metrics The geometric Riemannian the of domain. importance quantum the the to fillip new ieetnlmn aeaycneto otegeometry the to connection any have entanglement tite h unu tt pc r hrceie yteRie- [ the by corresponding Hilbert the characterized on are defined of metrics space mannian properties state geometric quantum the information theory, the quantum world quantum of In quantum field the of science. into nature particular, the in can into and, space insights state un- quantum new an of provide and geometry that theory, the hoped quantum of is derstanding of it formulation Similarly, reformu- geometric the a mechanics. in classical role of important lation an played has space-time yPnhrta [ Pancharatnam by nageethspae ioa oei h recent the in role pivotal a played has Entanglement nti ril,w oeteqeto:de multipar- does question: the pose we article, this In ntebgnigo h atcnuy h emtyof geometry the century, last the of beginning the In 13 9 ]. .Oeo h infiatotoe fti geo- this of outcomes significant the of One ]. 4 ii nTm-nryUcranywt utprieEntang Multipartite with Uncertainty Time-Energy on Limit emti esr fmliatt nageetfra arbi an for uncert entanglement time-energy multipartite the of that measure show geometric we particular, In glement. hsrlto od o uea ela o ie tts epr We states. saturation. mixed to for close multiparti as reaches the well bound as by th the pure which below of for for speed bounded holds time-averaged is relation the evolution This of the product of interval The system. partite ]. eetbiharlto ewe h emti ieeeg u time-energy geometric the between relation a establish We 1 aaedaNt ea .Pah,Au ua ai dt Sen(De Aditi Pati, Kumar Arun Prabhu, R. Bera, Nath Manabendra .INTRODUCTION I. – 6 aihCadaRsac nttt,Chta od Jhunsi Road, Chhatnag Institute, Research Harish-Chandra 2 .Tedsoeyo emti phase, geometric of discovery The ]. 7 , n hnb er [ Berry by then and ] 4 , 10 hc osrit h motion the constraints which ] 11 .O h prac- the On ]. 8 ,poie a provided ], 12 .An ]. etr namliatceHletspace Hilbert multiparticle a in vectors sn ieetmtis ial,w ocuei e.V. Sec. simi- in a conclude that we show Finally, states we quantum metrics. mixed IV, different for using obtained Sec. be with In can connection relation III. lar the Sec. illustrate in and examples, cor- states, GQUR the pure for the for measures between responding entanglement relation evolution multipartite the the quantum derive and We to rela- connection states. uncertainty pure in quantum (GQUR) geometric tion the introduce we chain. spin Heisenberg many-body the quantum by viz., realizable model, GQUR physically the a with considering entanglement multipartite genuine eetfaue r sdt ul h emti quantum geometric [ the build relation in- to uncertainty used their are and features properties, herent geometric interesting possesses eeepiyti yuigvrosmtissc s the as, such metrics various used. [ using Fubini-Study metric by specific this the generic, of exemplify is independent We is relation form the its that that show in we states, mixed the of In case fluctuation. energy path-integrated corresponding e erc [ metrics res rmxd orahatre nage tt sbounded is entangle- state multipartite [ of entangled ment measure target geometric (pure the a state by reach below quantum the to initial mul- that mixed) an to find or by same we taken the particular, time of minimum In connection a entanglement. provide tipartite time- to sys- geometric quantum and evolution. multiparty the tems, its for consider during relation to uncertainty mixed) naturally energy or us (pure leads state This trav- the distance by total the eled a and system in a establish prac- many-body multipartite We due its the of between information. only relationship because quantum not also in important relevance but is tical nature, fundamental question its This to in system? present entanglement the multipartite geo- the quan- the with connected, titatively, can be generally, relation uncertainty more quantum Or, metric evolution? quantum of I EMTI UNU UNCERTAINTY QUANTUM GEOMETRIC II. h tutr fteatcei sflos nSc II, Sec. In follows. as is article the of structure The h ibr pc omls fqatmmechanics quantum of formalism space Hilbert The 14 rr unu vlto faymulti- any of evolution quantum trary it eaini one eo ythe by below bounded is relation ainty , eetnlmn ftetre state. target the of entanglement te 15 vd xmlso hsclsystems physical of examples ovide unu vlto n h time the and evolution quantum e cranyadmliatt entan- multipartite and ncertainty c.[ (cf. ] 6 laaa 1 1,India 019, 211 Allahabad , , 2 10 , 5 .W hnivsiaeterlto of relation the investigate then We ]. ,teHletShit[ Hilbert-Schmidt the ], 16 poafco eedn pnthe upon depending factor a upto ) ] 1 RELATION – 4 , 6 ,Uja Sen Ujjwal ), , 17 .Let ]. lement {| Ψ 4 H i} ,adteBu- the and ], = eastof set a be ⊗ i N =1 H i . 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 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 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. (2). In addition, one Bargmann angle. One can also use the Fubini-Study dis- can also define the “minimum normed distance” between tance of Eq. (1) or the minimum normed distance of Eq. the states ψ1 and ψ2 as [1, 5] | i | i (4) to generate the metric, in which case weaker rela- 2 tions are obtained. Such relations are usually referred to S = 2(1 Ψ1 Ψ2 ) . (4) N − |h | i| as the geometric quantum uncertainty relation (GQUR). However, when the states differ infinitesimally, we get The equality in Eq. (9) holds only for the quantum tra- 4dS2 = dS2. jectories that coincide with the geodesics in . Quantum N P Let us now consider the parametrization of a curve in states which satisfy the equality are known as intelligent using the time t. During the unitary time evolution states [18]. In case, the Hamiltonian is time-dependent, ofH the quantum state by the Hamiltonian H, the path the GQUR is taken by the state should be guided by the property of τ the quantum state and the characteristics of the Hamil- ∆H(t)dt > ~ cos−1( Ψ(0) Ψ(τ) ). (10) |h | i| tonian. The important result obtained by Aharonov and Z0 3

III. GQUR AND MULTIPARTITE QUANTUM With the definition in Eq. (12) and remembering that the ENTANGLEMENT distance function is always positive, the above relation becomes Entanglement measures of multiparty quantum states cos−1 ( Ψ(0) Ψ(τ) ) > ( ) . (14) can be defined by considering the geometric distance |h | i| G EC as measured by a given metric from a specified set of Now, using the geometric quantum uncertainty relation non-entangled (separable) quantum states, in the Hilbert in Eq. (10) and the above equation, we obtain space. Depending on the set of non-entangled states, from which the distance is measured, the distance quan- ∆Hτ > ~ ( C) , (15) tifies different types of entanglement present in the sys- G E tem. For an N-party pure quantum state ΨN , such a where ∆H is the time-averaged energy fluctuation, de- | i measure of multipartite entanglement can be defined as fined as [14, 15] 1 τ 2 ∆H = ∆H(t)dt, (16) C( ΨN ) = min 1 Φm ΨN , (11) τ 0 E | i {|Φmi∈SC } − |h | i| Z  which can also be interpreted as the time-averaged speed where the minimization is carried over a certain set of of the evolution. The equality in Eq. (15) holds only separable states S . There is a hierarchy of such geomet- C when the Hamiltonian which drives the state follows the ric measures, depending on the set of states Φm SC geodesic, and the Ψ(0) coincides with the state Φmin chosen for the minimization. Among the set{| of suchi ∈ mul-} for which the minimization| i is achieved in . | i tipartite entanglement measures, , two are more promi- C C Note here that the “unentangled” initialE state in the nent, in that they are defined withE respect to two ex- statement of the theorem is from the same set of sepa- tremal choices of the set SC. More precisely, choosing SC rable states, SC, which is used in the definition of the to be the set, S0, of all fully separable states, i.e., states . of the form Φ1 Φ2 ... ΦN in the N-party tensor C E The theorem implies that for a given energy uncer- product Hilbert| i⊗| spacei⊗ of the⊗| N-partyi system, we obtain tainty ∆H, the minimum time τ required to connect the “geometric measure” (GM) of multiparty entangle- the initial and final states depends on the entanglement ment [14]. We denote it by 0. On the other hand, if the present in the final state. It requires more time, τ when set S is formed by the statesE which are not genuinely C the final state has more multipartite geometric entan- multipartite entangled, the “generalized geometric mea- glement. This result is potentially important in founda- sure” (GGM), is obtained [15]. An N-party pure quan- tional as well as practical aspects of quantum information tum state is said to be genuinely multiparty entangled and beyond. if it is entangled across every partition of the N parties Let us now illustrate the effectiveness of the above into two disjoint sets. We denote this set of separable bound by considering some specific Hamiltonians. Con- set as S and the GGM as . In general, if the mini- G G sider the N- product state in the standard product mization is carried out overE the set of (k 1)-separable form given by states, the measure will be called the geometric− measure N−k+1. Note that the GGM, G, gives the lowest value Ψ = ϕ ⊗N , (17) Eamong all the geometric multipartiteE entanglement mea- | i | i sures, for a given multipartite quantum state. We can where ϕ = cos θ 0 + exp( iφ) sin θ 1 with θ [0, π] define another geometric multipartite entanglement mea- and φ | i[0, 2π), and| i where −0 and 1| arei eigen∈ vectors ∈ | i | i sure, ( C), which is a monotonically increasing function of the σ Pauli matrix with the eigen values +1 and -1, G E z of C, as respectively. Suppose that the evolution of the state is E guided by the Ising Hamiltonian −1 ( C) = cos 1 C. (12) G E −E N−1 p J z z Note that ( C) also satisfies all the properties of entan- HI = (I σ )(I σ ) . (18) G E 4 − i − i+1 glement measures which C satisfies. i=1 ! Let us now build the connectionE between the geometric X quantum uncertainty relation and the geometric measure We are therefore considering a liner arrangement of N quantum spin-1/2 particles. σa (a = x,y,z) are the Pauli of entanglement, C. i Theorem: For anE arbitrary quantum evolution, the time spin matrices at the site i. J is a system parameter hav- interval multiplied by the time-averaged energy fluctua- ing the dimension of energy. This Hamiltonian has been tion is bounded below by the geometric measure of mul- used to obtain the “cluster state” which is an important tipartite entanglement, provided the initial state is unen- substrate for quantum computation [19]. Starting from the initial state Ψ with φ = 0, we now tangled. | i Proof: By definition, we have check whether the lower bound given in Eq. (15) for this Hamiltonian is tight or not. To do this, we generate all Ψ(0) Ψ(τ) 2 6 1 ( Ψ(τ) ) (13) possible initial states by varying the state parameter θ, |h | i| −EC | i 4 and evolve the system for each such initial state, by using the bound, given in Eq. (15), saturates also at large times the Ising Hamiltonian. of the evolution. Also, in the (τ, θ)-space the saturation For ease of demonstration, we set of the bound happens in a larger area for high magnetic fields, in comparison to low fields (see Fig. 2). τ∆H δ = ( G). (19) ~ − G E So the entanglement bound of the time-energy geometric IV. MIXED STATES GQUR AND uncertainty is saturated if δ = 0. In Fig. 1, we plot δ, MULTIPARTITE ENTANGLEMENT in which the evolution is governed by the Hamiltonian given in Eq. (18), consisting of three spins, with respect Until now, we have dealt with the relation of the time- to the evolution time τ and the initial state parameter θ. energy uncertainty with multipartite entanglement, for pure quantum states. In this section, we show that it is also possible to obtain similar uncertainty relations for mixed multipartite quantum states, by again involving geometric multipartite entanglement measures. Unlike for pure states, there are quantitatively distinct relations that are obtained for different metrics that can be utilized to metrize the space of density operators on the Hilbert space corresponding to the physical system under con- sideration. However, all metrics produce quantitatively similar relations.

A. GQUR and multipartite entanglement using Fubini-Study metric

τ∆H FIG. 1. (Color online) Plot of δ = ~ −G(EG) with respect The distance between two arbitrary mixed quantum Jτ states can be quantified in several ways. One of the to ~ and initial state parameter θ. The product state, given by Eq. (17), evolves according to the cluster Hamiltonian, prominent ones is the Hilbert-Schmidt distance, which given in Eq. (18). The bound is tight for small evolution is defined for two arbitrary density matrices, ρ1 and ρ2, Jτ time. While δ and ~ are dimensionless parameters, θ is as measured in radians. 2 SHS [ρ1,ρ2]= Tr (ρ1 ρ2) . (21) To investigate whether the saturation of the lower − q bound is potentially generic, we evolve the state, given The above distance is Riemannian although it, in general, in Eq. (17), by considering the anisotropic XYZ (Heisen- is not contractive under completely positive maps. If berg) Hamiltonian, given by two density matrices, obtained via time evolution with N the same Hamiltonian and from the same initial state, x x y y differ infinitesimally in the time parameter t, the metric HH = J (1 + γ)σ σ + (1 γ)σ σ i i+1 − i i+1 i=1 becomes X  z z z 2 +µσi σi+1 + hσi . (20) 2 dSHS = Tr [ρ(t + dt) ρ(t)] 2 − (22) Here, γ [0, 1] is the anisotropy parameter which = Tr (ρ ˙) dt2, modulates∈ the relative strengths of the xx and − yy interactions and µ determines the strength of the where theρ ˙ is the time-derivative of ρ(t). The time evo- − zz interaction. And h is the applied magnetic field. γ, µ lution, governed by the Hamiltonian H, following the − and h are the dimensionless system parameters. J is a Schr¨odinger-von Neumann equation, i~ρ˙ = [H,ρ], leads system parameter that has the dimension of energy. The to Hamiltonian, HH , again describes a system of N quan- tum spin-1/2 particles arranged in a ring. The case for dS2 4 HS = Tr (ρ2H2) (ρH)2 . (23) which γ = 1 and µ = 0 corresponds to the transverse dt2 ~2 − Ising Hamiltonian, while that for which γ = 0 and µ =1   corresponds to the isotropic Heisenberg Hamiltonian. By The expression 2Tr (ρ2H2) (ρH)2 has been argued as simulating the dynamics numerically, we observe that for an quantum mechanical fluctuation− in energy. We denote 2   any choice of γ, δ, and h, the bound is tight for all low it as ∆HQ(t) [4]. A similar expression (up to a fac- values of the time of evolution. However, there exist in- tor) also can be derived by using the Anandan distance, termediate values of initial state parameter θ, for which which is an extension of the Fubini-Study (FS) distance 5

Jτ FIG. 2. (Color online) Projection plots of δ with respect to the evolution time ~ (horizontal axes) and initial state parameter θ (vertical axes). (a) and (b) respectively represent δ with no magnetic field (i.e., h = 0) and with h = 1.5. (1) is the plot of δ for γ = 0and µ = 0.5, (2) corresponds to the same for γ = 0.5and µ = 0, and (3) represents the same for γ = 0.5and µ = 0.5. Clearly, in all the cases, the bound is tighter for low τ than for larger times. However, at large time, when the applied magnetic field is high, then the bound is more effective as compared to low magnetic fields. This feature remains the same for other choices of the parameters γ, µ, and h. The units are as in Fig. 1.

for mixed states [4]. In fact, the metric, dSFS, in this For a time-independent Hamiltonian H, the above in- case, is related to dSHS as equality reduces to Tr [ρ(t + dt)ρ(t)] 2 τ ∆HQ > ~ S0. (28) dSFS =4 1 − Tr [ρ(t)2] (24)  2  To relate the uncertainty relation to multipartite en- =2 dS , HS tanglement measures, let us introduce a distance-based where we assume that the Hilbert-Schmidt distance is measure of quantum entanglement, based on the Fubini- normalized by the trace of the square of the density ma- Study metric. For an arbitrary N-party quantum state trix at time t. Note here that the latter quantity (i.e., the ρA1...AN , the measure of multipartite entanglement, us- normalization) is time-independent. For pure states, the ing the FS metric, is defined as above expression reduces to Eq. (2). So, by using Eqs. S (23) and (24), we obtain a relation between the speed of FS Tr ρA1...AN ρA1...AN (ρ 1 N ) = min 1 C A ...A S 2 the quantum evolution of the mixed quantum state with E ρ ∈SC − Tr ρ A1...AN  A1...AN ! the quantum fluctuation in energy, as (29)   dSFS 2 where the minimization is carried over a certain class, = ∆HQ(t). (25) dt ~ C, of separable states. There is again a hierarchy S Therefore, if quantum fluctuation in energy is large, the in the geometric measures defined in this way, just S speed of quantum evolution will be fast. Following the like for pure states. For example, if the ρA1...AN are arguments given in the case of pure quantum states, the fully separable, the corresponding measure denoted here FS as (ρA1...AN ), can be called “geometric measure” geodesic distance S0 measured between two mixed states, E0 ρ1 and ρ2, with the help of the Fubini-Study metric, can of entanglement under the FS metric. If the mini- be given by mization is considered over all the states which are not (k 1)-separable, then the measure, denoted as Tr [ρ1ρ2] 2 S0 FS − = cos . (26) N−k+1(ρA1...AN ), quantifies the entanglement content of Tr [ρ2] 2 E 1   the class of states ρA1...AN which are (k 2)-separable, (k 3)-separable{ and so on} upto the set− of genuinely Let us now consider that an initial state ρ(0) evolves to − a final state ρ(τ) in the time interval τ. In that case, the multipartite entangled states. Like in the case of pure geometric uncertainty relation is given by states, we can also define a valid measure of multipartite entanglement, given by τ 1 −1 Tr [ρ(0)ρ(τ)] ∆H (t)dt > S0 = 2cos (27) ~ Q Tr [ρ(0)2] FS = cos−1 1 FS, (30) Z0 s G EC −EC  q 6

FS B which is a monotonically increasing function of C . By the corresponding measure is denoted as G (ρA1...AN ), ~ ~ FS E E definition, S0 > C . Hence from Eq. (28), we and quantifies the genuine multipartite entanglement obtain G E present in the quantum state. Therefore, time-energy ge-  ometric quantum uncertainty relation, in terms of mul- FS τ ∆HQ > ~ , (31) tipartite entanglement measure quantified by using the G EC Bures metric, is given by which is an uncertainty relation involving multipartite entanglement and the energy fluctuation of multipartite ~ B τ ∆H > C . (37) quantum states involved in a quantum dynamics. G E A similar relation, but involving the time integral of the energy fluctuation, holds for time-dependent Hamiltoni- B. Bures metric, GQUR, and entanglement ans. V. CONCLUSION The distance between two arbitrary quantum states can geometrically be measured also by the Bures metric, In quantum theory, the celebrated Heisenberg uncer- which is defined as tainty relation provides a bound on the position and mo- mentum uncertainties, in terms of Planck’s constant. In- dS2 = 2 2 F (ρ(t),ρ(t + dt)) , (32) B − deed, for most of the history of quantum theory, the  p  Planck’s constant has played a dominating role in pro- where the Uhlmann fidelity, F , is given by viding the ultimate bounds on our ability to measure 2 two incompatible observables and in a variety of other aspects. In the ground-breaking works on Bell inequality F (ρ1,ρ2)= Tr √ρ1ρ2√ρ1 . (33) and other quantum information tasks, it is quantum en-   q tanglement that seems to dominate most of the develop- The metric is Riemannian. Moreover in contrast to the ments. It is intriguing to ask whether quantum entangle- FS metric, it is also contractive under completely positive ment also sets a fundamental bound on the quantum fluc- 2 maps. In this case, the energy fluctuation, ∆H(t) = tuations. In this work, for the first time, we have shown Tr(ρH2) (TrρH)2, can be shown to be related to the − that it is not only the Planck’s constant but also quantum Uhlmann fidelity [10] as entanglement that plays an important role in setting the τ limits for the quantum uncertainties. This is attained by ∆H(t)dt > ~ cos−1 F (ρ(0),ρ(τ)). (34) using the geometry of space of quantum states. This un- Z0 derlines the power of geometric ideas in quantum theory – p they help to bring together two of the most fundamental where ρ(0) and ρ(τ) respectively correspond to the quan- ingredients of quantum theory, namely, the “quantum of tum states at times t = 0 and t = τ. For time- action” and the quantum entanglement”. To be specific, independent Hamiltonians, it again reduces to we have found a relation between the time-energy uncer- tainty and the geometric measure of multipartite entan- > ~ −1 τ ∆H cos F (ρ(0),ρ(τ)). (35) glement for both pure and mixed quantum states. The time-energy uncertainty relation is shown to be bounded The above relation is known asp the time energy geometric below by the multipartite entanglement. We find that quantum uncertainty relation using Bures metric. the minimum time taken for an initial quantum state to We can exploit the Bures metric to define distance reach a target entangled state is directly proportional to based measure of multipartite entanglement, and for an the geometric measure of multipartite entanglement. We arbitrary N-party quantum state ρ 1 , it reads A ...AN have given examples by which our results can be illus- B trated clearly. We believe that these findings will have C (ρA1...AN )= E important bearing in many areas of quantum theory, in- S S cluding quantum information processing, precession mea- min 1 Tr ρ (ρ 1 N ) ρ . S A1...AN A ...A A1...AN ρ ∈SC A1...AN − ! surements and quantum metrology. rq q (36) Acknowledgement– R.P. acknowledges support from the Department of Science and Technology, Government In particular, if the minimum is taken over the set of of India, in the form of an INSPIRE faculty scheme at states which are not genuinely multipartite entangled, the Harish-Chandra Research Institute (HRI), India.

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