Geometric approach to non-relativistic quantum dynamics of mixed states

Geometric approach to non-relativistic quantum dynamics of mixed states Vicent Gimeno and Jose M. Sotoca Citation: Journal of Mathematical Physics 54, 052108 (2013); doi: 10.1063/1.4807096 View online: http://dx.doi.org/10.1063/1.4807096 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/54/5?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 JOURNAL OF MATHEMATICAL PHYSICS 54,052108(2013) Geometric approach to non-relativistic quantum dynamics of mixed states Vicent Gimeno1,a) and Jose M. Sotoca2,b) 1Departament de Matem` atiques - Institute of New Imaging Technologies, Universitat Jaume I, Castell´ o, Spain 2Departamento de Lenguajes y Sistemas Inform´ aticos - Institute of New Imaging Technologies, Universitat Jaume I, Castell´ o, Spain (Received 15 February 2013; accepted 3 May 2013; published online 23 May 2013) In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann’s principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincar´ e quantum recurrence in a physical system with finite energy levels. C !2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4807096] I. INTRODUCTION The geometrization of physical theories is a successful and challenging area in theoretical physics. The most well known examples are Hamiltonian mechanics based on symplectic geometry, General Relativity based on semi-Riemannian geometry and classical Yang-Mills theory which uses fibre bundles.11 Geometric ideas have also found a clear utility in non-relativistic quantum mechanics problems because quantum theory can be formulated in the language of Hamiltonian phase-space dynamics.12 Hence, the quantum theory has an intrinsic mathematical structure equivalent to Hamiltonian phasespace dynamics. However, the underlying phase-space is not the same space of classical mechanics, but the space of quantum mechanics itself, i.e., the space of pure states or the space of mixed states. Unlike general relativity or gauge theory where the metric tensor or the connection is related with the physical interaction, the most usual geometric formulation of the geometry of non-relativistic quantum mechanics is not dynamic, in the sense that is insensitive to changes in the Hamiltonian of the system. Under these assumptions, that approach only makes use of the differential structure of the Hilbert space for quantum states and the Fubini-Study metric. See, for example, the geometric interpretation of Berry’s phase.3 From a more dynamical point of view, Kryukov16 has stated that the Schr¨ odinger equation19 for a pure state |αt!(Planck’s constant is set equal to 1) d dt |αt!=−iH|αt!,(1) can be considered as a geodesic flow in a certain Riemannian manifold with an accurate metric which depends on the Hamiltonian of the system. The goal of this paper is to generalize the work of Kryukov for mixed states. To this end, we provide an underlying differential manifold to describe mixed states and a dynamic-dependent Riemannian metric tensor to analyze their temporal evolution. a)Electronic mail: [email protected].es b)Electronic mail: [email protected] 0022-2488/2013/54(5)/052108/12/$30.00 C !2013 AIP Publishing LLC54,052108-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 052108-2 V. Gimeno and J. M. Sotoca J. Math. Phys. 54,052108(2013) FIG. 1. Since the Hamiltonian vector field is a Killing vector field, its flow ϕpreserves the volume (Liouville theorem) in the sphere S, where each point can be projected into the space of density matrices P+. The mixed states are characterized by density matrices and the equation which plays the role of the Schr¨ odinger one is the von Neumann equation23 dρt dt =−i[H,ρ t].(2) To obtain the underlying differential manifold following the Uhlmann’s geometrization for non-relativistic quantum mechanics,26–29 we make use of a principal fibre bundle such that its base manifold is the space of mixed states. Finally, to provide the Riemannian metric we choose an appropriate metric in the principal bundle, in such a way that the projection of the geodesic flow in the principal manifold to the base manifold is just the temporal evolution given by the von Neumann equation. Among the geometric properties that are observed due to the movement of this geodesic flow, in this paper we analyze the phase volume conservation according to the Liouville theorem (see Fig. 1). That allows us to show a geometric proof of the Poincar´ erecurrencetheoremrelatingit with the recurrence principle for physical systems with discrete energy levels. Let us emphasize that our geometric proof for the quantum Poincar´ erecurrenceisclosertotheclassicalmechanicsproof 1 (that also uses the conservation of the volume in the phase-space evolution) than the previous given in the quantum setting.5,22,24 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 052108-3 V. Gimeno and J. M. Sotoca J. Math. Phys. 54,052108(2013) II. DENSITY MATRICES SPACE AS A BASE OF A PRINCIPAL FIBRE BUNDLE The most general state, the so-called mixed state,isrepresentedbyadensity operator in the Hilbert space H.Inthispaperwealwaysassumethatdim(H)=n<∞,beingHa vector space on the complex field (H=Cn). The density operator ρis in fact a density matrix. Recall that a density matrix is a complex matrix ρthat satisfies the following properties: 1. ρis a Hermitian matrix, i.e., the matrix coincides with its conjugate transpose matrix: ρ=ρ†. 2. ρis positive ρ≥0. It means that any eigenvalue of Ais non-negative. 3. ρis normalized by the trace tr(ρ)=1. Let us denote by Pthe space of mixed quantum states. Note that the space of pure states P(H) is just P(H)=!ρ∈P|ρ2=ρ". Recall that the space of quantum pure states has an elegant interpretation as a U(1)-fibre bundle S(H)→P(H). Following Uhlmann26–29 and Bengtsson and Chru´ sci´ nski books,3,9we can use a similar argument to the case of quantum pure states. The key idea of Uhlmann’s approach is to lift the system density operator ρ,actingontheHilbertspaceH,toanextendedHilbertspace Hext :=H⊗H. In quantum information theory,18 the procedure of extension, H→Hext is known as attaching an ancilla living in H.Obviously,thespaceofsquaredmatricesMn,n(C)(nrows, ncolumns) over C(that is a 2n2real dimensional manifold) can be identified with Hext Mn,n(C)∼ =Hext. Since tr(WW†)isasmoothfunctioninthespaceofsquaredmatrices,bytheregularlevelset theorem,17 the set S0:=!W∈Mn,n(C):tr(WW†)=1",(3) is a smooth manifold of Mn,n(C). Actually, it is not hard to see that S0is diffeomorphic to the sphere S2n2−1.Ifρis a mixed state in P,weshalldenoteanelementW∈S0apurification of ρif ρ=WW†,(4) therefore, we get the space of density matrices Pby the projection π:S0→P, where the projection is given by π(W)=WW†.(5) Observe that, if uis a unitary matrix (i.e., uu†=u†u=In)then π(Wu)=π(W).(6) Moreover, to fix notation recall that the Lie group U(n)isaLie transformation group13 acting on S0on the right. In general, a principal fibre bundle13 will be denoted by P(M,G,π), being Pthe total space,Mthe base space,Gthe structure group,andπthe projection.Foreachx∈M,π−1(x) is a closed submanifold of P,calledthefibre over x.Ifpis a point of π−1(x), then π−1(x)istheset of points {pa,a∈G},anditiscalledfibrethroughp. At this point, an important question to answer, is if S0(P,U(n),π) is a principal fibre bundle over the base manifold Pof density matrices. Unfortunately, the answer is no, because U(n)does not act freely on S0.Ingeneral,Wu =Wfor W∈S0and u∈U(n)donotimplythatu=In,but observe that if det(W))=0 (i.e., Wis an invertible matrix ) U(n)wouldactfreelyonourspace.This should be the way to describe the space of density matrices. Instead of starting with Mn,n(C), we start with the subset of invertible matrices. That subset has the differentiable structure of the Lie This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 052108-4 V. Gimeno and J. M. Sotoca J. Math. Phys. 54,052108(2013) group GL(n,C). Then, we build a submanifold Sof GL(n,C)givenby S:=!W∈GL(n,C); tr(WW†)=1".(7) Finally, we obtain the base manifold P+using the projection π:S→P+given by π(W)=WW†(8) and therefore, S(P+,U(n),π)becomesaprincipalfibrebundle.Observethat P+={ρ∈P|ρ>0}, contains only strictly positive (or faithful) density operators. But Pcan be recovered from P+by continuity arguments.3 In short, we describe the geometry of density matrices as a base manifold of a principal fibre bundle consisting of a submanifold Sof the Lie group GL(n,C) diffeomorphic to the sphere S2n2−1 as a total space and the Lie group U(n)asstructuregroup. Since S(P+,U(n),π)admitsaglobalsection 20 τ:P+→S τ(ρ):=√ρ,(9) therefore S(P+,U(n),π)isatrivialbundlefromatopologicalpointofview,thatmeansthat 3,13 S=P+×U(n).(10) III. HAMILTONIAN VECTOR FIELD, DYNAMIC RIEMANNIAN METRIC, SHG-QUANTUM FIBRE BUNDLE AND MAIN THEOREM In this section, we define a Riemannian metric for dynamics systems and we study how this metric acts within the tangent vector space of S. We also discuss its relationship with other metrics such as the Bures metric or the metric proposed by Kryukov.16 A. Hamiltonian vector field, dynamic metric, and its relation with other metrics In order to provide explicit expressions for tangent vectors to Sand the metric tensor, we identify the tangent space TWMn,n(C)withMn,n(C)itself.SinceourtotalspaceSis a submanifold of the manifold Mn,n(C), where each point W∈Sis a matrix, and the tangent space TWSto Sin the point Wis a subspace of the tangent space TWMn,n(C). We can use a matrix to describe a point W∈Sand a matrix to describe a tangent vector X∈TWStoo. First of all, note that the Hamiltonian operator Hinduces a vector field h:S→TSon Sgiven by hW:=−iHW,(11) where hWdenotes the vector field in the point W∈S,i.e.,hW=h(W). That vector field hwill be denoted as the Hamiltonian vector field. For any point W∈S,andanytwotangentvectorsX,Y∈TWS,wedefinethedynamic Riemannian metric gH(X,Y)as gH(X,Y):=1 2tr(X†H−2Y+Y†H−2X).(12) It will be denoted by ∇Hthe Levi-Civita connexion (the sole metric torsion free connexion) given by gH.Indefinition(12)weuseH−2assuming that His an invertible matrix, but that in fact makes no restriction on the Hamiltonian of the system because we can set H→H+Inwithout changing the underlying physics. It is not hard to see that gHdefines a positive definite inner product in each tangent space TWS,beingthereforegHaRiemannianmetric. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 052108-5 V. Gimeno and J. M. Sotoca J. Math. Phys. 54,052108(2013) With that metric tensor gHthe (sub)manifold (S,gH)becomesaRiemannianmanifold.In order to fix the notation we denote !S(P+,U(n),π),h,gH"the SHg-quantum fibre bundle of dimension n. The rest of this section will examine the inherited metric in the base manifold (Theorem 1) from the dynamic metric in the principal manifold and its relation with other metrics. The tangent space TWSat the point W∈Scan be decomposed in its horizontal HWand vertical VWsubspaces: TWS=HW⊕VW. Observe, moreover, that the vertical subspaces VWare the vectors tangent to the fibres. Therefore, any vertical vector XV∈TWScan be written as XV=WA, where A∈u(n)(i.e.,Ais an anti-Hermitian matrix). Note that our metric gHdefines a natural connexion as follows: A tangent vector Xat Wis horizontal if it is orthogonal to the fibre passing through W, i.e., if gH(X,Y)=0, for all vertical vector Yat W(Y∈VW). Hence X∈TWSis horizontal if X†H−2W−W†H−2X=0.(13) Therefore, we can define a metric gP+ Hin the base manifold for any point ρ∈P+,givenby gP+ H(Y,Z):=gH(YHor,ZHor), where Y,Z∈TρP+and YHor (respectively, ZHor)arethehorizontalliftofY(respectively, Z). Theorem 1. The metric gP+ Hin the base manifold at any point ρ∈P+can be obtained as gP+ H(Y,Z):=1 2tr(H−1GYH−1Z), where GYis the unique Hermitian matrix satisfying H−1YH−1=GYH−1ρH−1+H−1ρH−1GY. Note that matrix GYexists and is unique by the existence and uniqueness of the solution of the Sylvester equation.2,25 Observe, moreover, that when His the identity matrix, then gP+ H(Y,Z):=1 2tr(GYZ), where GYis the (unique) solution of Y=GYρ+ρGY and that is the Bures metric.10 Proof. Let W:R→Sbe a curve, such that ˙ Wis a horizontal vector, then (˙ W)†H−2W=W†H−2˙ W. Let us define A=H−1W,thus ˙ A†A=A†˙ A. It is easy to see that the latter condition is fulfilled if ˙ A=GA,(14) where Gis a Hermitian matrix. Therefore ˙ W=HGH−1W. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 052108-6 V. Gimeno and J. M. Sotoca J. Math. Phys. 54,052108(2013) Hence applying Eq. (4) π∗(˙ W)=˙ WW†+W˙ W†=HGH−1ρ+ρH−1GH.(15) Suppose that π∗(˙ W)=Yπ∗(˙ V)=Z W(0)W(0)†=V(0)V(0)†=ρ, then gP+ H(Y,Z)=gH(˙ W,˙ V)=1 2tr( ˙ W†H−2˙ V+˙ V†H−2˙ W) =1 2tr(H−1GYGZH−1ρ+H−1GZGYH−1ρ), (16) where ˙ W=HGYH−1W˙ V=HGZH−1V. Applying Eq. (15)inπ∗(˙ V) Z=HGZH−1ρ+ρH−1GZH. Using the above expression (16)thetheoremfollows. ! In the case of pure states, our Hilbert space is Cnand the tangent space will be Cntoo. Following Kryukov work,14–16 we can define a metric gK(X,Y)foranytwotangentvectorsX=(x,x*), Y=(y, y*), by gK(X,Y):=Re #/H−1X,H−1Y!$, where /X,Y!=%n i=1xiy∗ i,therefore gK(X,Y):=1 2#/H−1X,H−1Y!+/H−1Y,H−1X!$=1 2tr(X†H−2Y+Y†H−2X). When His the identity, we recover the Fubini-Study metric. B. Geometric structure of the SHg-quantum fibre bundle As we have previously seen in the SHg-quantum fibre bundle !S(P+,U(n),π),h,gH"of dimension n,S(P+,U(n),π)isaprincipal(andtrivial)fibrebundle,Sis diffeomophic to the sphere of dimension 2n2−1, his a vector field on S,and(S,gH) is a Riemannian manifold. But the SHg-quantum fibre bundle has more geometric properties: Theorem 2 (Main theorem). Let !S(P+,U(n),π),h,gH"be a SHg-quantum fibre bundle of dimension n. Then: 1. h is a Killing vector field of (S,gH). 2. The integral curves γ:I⊂R→Sof h are geodesics of (S,gH). 3. The projection on the base manifold P+of the geodesic γsatisfies the von Neumann equation d dt π◦γ=−i[H,π◦γ].(17) Proof. Condition (1): In order to proof that his a Killing vector field, we only have to show that the flow ϕt:S→Sgiven by &ϕ0(W)=W,where W∈S d dt ϕt(W)|t=0=hW,(18) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 052108-7 V. Gimeno and J. M. Sotoca J. Math. Phys. 54,052108(2013) is an isometry, i.e., for any X,Y∈TWS gH(ϕt∗(X),ϕ t∗(Y)) =gH(X,Y).(19) Note that ϕt∗(X)=e−iHtX(20) and gH(ϕt∗(X),ϕ t∗(Y)) =gH(e−iHtX,e−iHtY) =1 2tr #(e−iHtX)†H−2e−iHtY+(e−iHtY)†H−2e−iHtX$ =1 2tr #X†eiHtH−2e−iHtY+Y†eiHtH−2e−iHtX$ =1 2tr #X†H−2Y+Y†H−2X$=gH(X,Y). (21) Conditions (2) and (3): First of all observe that if γis the integral curve of the vector field h, i.e., ˙γ=hγ=−iHγ.(22) The projection of γsatisfies d dt π(γ(t)) =d dt #γ(t)γ†(t)$=˙γ(t)γ†(t)+γ(t)˙γ†(t)=˙γ(t)γ†(t)+γ(t)( ˙γ(t))† =−iHγγ†(t)+γ(t)(−iHγ)†=−i[H,π(γ(t))]. (23) Hence, the projection of the integral curves of the vector field hsatisfies the von Neumann equation. So all we have to prove is that curves are actually geodesic curves ∇H hγhγ=0.(24) Since his a Killing vector field, we only have to proof that his a unitary vector field (as any unitary Killing vector field is a geodesic). Namely, the equality gH(hγ,hγ)=gH(−iHγ,−iHγ)=tr #(−iHγ)†H−2(−iHγ)$ =tr #γ†HH−2Hγ$=tr #γ†γ$=1. (25) Finally, since His a unitary Killing vector field and the integral curves of any Killing vector field of constant length is a geodesic (see Theorem 8 in the Appendix), the integral curves of Hare geodesics. ! Remark. Let us emphasize that for any Hermitian matrix A=A†,wecanbuildthevectorfield Aon Sgiven by −iAW for any W∈S.ItiseasytocheckasdoneinEq.(21)thatif[H,A]=0, A is a Killing vector field. Therefore, the set of operators compatible with the Hamiltonian is related to the set of isometries of (S,gH), and we can identify any conserved quantum observable with a Killing vector field. IV. GEOMETRIC APPROACH TO QUANTUM POINCAR ´ E RECURRENCE As we know from the main theorem, his a Killing vector field on the principal manifold (S,gH) endowed with the dynamic metric gH. Then, the transformations given by the 1-parametric subgroup This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.128.148.22 On: Wed, 05 Mar 2014 12:57:02 052108-8 V. Gimeno and J. M. Sotoca J. Math. Phys. 54,052108(2013) ϕt:S→Sof integral curves of hare distance-preserving and volume-preserving (see Theorem 9 in the Appendix). These two facts have the following consequences. Theorem 3 (Insensitivity to initial conditions theorem). Let !S(P+,U(n),π),h,gH"be a SHg-quantum principal bundle of dimension n. Then, for any two points W,V∈S dist(ϕt(W),ϕ t(V)) =dist(W,V),(26) being the ϕtthe 1-parametric subgroup of transformations given by the integral curves of the Killing field h. The classical Liouville theorem1states that the natural volume form on a symplectic manifold is invariant under the Hamiltonian flows. In our case, we have the 1-parametric subgroup of transformations ϕt:S→Sgiven by the integral curves of the Killing vector field hand we can set Theorem 4 (Liouville type theorem). Let !P+,U(n),π),h,gH"be a SHg-quantum principal bundle of dimension n. Then for any domain '⊂S Vol(ϕt(')) =Vol('),(27) being the ϕtthe 1-parametric subgroup of transformations given by the integral curves of the Killing vector field h. Using the above theorem, we can, therefore, state a similar theorem to the Poincar´ erecurrence theorem.1 Theorem 5 (Poincar´ etypetheorem).Let !S(P+,U(n),π),h,gH"be a SHg-quantum principal bundle of dimension n. For any domain '⊂Sand any time period T ∈R+there exists a point x∈'and a positive integer k >0such that ϕkT (x)∈',(28) being ϕt:S→Sthe 1-parametric subgroup of transformations given by the integral curves of the Killing field h. Proof. Consider the following sequence of domains ',ϕT('),ϕ 2T('),···,ϕ kT ('),··· All domain in the sequence belongs to the same volume Vol('). If the above domains never intersect S, an infinite volume would obtain, but Sis compact, so Vol(S)<∞. Then, there exist l ≥0andm>lsuch that ϕlT(')∩ϕmT ('))=∅,(29) so '∩ϕ(m−l)T('))=∅.(30) Setting k=m−lthe theorem is proven. ! Joining the above theorem with the insensitivity to the initial conditions we get This article is copyrighted as indicated in the article. 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