A metric for quantum states issued from von Neumann’s entropy (paper)

28/08/2013
Auteurs : Roger Balian
OAI : oai:www.see.asso.fr:2552:5134
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A metric for quantum states issued from von Neumann’s entropy (paper)

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A metric for quantum states issued from von Neumann’s entropy Roger Balian Institut de Physique Th´eorique CEA Saclay 91191 Gif-sur-Yvette Cedex roger@balian.fr Abstract. We introduce on physical grounds a natural metric structure for the space of states ˆD in quantum mechanics, in order to estimate, for instance, the quality of an approximation through the distance between a state and one of its approximations. This metric is issued naturally from quantum information theory. Its construction relies on an analysis of the mathematical foundations of quantum mechanics and on von Neumann’s entropy S=− Tr ˆD ln ˆD. Its form, ds2 =−d2 S( ˆD) deserves mathematical study. We present here how one can define on physical grounds a metric structure for the space of states in quantum mechanics, in order to estimate, for instance, the quality of an approximation through the distance between a state and one of its approximations. This metric is issued naturally from information theory. Its construction relies on an analysis of the mathematical foundations of the quantum theory, which we first recall. 1 Formalism of quantum mechanics. Although quantum mechanics is our most fundamental physical theory, it is irreducibly probabilistic. It does not allow us to describe an individual system at the microscopic scale, but provides information about statistical ensembles E that encompass a large number of systems prepared under similar conditions. Classical properties, such as the possibility for a physical quantity to take a well-defined value, emerge only at the macroscopic scale. Two concepts underlie the principles of quantum mechanics, that of “ob- servable” and that of “state”. An observable O is a mathematical object that describes a physical quantity pertaining to the considered system, such as the position or the angular momentum of a particle of the system, or the value of an electric field at some point. The values of observables are a priori undeter- mined; they play the role of random variables, except that they are elements of a non-commutative C -algebra of operators, the manifestation of their quantum nature. For finite systems on which we focus, they are represented by Hermitean matrices acting in a Hilbert space H with dimension depending on the system. What we term as a “state” of the considered system (or rather of the real or thought ensemble E in which it is embedded) is in fact a mathematical tool which gathers our information on the system and allows us to make predictions about it. It is characterised by a correspondence O → O associating with each observable a real number, which may be termed as a “quantum expectation value”. This definition sets a quantum state on the same footing as a probability distribution in classical physics, and in fact it is the same as the definition of a state in classical statistical mechanics, except for the non-commutation of the observables. When an observable is measured by letting the systems of an ensemble E interact with a macroscopic apparatus, different values may be found at each run, with relative frequencies depending on the state; then O is interpreted as an ordinary expectation value. However, this interpretation cannot be given simultaneously for two non-commuting observables (which cannot be measured with the same apparatus). This entails strange features for quantum states, which therefore differ from ordinary probability distributions. For instance, correlations between ordinary random variables satisfy inequalities which are violated by correlations between observables. When observables are represented by matrices in a Hilbert space H, the correspondence O → O , which is linear, can be implemented in terms of a matrix D in the space H, in the form of the trace O = Tr D O over H. The density matrix D, which is a representation of the state, is Hermitean (as O is real for O = O† ), normalised (as I = 1 for the unit observable I) and non-negative (as O2 ≥ 0 for any O). The bilinearity, in terms of observables and states, of the set of expectation values O = Tr D O leads us to regard the observables O as elements of a vector space over real numbers. Then, keeping aside the multiplication of observables and the matrix structure of observables and states, we identify each state as a linear mapping O = (D; O) of the vector space of observables onto real num- bers. Hence the density matrices D which characterise these mappings appear as elements of the dual vector space of that of observables. The mathematical representation of quantum mechanics can be changed, without any incidence on physics, by performing simultaneously in both vector spaces of observables and states a linear change of coordinates that preserves the scalar product (D; O). We thus obtain Liouville representations of quantum mechanics, more general than the representations in Hilbert spaces [1]. This mathematical transformation has no physical consequence, as it leaves invariant the expectation values O = (D; O). However, no other physically meaningful objects than the latter quantities are yet available. The desired construction of a physically meaningful metric in the space of states requires introducing scalar quantities that depend solely on this space and not on the two dual spaces; we should therefore resort to a further idea. 2 Von Neumann’s entropy. It is natural, since a state D gathers all the information available on an en- semble E of systems, to associate with it an entropy S. This quantity measures the amount of information which is missing because our predictions about the observables are only probabilistic. The entropy of a state represented by the density matrix D has been defined by von Neumann as S(D) ≡ − Tr D ln D. This expression was introduced in 1932 as an extension of the Boltzmann-Gibbs entropy of classical statistical mechanics; it has inspired Shannon when he built the theory of communications. Von Neumann showed its interest in physics, on the one hand in the theory of quantum measurements [2], on the other hand in quantum statistical mechanics; in particular; its maximum, for given values of the macroscopic extensive variables (such as energy or particle number), can be identified with the entropy of thermodynamics. More generally, von Neumann’s entropy is currently used to assign a state to a quantum system when only partial information about it is available [1, 3]. Among all possible states that are compatible with the known expectation values, we select the one which yields the maximum entropy under constraints on these data. This procedure has often been confirmed by the agreement with experiment of the statistical predictions yielded by the state thus constructed. The maximum entropy criterion has also been derived from Laplace’s principle of insufficient reason [4], by identifying expectation values in the Bayesian sense with averages over a large number of similar systems. Writing then that the ex- pectation value of each observable for a given system is equal to its average over E shows that equiprobability for E implies maximum entropy for each system. Such a derivation enforces the interpretation of von Neumann’s entropy as mea- sure of missing information: the state found by maximising this entropy can be regarded as the least biased one, as it contains no more information than needed to account for the given data. Other expressions than von Neumann’s entropy have been proposed for as- cribing an entropy to a quantum state D. For instance Renyi types of entropies have been introduced, involving a power of D instead of D ln D under the trace. However, the only physically satisfactory entropy is von Neumann’s, since it is characterised by the properties required for an interpretation as a lack of infor- mation or as a measure of disorder: additivity, subadditivity, extensivity for a macroscopic system, concavity [5]. These properties are also needed to ensure that the maximum entropy criterion follows from Laplace’s principle, since the latter principle provides an unambiguous construction of von Neumann’s entropy [4]. 3 Geometry derived from von Neumann’s entropy. We have at our disposal two types of physically meaningful scalar quantities, the expectation values Tr D O across the two dual spaces of observables and states, and the entropy − Tr D ln D within the space of states. Comparison between these expressions leads us to regard ln D as an element of the space of observables and S = −(D; ln D) as a scalar product. This will allow us to construct on physical grounds a geometric structure in these spaces. Indeed, the concavity of von Neumann’s entropy implies that the second differential d2 S, a quadratic form of the coordinates of D, is negative. It is then natural [6] to define the distance ds between the state D and a neighbouring state D + dD as the square root of ds2 = − d2 S. The corresponding Riemannian metric tensor is the Hessian of −S(D) as function of a set of independent coordinates of D. We have thus endowed the set of states with a Riemannian metric ds2 = − d2 S = Tr dD d ln D. (This set is not the full vector space of Hermitean operators, which is the dual of the space of observables, but its subset Tr D = 1, D ≥ 0.) Physical relevance therefore produces an elaborate geometric structure for the mathematical objects of quantum theory. We acknowledged first the C -algebraic structure of the space of observables, then the dual vector space structure of the two sets of observables and states, and finally the Riemannian metric structure ds2 = − d2 S of the space of states. Consequently, we can introduce in this space distances, geodesics, angles, curvatures, duality between covariant and contravariant vectors dD and d ln D, the latter living in the space of observables. This geometric duality between dD and d ln D is related to an algebraic du- ality between D and ln D based on a Legendre transformation. Consider the function F(X) ≡ ln Tr exp X of an arbitrary observable X. It can be regarded as a generalisation for operators of the Massieu potential f(β), which char- acterises the canonical thermodynamic equilibrium states, and to which it re- duces for X = −βH (where H is the Hamiltonian). The partial derivatives of F(X) with respect to the coordinates of X are given by dF = Tr DdX where D ≡ exp X/ Tr exp X. Being Hermitean, normalised and positive, D can be in- terpreted as a density matrix. We can then introduce the Legendre transform of F(X) with respect to the coordinates of X, to wit, the function F − Tr DX of the coordinates of D, which is readily identified with von Neumann’s entropy S(D) = − Tr D ln D. Such a correspondence can be regarded as a generalisa- tion of the Legendre transformation df = −udβ, s(u) = f + βu, ds = βdu, which relates the Massieu potential f(β) as function of the inverse temperature β to the entropy s(u) of equilibrium thermodynamics as function of the inter- val energy. Thus, X and D = exp X/ Tr exp X appear as conjugate variables in the Legendre transform that relates F(X) = ln Tr exp X and S(D). Accord- ingly, the Hessian metric ds2 = −d2 S in the space of states D takes the form ds2 = Tr dDdX = d2 F in the conjugate space of observables X. (Note that the normalisation of D implies Tr dD = Tr d2 D = 0.) 4 Relation with maximum entropy. As an application of the metric ds2 = − d2 S, we briefly recall how it enlight- ens the projection method of Nakajima and Zwanzig, a standard technique of non-equilibrium quantum statistical mechanics [1, 6]. The main ideas are the following. The dynamics is governed by the Liouville – von Neumann equation for the time-dependent density matrix D, which is too complicated to be solv- able, while we are interested only in the evolution of the expectation values ˆAk of some set of “relevant” observables { ˆA}, macroscopic, collective or slowly varying. We wish to get rid of the other, complicate or inaccessible, observables. Among the various density matrices that provide the values ˆAk = Tr D ˆAk, we therefore select the one, DR, which maximises the entropy; it contains only infor- mation about the set { ˆA} while D also contains the extra information about the irrelevant observables. We term it the relevant state. Its entropy SR( { ˆA} ) = − Tr DR ln DR, that we call the relevant entropy associated with the variables { ˆA} , is larger than S(D), while DR and D are equivalent as regards Tr DR ˆAk = Tr D ˆAk = ˆAk . The maximisation of SR under the latter constraints yields the exponential form DR = exp ( k λk ˆAk). The approximation consists therefore in restraining the operator X defined above to lie in the subspace of relevant observables. For arbitrary values of the set ˆAk , the relevant states lie, within the space of states, on some manifold R parameterised by the variables λk. The evolution of the set ˆAk is equivalent to the evolution of DR on R, so that the problem amounts to derive the dynamics of DR from that of D. This can be achieved in specific problems under some approximations. Depending on the choice of the set { ˆA}, the relevant entropy is identified with various entropies, such as the thermodynamic entropy or the Boltzmann entropy. The mapping D → DR, introduced above through maximum entropy, is a projection (obviously, DR → DR). The introduction of the above metric yields an interesting interpretation of this projection, since with this metric, it is an orthogonal projection in the following sense: the scalar product, evaluated with the metric tensor at DR, of the vector D − DR with any dDR in the manifold R vanishes. Thus, the relevant state DR associated with D may be regarded as the point of R that is closest to D, i. e., the best approximation within the considered scheme. 5 Properties of the natural quantum information metric. The Riemannian metric tensor ds2 = − d2 S = Tr dD d ln D can be expressed in the form ds2 = Tr ∞ 0 dξ dD 1 ξ + D 2 . If we diagonalise the density matrix D at the point where we wish to evaluate ds2 , denoting the eigenvalues of D as Di and the matrix elements of dD in the basis where D is diagonal as dDij, we find ds2 = i,j ln Di − ln Dj Di − Dj dDijdDji. For i = j, the ratio reduces to 1/Di, so that this metric appears as the quan- tum extension, issued from the entropy of von Neumann, of the Fisher infor- mation metric. Indeed, the diagonal elements of D play the role of discrete probabilities pi and the Fisher metric associated with the distribution pi would be i(dpi)2 /pi. The expressions of the associated Riemann and Ricci curvature tensors have been written long ago [6]; we do not reproduce them here. The curvature of the Riemannian manifold of quantum states reflects of the non-abelian nature of the observable algebra. For the commutative algebra of observables occurring in classical statistical mechanics, the space of states would be flat. Since the epoch when the curvature of the quantum metric ds2 = − d2 S has been evaluated, only a few allusions to this metric appeared in the literature. Now and then it may appear under the name of Bogoliubov–Kubo–Mori metric, although the latter authors did not dwell upon it. However, much more work has been devoted to another metric in the space of quantum states, the Bures metric [7], which has the form ds2 = i,j 2 Di + Dj dDijdDji in a basis where D is diagonal. It was introduced much earlier, as a matrix extension of the Fisher information metric, but its specific form relies only on a mathematical analogy. The metric ds2 = − d2 S on the manifold of quantum states, which derives from von Neumann’s entropy, has a more obvious physical relevance. Its mathematical properties deserve to be studied, as this tool might be of interest in the quantum information theory and maybe elsewhere. References [1] An elementary introduction can be found in R. Balian, Incomplete descriptions and relevant entropies, Am. J. Phys. 67 (1989) 1078-1090 and Information in statistical physics, Stud. Hist. Phil. Mod. Phys. 36 (2005) 323-353 [2] For a recent review, see A. Allahverdyan, R. Balian and Th. Nieuwenhuizen, Un- derstanding quantum measurement from the solution of dynamical models, Phys. Reports, 525 (2013)1-166. ArXiv: 1107,2138 [3] E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (1957) 620-630. R. Balian, From microphysics to macrophysics: Methods and applications of statistical physics, vol. I and II (Springer Verlag, 2007) [4] R. Balian and N. Balazs, Equiprobability, information and entropy in quantum theory, Ann. Phys. NY, 179 (1987) 97-144 [5] W. Thirring, A course of mathematical physics, vol. 4, Quantum mechanics of large systems (Springler Verlag, 1983) [6] R. Balian, Y. Alhassid and H. Reinhardt, Dissipation in many-body systems: a geometric approach based on information theory, Phys. Reports 131 (1986) 1-146 [7] D. Bures, An extension of Kakutani’s theorem, Trans. Am. Math. Soc. 135 (1969) 199-212 Many further references can be found in the above articles