The Erwin Schrodinger International Pasteurgasse ESI Institute for Mathematical Physics A Wien Austria Geometries of Quantum States Denes Petz Csaba Sudar Vienna Preprint ESI March Supp orted by Federal Ministry of Science and Research Austria Available via WWWESIACAT Geometries of Quantum States Denes Petz and Csaba Sudar Department of Mathematics Faculty of Chemical Engineering Technical University Budap est H Budap est XI Szto czek u Hungary Abstract The quantum analogue of the Fisher information met ric of a probability simplex is searched and several Rie mannian metrics on the set of p ositive denite density matrices are studied Some of them app eared in the lit erature in connection with CramerRao typ e inequali ties or the generalization of the Berry phase to mixed states They are shown to b e sto chastically monotone here All sto chastically monotone Riemannian metrics are characterized by means of op erator monotone func tions and it is proven that there exist a maximal and a minimal among them A class of metrics can b e extended to pure states and the FubiniStudy metric shows up there Also Mathematical Institute of the Hungarian Academy of Sciences H Budap est PF Hungary Email PETZCHBMEHU Email SUDARCHBMEHU I Intro duction The state space of a classical system with n alternatives is the simplex of proba bility distributions on the np ointspace The probability simplex is an n dimen sional manifold with b oundary and its ane structure is fairly trivial The extreme b oundary consists of n discrete p oints In quantum mechanics the state space of an n level system is identied with the set of all n n p ositive semidenite complex matrices of trace They are called density matrices The case n is easily visualized as the unit ball in the space x y iz x y z IR x y z x iy x The b oundary consists of noninvertible matrices and it is an innite set The case n is simple but for higher n the structure of the top ological b oundary is rather complicated The extreme b oundary consists of the density matrices of rank one and for n it is much smaller than the top ological b oundary As far as dimensionality concerned the top ological b oundary is n and the extreme one is n The extreme states are usually called pure and they are describ ed in the textb o oks by nonzero vectors of a complex Hilb ert space of linear dimension n The same state is describ ed by a vector as well as where is any complex numb er dierent from This means that pure states are in onetoone corresp ondence to rays f n Cg The rays form a smo oth manifold called complex pro jective space CP On the level of convex structure the dierence b etween the classical and quantum state space is wellundersto o d The classical one is a Cho quet simplex and dierent axiomatizations of the quantum one are available in the literature the reader may b e referred to the works for example Our main concern here is the p ossible Riemannian structure in the quantum case Before turning to that sub ject we review briey the classical case that is the Riemannian structure on the space of measures From the viewp oint of information geometry the spherical representation of the probability simplex is adequate b ecause the squared length of the tangent vector of p p a curve equals the Fisher information Indeed intro duce the parameters z i i P P where i n and p Then z and the probability simplex is i i i i parametrized with a p ortion of the nsphere Let xt b e a curve on the sphere The square of the length of the tangent is X X p t log p t x h x xi i t i t i t t i i which is the Fisher information The geo desic distance b etween two probability distri butions Q and R can b e computed along a great circle and it is a simple transform of the Hellinger distance The lecture notes contains further details as well as statistical applications of this geometric approach To the b est of our knowledge Riemannian metric on quantum states was rst considered by Helstrom in connec tion with state estimation theory Since Helstroms work several other metrics app eared in the literature see for example and Uhlmann approached Helstroms metric in a dierent way The present pap er is organized as follows In Section II we survey the work of Chentsov b oth in the probabilistic and in the quantum case We explain how he arrived at the study of invariant metrics on the space of probability measures motivated by decision theory and how far he could go towards the quantum gen eralization after his unicity result ab out the Fisher information in the probabilistic context Section IV reviewes dierent approaches to Riemannian metric on the quan tum state space The relation of Uhlmanns and Helstroms work to Chentsovs idea is enlightened and a concise description of the complex pro jective space is given The main results are contained in Sections IV and V We construct monotome metrics by means of op erator monotone functions and prove that all monotone metrics are obtained in this way Our result completes the program initiated by Chentsov It turns out that the symmetric logarithmic derivative metric of Helstron which is the same as the metric studied by Uhlmann is monotone Furthermore this metric is minimal among all monotone metrics The sub ject of Section V is the extension of monotone metrics to pure states We prove that if the extension exists then it coincides with the standard metric of pure states up to a constant factor I I The viewp oint of Chentsov Chentsov was led by decision theory when he considered a category whose ob jects are probability spaces and whose morphisms are Markov kernels Although he worked in with arbitrary probability spaces his idea can b e demonstrated very well on nite ones In this case a morphism from the probability nsimplex S to an m n simplex S is an n m sto chastic matrix If is such a matrix and P S then P m n S is considered more random than P Generally sp eaking the parametrized family m Q is more random than the parametrized family P with the same parameter i i set if there exists a sto chastic matrix such that P Q for every value of the i i parameter i Two parametric families P and Q are equivalent in the theory of i i statistical inferences if there are two sto chastic matrices and such that P Q and Q P i i i i for every i Chentsov said a numerical function f dened on pairs of measures to b e invariant if P P Q Q implies f P P f Q Q and monotone if f P P f P P for every sto chastic matrix A monotone function f is obviously invariant Statis tics and information theory know a lot of monotone functions relative entropy and its generalizations are so If a Riemannian metric is given on all probability sim plexes then this family of metrics is called invariant resp ectively monotone if the corresp onding geo desic distance is an invariant resp ectively monotone function Chentsovs greate achievement was that up to a constant factor the Fisher informa tion yields the only monotone family of Riemannian metrics on the class of nite probability simplexes see also A decade later Chentsov turned to the quan tum case where the probability simplex is replaced by the set of density matrices A linear mapping b etween two matrix spaces sends a density matrix into a density if the mapping preserves trace and p ositivity ie p ositive semidenitness By now it is wellundersto o d that completely p ositivity is a natural and imp ortant require ment in the noncommutative case Therefore we call a trace preserving completely p ositive mapping sto chastic One of the equivalent forms of the completely p ositivity of a map T is the following n n X X a T b b a i i i i i j for all p ossible choice of a b and n A completely p ositive mapping T satises the i i Schwarz inequality T a a T a T a Chentsov recognized that sto chastic mappings are the appropriate morphisms in the category of quantum state spaces The monograph contains more informa tion ab out sto chastic mappings see also The ab ove denitions of invariance and monotonicity make sense when sto chastic matrices are replaced by sto chastic mappings Chentsov with Morozova aimed to nd the invariant or monotone Riemannian metrics in quantum setting as well They obtained the following result Assume that a family of Riemannian metrics is given on all spaces of den sity matrices which is invariant then there exist a function cx y and a constant C such that the squared length of a tangent vector A A at a diagonal p oint ij D Diag p p p is of the form n n X X cp p jA j A C p j k j k k k k j k k Furthermore the function cx y is symmetric and cx y cx y This result of Morozova and Chentsov was not complete Although they had prop osals for the function cx y they did not prove monotonicity or invariance of any of the corresp onding metrics A complete result will b e given here but now a few comments on are in order Both the function cx y and the constant are indep endent from the matrix size n Restricting ourselves to diagonal matrices which is in some sense a step back to the probability simplex we can see that there is no ambiguity of the metric Lo osely sp eaking the unicity result in the simplex case survives along the diagonal and the odiagonal provides new p ossibilities for the denition of a sto chastically invariant metric I I I Riemannian metrics on quantum states The demand for Riemannian structure on the whole quantum state space or on a parametrized family of density op erators