Means, Covariance, Fisher Information:

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1 Vienna Erwin Schrödinger Institute Means, Covariance, Fisher Information: the quantum theory and uncertainty relations Paolo Gibilisco Department of Economics and Finance University of Rome Tor Vergata November 21, 2012 P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

2 History and credits What follows originated from some works by S. Luo, Quantum Fisher information and uncertainty relations, LMP, S. Luo, Wigner-Yanase information and uncertainty relations, PRL, Other people involved: Z. Zhang, Q. Zhang, Kosaki, Yanagi, Furuichi, Kuriyama, Gibilisco, Imparato, Isola, Hansen, Andai, Petz, Hiai, Szabo, Audenaart, Cai. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

3 Heisenberg uncertainty principle A, B M n,sa (C), ρ density matrix [A, B] := AB BA E ρ (A) := Tr(ρA) Var ρ (A) := E ρ (A 2 ) E ρ (A) 2 Heisenberg uncertainty principle (1927) reads as Var ρ (A) Var ρ (B) 1 4 Tr(ρ[A, B]) 2. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

4 Classical doubts In classical probability, let (X, Y ) be a r.v. on (Ω, G, p). The covariance matrix of (X, Y ) is symmetric and semidefinite positive so its determinant is non-negative and therefore Var p (X ) Var p (Y ) Cov p (X, Y ) 2. So to have a general bound for Var p (X ) Var p (Y ) does not seems such a quantum phenomenon. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

5 Quantum covariance Quantum covariance (Schrödinger & Robertson) Cov ρ (A, B) := 1 Tr(ρ(AB + BA)) Tr(ρA) Tr(ρB) = 2 [( ) ] Lρ + R ρ = Tr (A 0 )B 0. 2 where A 0 := A Tr(ρA) I and L ρ (A) := ρa R ρ := Aρ P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

6 Schrödinger Robertson UP Schrödinger and Robertson ( ) improved UP Namely Var ρ (A) Var ρ (B) Cov ρ (A, B) Tr(ρ[A, B]) 2. Var ρ (A) Cov ρ (A, B) i 2Tr(ρ[A, A]) det det Cov ρ (B, A) Var ρ (B) i 2Tr(ρ[B, A]) i 2 Tr(ρ[A, B]) i 2Tr(ρ[B, B]) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

7 Robertson general UP (1934) Let A 1..., A N M n,sa (C). for h, j = 1,..., N det {Cov ρ (A h, A j )} det { i } 2 Tr(ρ[A h, A j ]), det {Cov ρ (A h, A j )} is the generalized variance of the random vector (A 1,..., A n ). P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

8 Robertson general UP (2nd version) The matrix { i 2 Tr(ρ[A h, A j ])} is anti-symmetric. Therefore, the Robertson UP reads as { 0, N odd det {Cov ρ (A h, A j )} det{ i 2 Tr(ρ[A h, A j ])}, N even, Remark If N = 2m + 1, UP says (classically!) that the generalized variance is non-negative. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

9 First problem: searching an UP for N odd Robertson UP is based on the commutator [A h, A j ]. If N = 1 this structure becomes meaningless! Intuitively, an UP for N odd should be based on a structure which involves [ρ, A]. This commutator appears in quantum dynamics. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

10 (Elementary) Quantum dynamics Let ρ(t) be a curve in D 1 n and let H M n,sa ; ρ(t) satisfies Schrödinger equation w.r.t. H if ρ(t) = d ρ(t) = i[ρ(t), H]. dt Equivalently, ρ H (t), the time evolution of ρ = ρ H (0) determined by H, evolves according to the formula ρ H (t) := e ith ρe ith. Therefore ρ H (0) = i[ρ, H] P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

11 Second problem: different quantum covariances? Cov ρ (A, B) := 1 Tr(ρ(AB + BA)) Tr(ρA) Tr(ρB) = 2 [( ) ] Lρ + R ρ = Tr (A 0 )B 0. 2 Is the above definition natural? Certainly it coincides with the classical covariance in a commutative setting. It uses the arithmetic mean of the left and right multiplication operator m arith (L ρ, R ρ ) := L ρ + R ρ 2 This suggest that we may consider other noncommutatitive means. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

12 Harmonic covariance? If we consider the harmonic covariance Cov har ρ (A, B) := Tr (( 2(L 1 ρ + Rρ 1 ) 1) ) (A 0 )B 0, also this coincides with the classical definition where there is no difference between L ρ and R ρ! Is there a quantum criterion to prefer a certain covariance (mean)? P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

13 Means for numbers... Let R + = (0, + ). A mean for pair of positive numbers is a function m(, ) : R + R + R + such that i) m(x, x) = x ; ii) m(x, y) = m(y, x) ; iii) x x y y = m(x, y) m(x, y ) ; iv) for t > 0 one has m(tx, ty) = t m(x, y); v) m(, ) is continuous. M nu := {m(, ) : R + R + R + m is a mean } P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

14 ... from certain functions F nu is the class of functions f ( ) : R + R + such that iii) f (1) = 1; iv) tf (t 1 ) = f (t); iii) x x = f (x) f (x ); iv) f is continuous. Proposition There is bijection betwen M nu and F nu given by the formula m f (x, y) := yf (xy 1 ) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

15 Operator means Kubo-Ando 1980 Let D n := {A M n A > 0}. A mean is a function m : D n D n D n such that (i) m(a, A) = A, (ii) m(a, B) = m(b, A), (iii) A < A, B < B = m(a, B) < m(a, B ), (iv) m is continuous, (v) Cm(A, B)C m(cac, CBC ), for every C M n. Property (vi) is the transformer inequality. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

16 Operator monotone functions M n = complex matrices Definition f : (0, + ) R is operator monotone iff A, B M n and n = 1, 2,... 0 A B = 0 f (A) f (B). Definition ϕ is a Pick function if it is analytic in the upper half plane and map the latter into itself. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

17 Löwner Theorem Löwner 1932 Theorem f is operator monotone iff it is the restriction of a Pick function. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

18 F op Usually one consider o.m. functions that are: i) normalized i. e. f (1) = 1; ii) symmetric i.e. tf (t 1 ) = f (t). F op := family of normalized symmetric o. m. functions. Examples 1 + x 2, x, 2x 1 + x. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

19 Kubo Ando theorem M op := family of matrix means. Kubo and Ando (1980) proved the following, fundamental result. Theorem There exists a bijection between M op and F op given by the formula m f (A, B) := A 1 2 f (A 1 2 BA 1 2 )A 1 2. [A, B] = 0 = m f (A, B) := Af (BA 1 ). P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

20 Kubo Ando inequality Examples of operator means Fundamental inequality A + B 2 A 1 2 (A 1 2 BA ) 2 A 2 2(A 1 + B 1 ) 1 2(A 1 + B 1 ) 1 m f (A, B) A + B 2 f F op P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

21 g-covariance To each operator monotone g F op one associate the means m g (, ). Define the g-covariance as Cov g ρ(a, B) := Tr(m g (L ρ, R ρ )(A 0 )B 0 ) If g(x) = 1 + x 2 then m g = arithmetic mean and Cov g ρ(a, B) is the standard covariance introduced by Schrödinger and Roberston. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

22 Fisher information for densities X : Ω R real r. v. with diff. density ρ The score is J ρ := ρ ρ The Fisher information is I X := I ρ = Var ρ (J ρ ) = R (ρ ) 2 ρ P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

23 FI as Riemannian metric (Rao 1945) M statistical model (set of densities) M can be considered as a manifold where the ρ s play the role of tangent vectors. I ρ is associated to a Riemannian metrics as in the formula g ρ,f (ρ, ρ ρ ρ ) = = I ρ ρ R P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

24 FI on the simplex P 1 n := {ρ R n i ρ i = 1, ρ i > 0}. T P 1 n = {u R n i u i = 0}. g ρ,f (u, v) := i u i v i ρ i This will be the Fisher-Rao metric ( ) Geodesic distance (Bhattacharya): d F (ρ, σ) = 2 arccos i ρ 1 2 i σ 1 2 i P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

25 The link with entropy i) Hessian of Kullback-Leibler relative entropy S(ρ, σ) := i ρ i (log ρ i log σ i ); 2 t s S(ρ + tu, ρ + sv) t=s=0 = n i=1 u i v i ρ i + sv i t=s=0 = n i=1 u i v i ρ i = g ρ,f (u, v). P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

26 The link with the sphere FI as a spherical geometry (Rao, Dawid) ii) pull-back of the map ϕ(ρ) = ϕ(ρ 1,..., ρ n ) = 2( ρ 1,..., ρ n ) g ϕ ρ (u, v) = g ϕ(ρ) (D ρ ϕ(u), D ρ ϕ(v)) = M ρ 1/2(u), M ρ 1/2(v) n u i v i = = g ρ,f (u, v). ρ i i=1 (1) This explains the geodesic ( distance ) (Bhattacharya): d F (ρ, σ) = 2 arccos i ρ 1 2 i σ 1 2 i P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

27 From the properties of Fisher information... Look at Fisher information in different ways: i) Hessian of Kullback-Leibler relative entropy K(ρ, σ) := i ρ i (log ρ i log σ i ); ii) pull-back of the map ρ ρ; iii) get the scores using the (Symmetric) Logarithmic Derivative ρ(θ) = 1 ( ) θ 2 θ log(ρ(θ)) ρ(θ) + ρ(θ) θ log(ρ(θ)) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

28 ... to a zoo of examples of QFI Examples of quantum Fisher informations Hessian of Umegaki relative entropy Tr(ρ(log ρ log σ)) BKM metric Pull-back of the map ρ ρ WY metric Symmetric logarithmic derivative Bures-Uhlmann metric (SLD) Can we have a unified quantum approach? P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

29 Chentsov Theorem Yes! Using the classical Chentsov theorem. On the simplex P 1 n the Fisher information is the only Riemannian metric contracting under an arbitrary coarse graining T, namely for any tangent vector X at the point ρ we have g m T (ρ) (TX, TX ) g n ρ (X, X ) Remark Coarse graining = stochastic map = linear, positive, trace preserving. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

30 Monotone metrics (or QFI) according Chensov-Morozova D 1 n := {ρ M n Tr(ρ) = 1 ρ > 0} = faithful states Definition A quantum Fisher information is a Riemaniann metric on D 1 n contracting under an arbitrary coarse graining T, namely g m T (ρ) (TA, TA) g n ρ (A, A). (quantum) coarse graining = linear, (completely) positive, trace preserving map. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

31 Petz theorem Petz theorem L ρ (A) := ρa R ρ (A) := Aρ There is bijection among quantum Fisher information and operator monotone functions (and/or operator means) given by the formula A, B ρ,f := Tr(A m f (L ρ, R ρ ) 1 (B)). P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

32 Geodesic distances Closed form of geodesics distances are known only in two cases D Bures (ρ, σ) = 2 arccos Tr(ρ σρ 2 ) 2 (Gibilisco-Isola) D WY = 2 arccos Tr(ρ 1 2 σ 1 2 ) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

33 Summary Kubo-Ando-Petz ( Löwner) f m f (A, B) := A 1 2 f (A 1 2 BA 1 2 )A 1 2. A, B ρ,f := Tr(A m f (L ρ, R ρ ) 1 (B)). P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

34 Decomposition of tangent space I T ρ D 1 n = {A M n,sa A = A, Tr(A) = 0}. T ρ D 1 n = (T ρ D 1 n) c (T ρ D 1 n) o where (T ρ D 1 n) c := {A T ρ D 1 n [ρ, A] = 0 } (T ρ D 1 n) o := orth. compl. of (T ρ D 1 n) c resp. to H-S P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

35 Decomposition of tangent space II For each QFI and for each A (T ρ D 1 n) c one has A, A ρ,f = Tr(ρ 1 A 2 ). To evaluate a QFI one has just to know what happens for (T ρ D 1 n) o whose typical element has the form i[ρ, A] A s.a. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

36 Regular and non-regular QFI F op := {f op. mon. f (1) = 1, tf (t 1 ) = f (t)} F r op := {f F op f (0) := lim t 0 f (t) > 0} F n op := {f F op f (0) = 0} F op = F r op F n op Remark The word non-regular should lead to a negative attitude: the BKM metric is non-regular but widely used in quantum statistical mechanics. Why is this decomposition relevant? P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

37 Riemannian metrics on the sphere B 3 := {(x, y, z) R 3 x 2 + y 2 + z 2 1} S 2 := B 3 0 := (0, 0, 0) M := B 3 /(S 2 {0}) M is a fiber bundle over S 2 with projection π(x, y, z) := π : M S 2 1 (x, y, z) x 2 + y 2 + z2 P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

38 Riem. metrics on the sphere II M D n ρ S 2 radially iff π(d n ) = ρ n and lim D n = ρ Differential T π : T M TS 2 Horizontal-Vertical decomposition T D M = Ker(T D π) H D H D = horizontal tangent vectors at the point D P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

39 Riem. metrics on the sphere III Restriction T D π = H D T ρ S 2 is a linear isomorphism between H D and T ρ S 2 (where ρ = T (D)). We may lift tangent vectors u, v T ρ S 2 to u D, v D T D M. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

40 Radial extensions Suppose we have: i) a Riemannian metric g(, ) on M; ii) a Riemannian metric h(, ) on S 2. h is the radial extension of g if D n ρ radially g(u Dn, v Dn ) k(u, v) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

41 The Bloch sphere 2 2 matrices, I identity, σ 1, σ 2, σ 3 Pauli matrices σ 1 = ( ) 0 1, σ = Stokes parametrization of qubits ( ) ( ) 0 i 1 0, σ i 0 3 =, 0 1 ρ = 1 2 (I + (x, y, z), (σ 1, σ 2, σ 3 ) ) x 2 + y 2 + z 2 1 P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

42 Petz-Sudar theorem Pure states x 2 + y 2 + z 2 = 1 (the sphere S 2 ) Faithful mixed states x 2 + y 2 + z 2 < 1 (manifold M plus the origin) Theorem If, FS denotes the standard Riemannian metric on the sphere S 2 (pure states), then a QFI, ρ,f has a radial extension iff it is regular. The extension is given by 1 2f (0), FS P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

43 General P-S theorem Remark True in general using the Fubini Study metric on the projective space CP n. More delicate because for n > 2: extreme boundary (pure states) topological boundary (detρ = 0). P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

44 The function f f (x) := 1 2 [ (x + 1) (x 1) 2 f (0) ] f (x) Theorem f F r op (f is a regular n. s. o. m. function) f F n op ( f is a non-regular n. s. o. m. function) Moreover f f is bijection. Gibilisco-Imparato-Isola-Hansen P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

45 Regular and non-regular means f f Examples m f x + y 2 m f 2 1 x + 1 y ( ) 2 x + y xy 2 P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

46 β example x > 0, β (0, 1 2 ) (x 1) 2 f β (x) := β(1 β) (x β 1)(x 1 β 1) f β (x) = x β + x 1 β Fix x > 0. Then f β (x) is decreasing as a function of β. This remark allows a great simplification of a preceding result by Kosaki. 2 P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

47 Fundamental formula Theorem If f is regular then f (0) 2 i[ρ, A], i[ρ, B] ρ,f = Cov ρ (A, B) Cov f ρ (A, B). An immediate consequence is Proposition Var ρ (A) f (0) 2 i[ρ, A] 2 ρ,f The above is the case N = 1 of the following... P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

48 The dynamical UP Theorem Let A 1..., A N M n,sa (C). for h, j = 1,..., N, for all f F op. det {Cov ρ (A h, A j )} det { } f (0) 2 i[ρ, A h], i[ρ, A j ] ρ,f Nontrivial bound also if N is odd! P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

49 Interlude: the case N = 1 and WYD-information Wigner-Yanase-Dyson information A s.a. matrix (observable in QM) ρ density matrix (state in QM) I β ρ (A) := 1 2 Tr([ρβ, A][ρ 1 β, A]) β (0, 1 2 ] P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

50 Interlude: WYD applications strong subadditivity of entropy (Lieb-Ruskai,1973) homogeneity of the state space of factors of type III 1 (Connes-Stormer,1978); measures for quantum entanglement (Chen,2005; Klyachko-Oztop-Shumovsky,2006); uncertainty relations ; quantum hypothesis testing (Calsamiglia et al., 2008) (Explanation: WYD is a QFI) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

51 Interlude: Lieb convexity Theorem (case β = 1/2: Wigner & Yanase general case: Lieb 1973) I β ρ (A) is convex as function of ρ. If two ensembles are united, the information content of the resulting ensemble should be smaller than the average information content of the component ensembles (Wigner-Yanase original paper) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

52 Interlude: convexity or concavity? Remark Since Iρ β (A) = 1 2 Tr(ρA2 ) Tr(ρ β Aρ 1 β A) convexity of Iρ β (A) is equivalent to concavity of ρ Tr(ρ β Aρ 1 β A) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

53 Lieb convexity and SSA Applications: Strong subadditivity of von Neumann quantum entropy (Lieb & Ruskai 1975) S(ρ) := Tr(ρ log ρ) SSA implies strong results about to thermodynamic limit of entropy per unit volume (Robinson, Ruelle, & Lanford ) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

54 Interlude: WYD as quantum Fisher information Definition The metric adjusted skew information is Proposition In the case x > 0, β (0, 1 2 ] I f ρ (A) := f (0) 2 i[ρ, A] 2 ρ,f we have (x 1) 2 f β (x) := β(1 β) (x β 1)(x 1 β 1) I f β ρ (A) := f β(0) 2 i[ρ, A] 2 ρ,f β = 1 2 Tr([ρβ, A][ρ 1 β, A]) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

55 Interlude: concavity-convexity trick T ( ),S( ) real functions on states S( ) concave, T ( ) convex S( ) = T ( ) on pure states S(ρ) T (ρ) ρ D 1 n P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

56 Interlude: N = 1, the DUP from Lieb convexity Theorem (F. Hansen 2006) Var ρ (A) I f ρ (A) Proof Var ρ (A) is concave Iρ f (A) is convex ρ pure implies Var ρ (A) = Iρ f (A) - Warning: this elegant proof do not work in the general case of the DUP! P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

57 Why dynamical? Let ρ > 0 be a state and H, K M n,sa. Suppose that ρ = ρ H (0) = ρ K (0). Then, for any f F op, one has (taking the square root of both sides of the DUP) Area Cov ρ (H, K) f (0) 2 Areaf ρ( ρ H (0), ρ K (0)). The bound on the right side of the inequality can be seen as a measure of the difference between the dynamics generated by H and K. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

58 How the bound in the DUP depends on f Theorem Define for f Fop r { } f (0) S(f ) := det 2 i[ρ, A h], i[ρ, A j ] ρ,f f (x) := 1 2 Then, for any f, g F r op [ (x + 1) (x 1) 2 f (0) ]. f (x) f g = S(f ) S(g). P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

59 The optimal bound Let f SLD (x) := 1 + x 2. Since for any f F r op then 2x 1 + x = f SLD f S(f SLD ) S(f ) namely the optimal bound is given by Bures-Uhlmann metric. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

60 Relation with standard UP - I Let f Fop. r The inequality { } f (0) det 2 i[ρ, A h], i[ρ, A j ] ρ,f is (in general) false for any N = 2m. Remark The proof is a consequence of Hadamard inequality: for any H M N,sa. det(h) { det i } 2 Tr(ρ[A h, A j ]) N j=1 h jj P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

61 Relation with standard UP - II Let f Fop. r The inequality { } f (0) det 2 i[ρ, A h], i[ρ, A j ] ρ,f is (in general) false for any N = 2m. { det i } 2 Tr(ρ[A h, A j ]) P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

62 The dynamical UP (g-version) Theorem Let A 1..., A N M n,sa (C). det { Cov g ρ(a h, A j ) } det {g(0)f (0) i[ρ, A h ], i[ρ, A j ] ρ,f } for h, j = 1,..., N, for all g, f F op. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

63 Conclusion - Dynamical UP case For the dynamical UP quantum g-covariances coming from regular g (constant g(0) 0) do have uncertainty relations. Quantum g-covariances coming from nonregular g (constant g(0) = 0) do NOT have (non-trivial) uncertainty relations. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

64 g-version of Robertson UP Theorem Let A 1..., A N M n,sa (C). det { Cov g ρ(a h, A j ) } det { i g(0) Tr(ρ[A h, A j ])}, for h, j = 1,..., N, for all g F op. Remark: g(0) is the best constant in the above inequalities. P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

65 The end Quantum g-covariances coming from regular g (constant g(0) 0) do have uncertainty relations. Quantum g-covariances coming from nonregular g (constant g(0) = 0) do NOT have uncertainty relations. The usual quantum covariance has the most demanding one (since g(0) = 1 2 only for the arithmetic mean). After all Schrödinger and Robertson were right... Recent reference: P. Gibilisco, T. Isola. How to distinguish quantum covariances using uncertainty relations. Journal of Mathematical Analysis and Applications, 384: p , P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1

66 Post scriptum: The DUP on von Neumann algebras While the first papers on the DUP where forced to use eigenvalues (see the complicated calculations in the paper by Kosaki) now one has to generalize something like the mean of the operators L ρ and R ρ. These are commuting operator therefore m f (L ρ, R ρ ) = L ρ f (Rρ L 1 ρ ) = L ρ f ( ρ ) So we are dealing with the modular operator and this construction makes sense in the general setting of von Neumann algebras. Gibilisco-Isola Petz-Szabo P. Gibilisco (Rome Tor Vergata ) Means, Covariance, Fisher Information November / 1