Vienna – Erwin Schr¨ odinger Institute Means, Covariance, Fisher Information: the quantum theory and uncertainty relations Paolo Gibilisco Department of Economics and Finance University of Rome “Tor Vergata” [email protected]November 21, 2012 P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 1/1
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Vienna – Erwin Schrodinger Institute
Means, Covariance, Fisher Information:the quantum theory and uncertainty relations
Paolo Gibilisco
Department of Economics and FinanceUniversity of Rome “Tor Vergata”[email protected]
November 21, 2012
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 1 / 1
History and credits
What follows originated from some works by
S. Luo, Quantum Fisher information and uncertainty relations, LMP,2000.
S. Luo, Wigner-Yanase information and uncertainty relations, PRL,2003.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 2 / 1
Heisenberg uncertainty principle
A,B ∈Mn,sa(C), ρ density matrix
[A,B] := AB − BA Eρ(A) := Tr(ρA)
Varρ(A) := Eρ(A2)− Eρ(A)2
Heisenberg uncertainty principle (1927) reads as
Varρ(A) ·Varρ(B) ≥ 1
4|Tr(ρ[A,B])|2.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 3 / 1
“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 soits determinant is non-negative and therefore
Varp(X ) ·Varp(Y ) ≥ Covp(X ,Y )2.
So to have a general bound for Varp(X ) ·Varp(Y ) does not seems such a“quantum” phenomenon.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 4 / 1
Quantum covariance
Quantum covariance (Schrodinger & Robertson)
Covρ(A,B) :=1
2Tr(ρ(AB + BA))− Tr(ρA) · Tr(ρB) =
= Tr
[(Lρ + Rρ
2
)(A0)B0
].
where A0 := A− Tr(ρA) · I and
Lρ(A) := ρA Rρ := Aρ
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 5 / 1
Schrodinger – Robertson UP
Schrodinger and Robertson (1929-1930) improved UP
Varρ(A) ·Varρ(B) ≥ Covρ(A,B)2 +1
4|Tr(ρ[A,B])|2.
Namely
det
Varρ(A) Covρ(A,B)
Covρ(B,A) Varρ(B)
≥ det
− i2Tr(ρ[A,A]) − i
2Tr(ρ[A,B])
− i2Tr(ρ[B,A]) − i
2Tr(ρ[B,B])
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 6 / 1
Robertson general UP (1934)
Let A1 . . . ,AN ∈Mn,sa(C).
det Covρ(Ah,Aj) ≥ det
− i
2Tr(ρ[Ah,Aj ])
,
for h, j = 1, . . . ,N
det Covρ(Ah,Aj) is the generalized variance of the random vector(A1, ...,An).
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 7 / 1
Robertson general UP (2nd version)
The matrix − i2Tr(ρ[Ah,Aj ]) is anti-symmetric. Therefore, the
Robertson UP reads as
det Covρ(Ah,Aj) ≥
0, N odd
det− i2Tr(ρ[Ah,Aj ]), N even,
Remark If N = 2m + 1, UP says (classically !) that the generalizedvariance is non-negative.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 8 / 1
First problem: searching an UP for N odd
Robertson UP is based on the commutator [Ah,Aj ]. If N = 1 thisstructure becomes meaningless !
Intuitively, an UP for N odd should be based on a structure whichinvolves [ρ,A] .
This commutator appears in quantum dynamics.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 9 / 1
(Elementary) Quantum dynamics
Let ρ(t) be a curve in D1n and let H ∈ Mn,sa; ρ(t) satisfies Schrodinger
equation w.r.t. H if
ρ(t) =d
dtρ(t) = i [ρ(t),H].
Equivalently, ρH(t), the time evolution of ρ = ρH(0) determined by H,evolves according to the formula
ρH(t) := e−itHρe itH .
ThereforeρH(0) = i [ρ,H]
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 10 / 1
Second problem: different quantum covariances?
Covρ(A,B) :=1
2Tr(ρ(AB + BA))− Tr(ρA) · Tr(ρB) =
= Tr
[(Lρ + Rρ
2
)(A0)B0
].
Is the above definition “natural”?Certainly it coincides with the classical covariance in a commutativesetting.It uses the ”arithmetic mean” of the left and right multiplication operator
marith(Lρ,Rρ) :=Lρ + Rρ
2
This suggest that we may consider other noncommutatitive ”‘means”.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 11 / 1
Harmonic covariance?
If we consider the “harmonic” covariance
Covharρ (A,B) := Tr((
2(L−1ρ + R−1ρ )−1)
(A0)B0
),
also this coincides with the classical definition where there is no differencebetween Lρ and Rρ!
Is there a quantum criterion to prefer a certain covariance (mean)?
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 12 / 1
Means for numbers ...
Let R+ = (0,+∞).A mean for pair of positive numbers is a function m(·, ·) : R+ × R+ → R+
such thati) 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.
Mnu := m(·, ·) : R+ × R+ → R+|m is a mean
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 13 / 1
... from certain functions
Fnu is the class of functions f (·) : R+ → R+ such thatiii) f (1) = 1;iv) tf (t−1) = f (t);iii) x ≤ x ′ =⇒ f (x) ≤ f (x ′);iv) f is continuous.
PropositionThere is bijection betwen Mnu and Fnu given by the formula
mf (x , y) := yf (xy−1)
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 14 / 1
Operator means
Kubo-Ando 1980Let Dn := A ∈ Mn|A > 0.A mean is a function m : Dn ×Dn → Dn 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 ∈ Mn.Property (vi) is the transformer inequality.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 15 / 1
Operator monotone functions
Mn = complex matricesDefinitionf : (0,+∞)→ R is operator monotone iff ∀A,B ∈ Mn and ∀n = 1, 2, ...
0 ≤ A ≤ B =⇒ 0 ≤ f (A) ≤ f (B).
Definitionϕ is a Pick function if it is analytic in the upper half plane and map thelatter into itself.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 16 / 1
Lowner Theorem
Lowner 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 2012 17 / 1
Fop
Usually one consider o.m. functions that are:i) normalized i. e. f (1) = 1;ii) symmetric i.e. tf (t−1) = f (t).
Fop:= family of normalized symmetric o. m. functions.
Examples1 + x
2,√
x ,2x
1 + x.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 18 / 1
Kubo–Ando theorem
Mop:= family of matrix means.
Kubo and Ando (1980) proved the following, fundamental result.
Theorem
There exists a bijection between Mop and Fop given by the formula
mf (A,B) := A12 f (A−
12 BA−
12 )A
12 .
[A,B] = 0 =⇒ mf (A,B) := Af (BA−1).
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 19 / 1
Kubo–Ando inequality
Examples of operator means
A + B
2
A12 (A−
12 BA−
12 )
12 A
12
2(A−1 + B−1)−1
Fundamental inequality
2(A−1 + B−1)−1 ≤ mf (A,B) ≤ A + B
2∀f ∈ Fop
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 20 / 1
g -Covariance
To each operator monotone g ∈ Fop one associate the means mg (·, ·).
Define the g -covariance as
Covgρ(A,B) := Tr(mg (Lρ,Rρ)(A0)B0)
If
g(x) =1 + x
2
thenmg = arithmetic meanand Covgρ(A,B) is the standard covariance introduced by Schrodinger andRoberston.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 21 / 1
Fisher information for densities
X : Ω→ R real r. v. with diff. density ρThe score is
Jρ :=ρ′
ρ
The Fisher information is
IX := Iρ = Varρ(Jρ) =
∫R
(ρ′)2
ρ
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 22 / 1
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 tangentvectors.Iρ is associated to a Riemannian metrics as in the formula
gρ,F (ρ′, ρ′) =
∫R
ρ′ · ρ′
ρ= Iρ
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 23 / 1
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 30 / 1
Petz theorem
Lρ(A) := ρA Rρ(A) := Aρ
Petz theorem
There is bijection among quantum Fisher information and operatormonotone functions (and/or operator means) given by the formula
〈A,B〉ρ,f := Tr(A ·mf (Lρ,Rρ)−1(B)).
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 31 / 1
Geodesic distances
Closed form of geodesics distances are known only in two cases
DBures(ρ, σ) = 2 arccosTr(ρ12σρ
12 )
12
DWY = 2 arccosTr(ρ12σ
12 )
(Gibilisco-Isola)
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 32 / 1
Summary
Kubo-Ando-Petz ( Lowner)
f
l
mf (A,B) := A12 f (A−
12 BA−
12 )A
12 .
l
〈A,B〉ρ,f := Tr(A ·mf (Lρ,Rρ)−1(B)).
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 33 / 1
Decomposition of tangent space I
TρD1n = A ∈ Mn,sa|A = A∗, Tr(A) = 0.
TρD1n = (TρD1
n)c ⊕ (TρD1n)o
where(TρD1
n)c := A ∈ TρD1n| [ρ,A] = 0
(TρD1n)o := orth. compl. of (TρD1
n)c resp. to H-S
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 34 / 1
Decomposition of tangent space II
For each QFI and for each A ∈ (TρD1n)c one has
〈A,A〉ρ,f = Tr(ρ−1A2).
To evaluate a QFI one has just to know what happens for (TρD1n)o whose
typical element has the form
i [ρ,A] A s.a.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 35 / 1
Regular and non-regular QFI
Fop := f op. mon.|f (1) = 1, tf (t−1) = f (t)
F rop := f ∈ Fop|f (0) := lim
t→0f (t) > 0
F nop := f ∈ Fop|f (0) = 0
Fop = F rop ∪ Fn
op
Remark The word non-regular should lead to a negative attitude: theBKM metric is non-regular but widely used in quantum statisticalmechanics.Why is this decomposition relevant?
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 36 / 1
Riemannian metrics on the sphere
B3 := (x , y , z) ∈ R3|x2 + y2 + z2 ≤ 1
S2 := B3 0 := (0, 0, 0)
M := B3/(S2 ∪ 0)
M is a fiber bundle over S2 with projection
π :M→ S2
π(x , y , z) :=1√
x2 + y2 + z2(x , y , z)
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 37 / 1
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 38 / 1
Riem. metrics on the sphere III
Restriction
TDπ = HD → TρS2
is a linear isomorphism between HD and TρS2 (where ρ = T (D)).
We may “lift” tangent vectors u, v ∈ TρS2 to uD , vD ∈ TDM.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 39 / 1
Radial extensions
Suppose we have:i) a Riemannian metric g(·, ·) on M;ii) a Riemannian metric h(·, ·) on S2.
h is the radial extension of g if
Dn → ρ radially
⇓
g(uDn , vDn)→ k(u, v)
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 40 / 1
The Bloch sphere
2× 2 matrices, I identity, σ1, σ2, σ3 Pauli matrices
σ1 =
(0 11 0
), σ2 =
(0 −ii 0
), σ3 =
(1 00 −1
),
Stokes parametrization of qubits
ρ =1
2(I + 〈(x , y , z), (σ1, σ2, σ3)〉)
x2 + y2 + z2 ≤ 1
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 41 / 1
Petz-Sudar theorem
Pure states → x2 + y2 + z2 = 1 (the sphere S2)Faithful mixed states → x2 + y2 + z2 < 1(manifold M plus the origin)TheoremIf 〈·, ·〉FS denotes the standard Riemannian metric on the sphere S2 (purestates), then a QFI 〈·, ·〉ρ,f has a radial extension iff it is regular. Theextension is given by
1
2f (0)〈·, ·〉FS
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 42 / 1
General P-S theorem
Remark
True in general using the Fubini–Study metric on the projective space CPn.
More delicate because for n > 2:extreme boundary (pure states) 6= topological boundary (detρ = 0).
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 43 / 1
The function f
f (x) :=1
2
[(x + 1)− (x − 1)2
f (0)
f (x)
]Theoremf ∈ F r
op (f is a regular n. s. o. m. function)
⇓
f ∈ F nop (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 2012 44 / 1
Regular and non-regular means
f → f
mf → mf
Examples
x + y
2→ 2
1x + 1
y(√x +√
y
2
)2
→ √xy
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 45 / 1
β example
x > 0, β ∈ (0, 12)
fβ(x) := β(1− β)(x − 1)2
(xβ − 1)(x1−β − 1)
fβ(x) =xβ + x1−β
2
Fix x > 0. Then fβ(x) is decreasing as a function of β.This remark allows a great simplification of a preceding result by Kosaki.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 46 / 1
Fundamental formula
Theorem
If f is regular then
f (0)
2〈i [ρ,A], i [ρ,B]〉ρ,f = Covρ(A,B)− Covfρ(A,B).
An immediate consequence is
Proposition
Varρ(A) ≥ f (0)
2||i [ρ,A]||2ρ,f
The above is the case N = 1 of the following ...
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 47 / 1
The dynamical UP
Theorem
Let A1 . . . ,AN ∈Mn,sa(C).
det Covρ(Ah,Aj) ≥ det
f (0)
2〈i [ρ,Ah], i [ρ,Aj ]〉ρ,f
for h, j = 1, . . . ,N,for all f ∈ Fop.
Nontrivial bound also if N is odd!
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 48 / 1
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 βρ (A) := −1
2Tr([ρβ,A][ρ1−β,A]) β ∈ (0,
1
2]
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 49 / 1
Interlude: WYD applications
strong subadditivity of entropy (Lieb-Ruskai,1973)
homogeneity of the state space of factors of type III1(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 2012 50 / 1
“If two ensembles are united, the information content of the resultingensemble should be smaller than the average information content of thecomponent ensembles” (Wigner-Yanase original paper)
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 51 / 1
Interlude: convexity or concavity?
Remark
Since
I βρ (A) =1
2Tr(ρA2)− Tr(ρβAρ1−βA)
convexity of I βρ (A) is equivalent to concavity of
ρ→ Tr(ρβAρ1−βA)
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 52 / 1
Lieb convexity and SSA
Applications: Strong subadditivity of von Neumann quantum entropy
S(ρ) := −Tr(ρ log ρ)
(Lieb & Ruskai 1975)
SSA implies strong results about to thermodynamic limit of entropy perunit volume (Robinson, Ruelle, & Lanford 1967-8)
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 53 / 1
Interlude: WYD as quantum Fisher informationDefinitionThe metric adjusted skew information is
I fρ (A) :=f (0)
2||i [ρ,A]||2ρ,f
PropositionIn the case x > 0, β ∈ (0, 12 ]
fβ(x) := β(1− β)(x − 1)2
(xβ − 1)(x1−β − 1)
we have
Ifβρ (A) :=
fβ(0)
2||i [ρ,A]||2ρ,fβ = −1
2Tr([ρβ,A][ρ1−β,A])
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 54 / 1
Interlude: concavity-convexity trick
T (·),S(·) real functions on states
S(·) concave, T (·) convex
S(·) = T (·) on pure states⇓
S(ρ) ≥ T (ρ) ∀ρ ∈ D1n
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 55 / 1
Interlude: N = 1, the DUP from Lieb convexity
Theorem (F. Hansen 2006)
Varρ(A) ≥ I fρ (A)
ProofVarρ(A) is concaveI fρ (A) is convex
ρ pure implies Varρ(A) = I fρ (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 2012 56 / 1
Why “dynamical”?
Let ρ > 0 be a state and H,K ∈ Mn,sa. Suppose that ρ = ρH(0) = ρK (0).Then, for any f ∈ Fop, one has (taking the square root of both sides ofthe DUP)
AreaCovρ (H,K ) ≥ f (0)
2·Areafρ(ρH(0), ρK (0)).
The bound on the right side of the inequality can be seen as a measure ofthe difference between the dynamics generated by H and K .
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 57 / 1
How the bound in the DUP depends on f
Theorem
Define for f ∈ F rop
S(f ) := det
f (0)
2〈i [ρ,Ah], i [ρ,Aj ]〉ρ,f
f (x) :=1
2
[(x + 1)− (x − 1)2
f (0)
f (x)
].
Then, for any f , g ∈ F rop
f ≤ g =⇒ S(f ) ≥ S(g).
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 58 / 1
The optimal bound
Let fSLD(x) :=1 + x
2. Since for any f ∈ F r
op
2x
1 + x= fSLD ≤ f
then
S(fSLD) ≥ S(f )
namely the optimal bound is given by Bures-Uhlmann metric.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 59 / 1
Relation with standard UP - I
Let f ∈ F rop. The inequality
det
f (0)
2〈i [ρ,Ah], i [ρ,Aj ]〉ρ,f
≥ det
− i
2Tr(ρ[Ah,Aj ])
is (in general) false for any N = 2m.RemarkThe proof is a consequence of Hadamard inequality:
det(H) ≤N∏j=1
hjj
for any H ∈ MN,sa.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 60 / 1
Relation with standard UP - II
Let f ∈ F rop. The inequality
det
f (0)
2〈i [ρ,Ah], i [ρ,Aj ]〉ρ,f
≤ det
− i
2Tr(ρ[Ah,Aj ])
is (in general) false for any N = 2m.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 61 / 1
The dynamical UP (g -version)
Theorem
Let A1 . . . ,AN ∈Mn,sa(C).
detCovgρ(Ah,Aj)
≥ det g(0)f (0)〈i [ρ,Ah], i [ρ,Aj ]〉ρ,f
for h, j = 1, . . . ,N,for all g , f ∈ Fop.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 62 / 1
Conclusion - Dynamical UP case
For the dynamical UP quantum g -covariances coming from regular g(constant g(0) 6= 0) do have uncertainty relations.
Quantum g -covariances coming from nonregular g (constant g(0) = 0) doNOT have (non-trivial) uncertainty relations.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 63 / 1
g -version of Robertson UP
Theorem
Let A1 . . . ,AN ∈Mn,sa(C).
detCovgρ(Ah,Aj)
≥ det −i · g(0) · Tr(ρ[Ah,Aj ]) ,
for h, j = 1, . . . ,N,for all g ∈ Fop.
Remark: g(0) is the best constant in the above inequalities.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 64 / 1
The end
Quantum g -covariances coming from regular g (constant g(0) 6= 0) dohave uncertainty relations.
Quantum g -covariances coming from nonregular g (constant g(0) = 0) doNOT have uncertainty relations.
The usual quantum covariance has the most demanding one (sinceg(0) = 1
2 only for the arithmetic mean).
After all Schrodinger and Robertson were right ...
Recent reference:P. Gibilisco, T. Isola. How to distinguish quantum covariances usinguncertainty relations. Journal of Mathematical Analysis and Applications,384: p. 670-676, 2011.
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 65 / 1
Post scriptum: The DUP on von Neumann algebras
While the first papers on the DUP where forced to use eigenvalues (seethe complicated calculations in the paper by Kosaki) now one has togeneralize something like the mean of the operators Lρ and Rρ. These arecommuting operator therefore
mf (Lρ,Rρ) = Lρf (RρL−1ρ ) = Lρf (∆ρ)
So we are dealing with the modular operator and this construction makessense in the general setting of von Neumann algebras.Gibilisco-IsolaPetz-Szabo
P. Gibilisco (Rome “Tor Vergata”) Means, Covariance, Fisher Information November 2012 66 / 1