North-Holland An energy dispersion estimate

G~ven the densRy operator p, as an mltml value of a Hamiltoman motion that evolves in a time interval At to P2. Then At AE, AE being the energy dispersion (or energy uncertainty) oftbe motion, can be esumated from below by comparing the length of the Hamil

Physics LettersA 161 (1992) 329-331 North-Holland

PHYSICS LETTERS A

An energy dispersion estimate Armin Uhlmann

Departmentof Phystcs, Unlversttyof Letpztg. Am Augustusplatz, O-7010Letpztg, Germany Received 28 August 1991; revised manuscript received 1 November 1991, accepted for publlcatmn 4 November 1991 Communicated by A.P. Fordy G~ven the densRy operator p, as an mltml value of a Hamiltoman motion that evolves in a time interval At to P2. Then At AE, AE being the energy dispersion (or energy uncertainty) oftbe motion, can be esumated from below by comparing the length of the Hamiltonlan curve with a geodesic joimng the mltml and the final density operator The lengths are calculated m the Bures metric.

Let the curve o f the density operators t~p=p(t) be a solution o f

At AE>_.h arccos~ 2,

At=t2-q,

A E= x//~ 2 - (/~)2 .

(4)

ihb=[H,p],

H=H(t),

(1)

and assumepj=p(tj) for j= 1, 2 with t~<t2. Using an idea o f ref.[ 1] it is my aim to derive the a priori inequality 12

A further important special case appears if p ( t )= I~ (t) ) (~ (t)] describes a curve of pure states which is a solution o f a Schri~dinger equation ih~b=H~. N o w eq. (4) looks like[3]

AtAE>~h arccosl (~ ( t, ),~ ( t 2 ) )[.

(5)

f x/trpH 2- ( t r p H ) 2 dt>~hy~2, tl

(2)

where 0~< Yt2~½re, COS YI2~--"t'12:= t r x/pl/2p2pl/2 . (3)

The physical meaning o f the quantity r~2 is as follows[2]: Let~u~ and~'2 be two pure vector states o f a (possibly fictitious) larger quantum system and let their reduced density operators coincide with p~ and P2. Then I (~u,, I//2)1 should be not larger than r~2, and Zl2 is the smallest n u m b e r with that property. In short, zt2 is the supremum o f I (¢~,¥2) I if the pair of unit vectors¥~ and¥2 runs through all possible simultaneous purifications o f the pair pj and P2. In case the Hamiltonian is time independent, the expectation values L and~72 o f H and H 2 are cone stants o f motion and inequality (2) simplifies to

Finally, if~(tt ) and~(t2) are orthogonal then the right-hand side o f (5) takes its maximal value~h. This is a result o f Anandan and Aharonov[ 1]. Theproofof (2) and (3) is in two steps, and will be done for non-singular density operators. The general case follows by continuity. The first step starts by lifting the curve o f density operators into the Hilbert space o f Hilbert-Schmidt operators, W, with scalar product

(W, W ' )= t r W ' W *, i.e. we purify this curve by an ansatz

(6)

t~W=W(t),

withp(t)=WW*.

(7)

There is a gauge freedom IV(t )~ IV(t ) U(t) with arbitrary unitaries U(t) for these purifications. The freedom can be diminished by choosing a curve (7) which is as short as possible in the metric given by the scalar product (6). This variational d e m a n d produces the paraUelity condition[ 4 - 6] W'W= W'IV. (8) 329

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