Fundamental Limits on the Speed of Evolution of Quantum States

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by  Ulvi Yurtsever

Fundamental Limits on the Speed of Evolution of Quantum States
Ulvi Yurtsever∗
MathSense Analytics, 1273 Sunny Oaks Circle, Altadena, CA 91001 and
Hearne Institute for Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803
(Dated: January 21, 2010)
ThispaperreportsonsomenewinequalitiesofMargolus-Levitin-Mandelstam-Tamm-typeinvolv-
ing the speed of quantum evolution between two orthogonal pure states. The clear determinant
of the qualitative behavior of this time scale is the statistics of the energy spectrum. An often-
0 overlooked correspondence between thereal-time behavior of a quantumsystem and thestatistical
1 mechanicsofatransformed(imaginary-time)thermodynamicsystemappearspromisingasasource
0 of qualitative insights into thequantum dynamics.
2

n
Motivation:
a
J As quantum information processors evolve in architecturalcomplexity, it will be increasingly important to develop

1 qualitative, architecture-independent methods that can answer fundamental questions about processor properties as
2 a function of growing complexity without recourse to detailed, large-scale simulations, and independently of the
details of processor architecture. The key questions will be speed of quantum evolution (analogous to clock speed),
]
h decoherence rates, and stability through control of decoherence.
p Basic limits on the speed of quantum evolution can be deduced directly from the Schr¨odinger equation. Consider,
- for example, a system with Hamiltonian H in a pure quantum state evolving in time according to
t
n
a ψ =e−iHt ψ , (1)
t 0
u | i | i
q where werescaledthe time parametert ininverse-energyunits as t time/~(or,equivalently, set~=1). Expanding
[ ≡
in the energy eigenbasis En of the Hamiltonian we obtain
| i
3
∞
v
5 Φ(t) ψ0 ψt =e−iE0t cn 2e−i(En−E0)t , (2)
≡h | i | |
0 nX=0
5
where E is the lowest-energy (ground) state of H (so E E >0 n). According to Eq.(2)
4 0 n 0
| i − ∀
.
2 ∞
1 Re[eiE0tΦ(t)]= cn 2cos((En E0)t), (3)
9 nX=0| | −
0
: and making use of the elementary inequality (see Fig.1)
v
i
X 2
cosx>1 (x+sinx) x>0, (4)
r − π ∀
a
∞
2 2
Re[eiE0tΦ(t)] > cn 2 1 (En E0)t sin((En E0)t)
| | (cid:18) − π − − π − (cid:19)
nX=0
2t 2
= 1 H E + Im[eiE0tΦ(t)], (5)
0
− πh − i π
where F ∞ c 2F = ψ F ψ denotes the expectation value of an observable F in the initial state ψ . If
h i≡ n=0| n| n h 0| | 0i | 0i
T0 denotes thPe first zero of the overlapΦ(t) (i.e. the earliest time t at which ψ0 evolves to an orthogonalstate), after
restoring to natural time units Eq.(5) yields the inequality
π~
T > . (6)
0
2 H E
0
h − i
∗Electronicaddress: [email protected]
1
FIG. 1: Proof of theinequality cos(x)>1−(2/π)(x+sin(x)).
On the other hand, the Schr¨odinger equation in the Heisenberg form for an operator A
dA 1
= [H,A] (7)
dt i~
combined with the uncertainty inequality (for any state vector s )
| i
1
(∆A)2(∆H)2 > s[A,H]s 2 (8)
4|h | | i|
where(∆A)2 (A A )2 = sA2 s sAs 2,givesrisetotheMandelstam-Tamm-Margolus-Levitininequality[1]
≡h −h i i h | | i−h | | i
π ~
T > . (9)
0
2∆H
Lower bounds on the first zero T probe the “long-time” behavior of the overlap Φ(t) = ψ ψ , and can be
0 0 t
h | i
interpreted as “speed limits” on the rate of quantum evolution [2]. On the other hand, the short-time behavior of
Φ(t) is completely determined by the variance ∆H since it can be shown easily that
(∆H)2
Φ(t)2 =1 t2+O(t3). (10)
| | − ~2
As a quick application, Eq.(10) implies that the quantum Zeno effect [3] for the initial state ψ , which requires the
0
condition lim Φ(t/n)2n =1, is in principle always present as long as the variance ∆H i|s fiinite. More precisely,
n→∞
| |
if n projective measurementsof the operator ψ ψ are performedsuccessively atequal intervals t/n,the enhanced
0 0
| ih |
probability of finding the system in the initial state ψ at time t is given by
0
| i
Φ(t/n)2n e−(∆~H2)2tn2 . (11)
| | ≈
Consequently, this “stasis” probability can be made close to unity provided
(∆H)2 t 2
n & t2 & . (12)
~2 (cid:18)T (cid:19)
0
While the Zeno effect is significant for applications involving quantum control of decoherence, long-time behavior
of the overlap Φ(t) has significance for exploring ultimate limits on the speed of quantum information processing.
In fact, limits on the speed of quantum evolution are connected to another widely known fundamental limit on the
power of information processing, the holographic entropy bound, as suggested by the following arguments: Consider
ageneralphysicalsystemofapproximatespatialsizeL(hence surfaceareaL2)andtotalenergyE. Bythe discussion
2
above, there is a fundamental bound given by an inequality of the type Eq.(6) or Eq.(9) on the time τ during which
a quantum state of the system can evolve to an orthogonalstate:
~
τ > . (13)
E
Let S be the entropy the system, which is, equivalently, the Boltzmann constant k times the number of (classical)
B
bits of information that can be stored. Since a physical signal propagating across the system can be used to interact
with each classical “bit” successively and change its state, the shortest time in which this could be done, (S/k )τ,
B
must be no longer than the light crossing time across the system:
S L
τ < . (14)
k c
B
Combining Eq.(14) with Eq.(13) yields
S L EL
< < . (15)
k cτ ~c
B
The entropy bound Eq.(15) (which was first discovered by Bekenstein [4]) can also be derived, independently, by an
argument based on the system’s formation history. The speed-of-evolution bound Eq.(13) imposes an ultimate limit
on the time rate of entropy change:
1 ∂S 1 E
< < . (16)
k (cid:12)∂t(cid:12) τ ~
B (cid:12) (cid:12)
(cid:12) (cid:12)
But no matter what the system’s actual formation(cid:12)hist(cid:12)ory and its final state are, a state of zero entropy must be
reachablefromthatfinalstatewithin atime onthe orderofthe lightcrossingtime L/c. Hence the Bekensteinbound
Eq.(15) on the final entropy follows once again, this time from Eq.(16). Since black-hole formation (gravitational
collapse) imposes the Schwarzschild limit E < c4L/(2G) on the maximum energy E of the system, Eq.(15) implies
the holographic entropy bound [5]
S c3L2 1L2
< = , (17)
kB 2~G 2lp2
where l ~G/c3 is the Planck length.
p
≡
p
New inequalities:
A keyobservationaboutthe overlapfunction Φ(t) (Eq.(2)) is that itcanbe expressedasthe Fouriertransform(or
“characteristic function”) of a positive probability distribution function ρ(E), characterizing the energy spectrum of
the system in the initial quantum state ψ :
0
| i
Φ(t)= e−itEρ(E)dE , (18)
Z
where again we used the rescaled time parameter t time/~. For example, if the Hamiltonian H has a discrete
≡
spectrum E , the state ψ can be expanded as
i 0
{ } | i
ψ = c E , (19)
0 j j
| i | i
Xj
and the distribution function and the overlaphave the discrete forms
ρ(E)= cj 2δ(E Ej), Φ(t)= cj 2e−iEjt . (20)
| | − | |
Xj Xj
More generally, using the spectral theorem [6], the Hamiltonian H can be expressed in the form
H = EdP(E), (21)
Z
3
where dP(E) denotes integration with respect to a projection-valued measure on the Hilbert space, and the energy
probability distribution (“density of states”) ρ(E) can be defined via the identity
ρ(E)dE = ψ dP(E) ψ . (22)
0 0
h | | i
From a practical point of view, the distribution function ρ(E) is a key design parameter since it is relatively easy to
manipulate. This distribution is related to the overlap function Φ(t) via the inverse Fourier transform
1 ∞
ρ(E)= eiEtΦ(t)dt. (23)
2π Z
−∞
Itis usefultoextendthe function Φ(t)to the complexdomain;infact, wewillassumeaPaley-Wiener[7]condition
on the distribution ρ(E) such that Φ(t) as defined by Eq.(18) is an entire analytic function on the complex plane
t C. For example, this is the case if ρ(E) is of compact support, or, more generally, falls off faster than any
∈
exponential as E . In either case, one can reasonably expect such Paley-Wiener conditions to hold for most
→ ±∞
physical systems.
Denoting complex time with the commonly-usedsymbol z,Eq.(18)implies thatthe analytic function Φ(z)has the
power series expansion
∞ ( i)n En
Φ(z)=1+ − h izn , (24)
n!
nX=1
which is a rephrasing of the generating-function identities
∂nΦ
En =in =inΦ(n)(0). (25)
h i ∂tn (cid:12)
(cid:12)t=0
(cid:12)
Consequently, the function logΦ(z) can be expanded in a(cid:12)power series around z =0:
∞
logΦ(z)= γ zn , (26)
n
nX=1
where
n ( 1)k−1 n ( i)l1 El1 ( i)l2 El2 ( i)lk Elk
γ − − h i − h i − h i
n
≡ k l ! l ! ··· l !
Xk=1 lXi>1 1 2 k
l1+l2+···+lk=n
( i)n n ( 1)k−1 n n
= − − El1 El2 Elk . (27)
n! k (cid:18)l l l (cid:19)h ih i···h i
Xk=1 lXi>1 1 2 ··· k
l1+l2+···+lk=n
Clearly, T cannot be less than the radius of convergence of the power series Eq.(26), which gives us our first new
0
inequality:
~
T > , (28)
0 limn→∞ γn n1
| |
where
n ( 1)k−1 n El1 El2 Elk
γ =( i)n − h ih i··· h i . (29)
n
− k l !l ! l !
Xk=1 lXi>1 1 2 ··· k
l1+l2+···+lk=n
For our next set of new inequalities, we will rely on Bochner’s Theorem from real analysis [8]. First, a complex-
valued function f on R is called positive-definite if for any choice of complex numbers α ,α , ,α and for any
1 2 r
x ,x , ,x R, we have ···
1 2 r
··· ∈
r
α α f(x x )>0. (30)
i j i j
−
iX,j=1
4
Observe that consequences of positive-definiteness are: (i) f(0) > 0 (derived from Eq.(30) with r = 1), and (ii)
f( x) = f(x) and f(0)> f(x) x R (derived from Eq.(30) with r = 2). It turns out that positive-definiteness,
− | | ∀ ∈
along with the normalization condition Φ(0) = 1, completely characterizes functions Φ that are expressible as the
Fourier transform of a positive density of states ρ(E) as in Eq.(18):
Bochner’s Theorem: A continuous complex-valued function f : R C with f(0) = 1 is the Fourier transform
f(x)= e−ixEρ(E)dE of a finite, normalized, positive Borel measure−ρ→(E) if and only if f is positive-definite.
Since BRochner’s theorem completely characterizes a general overlap function Φ(t), the (in general infinite) class of
inequalities
r
α α Φ(t t )>0 α ,α , ,α C, t ,t , ,t R, (31)
i j i j 1 2 r 1 2 r
− ∀ ··· ∈ ··· ∈
iX,j=1
arethemostgeneral,universalsetofinequalitiesconstraininganoverlapfunctionΦ(t)definedasinEq.(2)forapure
state. Consequently, every inequality of the Margolus-Levitin-Mandelstam-Tamm-type constraining the first zero T
0
of Φ(t) is necessarily a consequence of Eqs.(31).
To derive some new examples of such inequalities for the first zero-crossing time T from Eqs.(31), observe that
0
if Φ(t) is an overlap function (hence is positive-definite), then Eqs.(18) and (25) combined with Bochner’s theorem
imply that for any positive integer n,
Φ(2n)
f(t) ( 1)n (32)
≡ − E2n
h i
is also a positive-definite function satisfying f(0)=1. For ease of calculation, let us redefine the hamiltonian as
H H H , (33)
−→ −h i
so that the average (first-moment) of H vanishes: H = E = 0. Taking n = 1 in Eq.(32) and applying the basic
h i h i
consequence 1=f(0)> f(t) of Eqs.(30) gives
| |
Φ′′(t)6 E2 . (34)
− h i
Integrating Eq.(34) once from t=0 to t and using Φ′(0)= i E =0 we obtain the inequality
− h i
Φ′(t)6 E2 t. (35)
− h i
Integrating Eq.(35) once more from t=0 to t=T (the first zero-crossing)gives
0
T 2
16 E2 0 . (36)
h i 2
After restoring to natural time units and undoing the rescaling Eq.(33), Eq.(36) becomes the inequality
√2~
T > . (37)
0
(E E )2
h −h i i
p
Applying anentirelyparallelstreamofargumentsandstartingfromn=2, 3, ... inEq.(32),weobtainthe following
infinite series of new inequalities involving the first zero-crossingtime T :
0
T 2
(E E )2 0 > 1
h −h i i2~2
T 4 T 2
(E E )4 0 > (E E )2 0 1
h −h i i24~4 h −h i i2~2 −
T 6 T 4 T 2
(E E )6 0 > (E E )4 0 (E E )2 0 +1
h −h i i6!~6 h −h i i24~4 −h −h i i2~2
. .
. .
. .
T 2n n T 2(n−s)
(E E )2n 0 > ( 1)s (E E )2(n−s) 0 . (38)
h −h i i(2n)!~2n − h −h i i[2(n s)]!~2(n−s)
Xs=1 −
5
Connection with thermodynamics:
As we arguedabove,for a wide class ofphysicalsystems the overlapfunction Φ(t) givenby Eq.(18)is holomorphic
on the complex t-plane. It is straightforwardto observe that at imaginary times the overlap function is equal to the
canonical partition function of a thermodynamic system: for β >0:
Z(β) Φ( iβ)= e−βEρ(E)dE . (39)
≡ − Z
Introducing the canonical probability distribution
e−βEρ(E)
ρ (E) (40)
c
≡ Z(β)
completes the correspondence with the thermodynamics of a (in general, abstract) physical system whose density of
states is given by ρ(E).‡ For example, the thermodynamic entropy is given by
S
ρ (E) logρ (E)dE =logZ(β)+β E logρ(E) , (41)
c c c c
kB ≡−Z h i −h i
wherek isBoltzmannsconstant,and denotesexpectationwithrespecttothecanonicalprobabilitydistribution
B c
h···i
function ρ (E): ρ (E)dE. Real zeros of the overlap function Φ(t) correspond to pure-imaginary zeros
c c c
h···i ≡ ···
of the partition function ZR(β).
The promise of the thermodynamic correspondencelies in the fact that the qualitative behavior of thermodynamic
systemshavebeenextensivelyinvestigated,andcontainavastpanoplyofresultsandtechniquesthatmightpotentially
translate to insights into the real-time behavior of Φ(t) via the imaginary inverse-temperature correspondence with
Z(β). For example, an impressive array of qualitative and quantitative results are already known for the behavior
of the complex zeros of Z(β) for a variety of lattice models in statistical mechanics [9, 10]. The condensation of the
complexzerosofthevolume-scaledpartitionfunctionontotherealaxisinthethermodynamicscalinglimitisdirectly
related to the phenomenon of phase transitions according to the classical Lee-Yang theory [11].
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(1998).
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A.Plastino, European Physics Letters 81, 30007 (2008).
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70, 062302 (2004).
[4] J. Bekenstein, Phys. Rev.D 23, 287 (1981), and Phys.Rev.Lett. 46, 623 (1981).
[5] G.‘t Hooft inSalam Festschrifft, editedbyA.Aly,J. Ellis, and S.Randjbar-Daemi(World Scientific,Singapore 1993); J.
Bekenstein in Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity, edited by R. Ruffini and M.
Keiser (World Scientific, RiverEdge, NewJersey 1996); U. Yurtsever,Phys. Rev.Lett. 91, 041302 (2003).
[6] W. Rudin,Theorem 12.23 in Functional Analysis, 2nd edition, McGraw-Hill, New York,1991.
[7] Y.Katznelson, An Introduction to Harmonic Analysis, Cambridge UniversityPress (3rd edition, 2004).
[8] W. Rudin,p. 303 in Functional Analysis, 2nd edition, McGraw-Hill, New York,1991.
[9] M. Biskup, C. Borgs, J. T. Chayes, L.J. Kleinwaks, and R.Kotecky´,Phys. Rev.Letters 84, 4794 (2000).
[10] M. Biskup, C. Borgs, J. T. Chayes, L.J. Kleinwaks, and R.Kotecky´,Commun. Math. Phys.251, 79131 (2004).
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‡ This system does not have to be quantum mechanical; in general, a classical system could be designed to have the density of states
ρ(E)dE lyingbetweenitsconstant-energy shells{H=E}and{H =E+dE}.
6