We show that discretization of spacetime naturally suggests discretization of
Hilbert space itself. Specifically, in a universe with a minimal length (for
example, due to quantum gravity), no experiment can exclude the possibility
that Hilbert space is discrete. We give some simple examples involving qubits
and the Schrodinger wavefunction, and discuss implications for quantum
information and quantum gravity.Comment: 4 pages, revtex, 1 figur

A. Zee

Bennett

Bombelli

Calmet

Feynman

Garay

Holevo

Mead

Padmanabhan

Penrose

Regge

Roman V. Buniy

Shor

Srinkanth

Stephen D.H. Hsu

Wheeler

Wolfram

Physics Letters B

English

arXiv

Crossref

Is Hilbert space discrete?

AbstractWe show that discretization of spacetime naturally suggests discretization of Hilbert space itself. Specifically, in a universe with a minimal length (for example, due to quantum gravity), no experiment can exclude the possibility that Hilbert space is discrete. We give some simple examples involving qubits and the Schrödinger wavefunction, and discuss implications for quantum information and quantum gravity

Buniy, Roman V.

Hsu, Stephen D.H.

Zee, A.

Elsevier - Publisher Connector 

Is Hilbert space discrete? 

Physics Letters B 630 (2005) 68–72
www.elsevier.com/locate/physletb
Is Hilbert space discrete?
Roman V. Buniy a, Stephen D.H. Hsu a,∗, A. Zee b
a Institute of Theoretical Science University of Oregon, Eugene, OR 97403, USA
b Kavli Institute for Theoretical Physics UCSB, Santa Barbara, CA 93106, USA
Received 21 August 2005; accepted 9 September 2005
Available online 7 October 2005
Editor: H. Georgi
Abstract
We show that discretization of spacetime naturally suggests discretization of Hilbert space itself. Specifically, in a universe
with a minimal length (for example, due to quantum gravity), no experiment can exclude the possibility that Hilbert space
is discrete. We give some simple examples involving qubits and the Schrödinger wavefunction, and discuss implications for
quantum information and quantum gravity.
 2005 Elsevier B.V. Open access under CC BY license.One of the most fundamental questions is whether
our universe is ultimately discrete or continuous [1,2].
It is an ancient idea that matter and perhaps even space
and time are constructed from denumerable compo-
nents. Quantum mechanics introduces new objects to
the description of nature, such as the set of all pos-
sible quantum states—Hilbert space. Conventionally,
Hilbert space is taken as continuous, but in this Letter
we will discuss the possibility that it is in fact discrete.
There are many indications that spacetime may be
discrete rather than continuous [3]. For example, met-
ric fluctuations due to quantum gravity might preclude
* Corresponding author.
E-mail addresses: roman@uoregon.edu (R.V. Buniy),
hsu@duende.uoregon.edu (S.D.H. Hsu), zee@itp.ucsb.edu
(A. Zee).0370-2693  2005 Elsevier B.V.
doi:10.1016/j.physletb.2005.09.084
Open access under CC BY license.any notion of distances less than of order the Planck
length lP . Recent work [4] has shown that no macro-
scopic experiment can be sensitive to discreteness of
position on scales less than the Planck length. Any
device (such as an interferometer) capable of such
resolution would be so massive that it would have
already collapsed into a black hole. Relativistic co-
variance suggests that discretization of space implies
discretization of time. Indeed, minimal length proba-
bly makes it impossible to construct a clock capable
of measuring time differences less than of order the
Planck time. (Consider, for example, a bouncing pho-
ton between two mirrors as the ticking of the clock.)
As an explicit but crude toy model of discreteness, one
might imagine that our universe lives on a spacetime
lattice with spacing lP . More sophisticated models
have been proposed [5] in which spacetime is discrete,
R.V. Buniy et al. / Physics Letters B 630 (2005) 68–72 69but not necessarily regular. In our discussion discrete-
ness should not be taken to imply regularity, either in
spacetime or the structure of Hilbert space. We are
not suggesting that continuous Hilbert space necessar-
ily be replaced by a lattice; instead, for example, the
discreteness might be due to an intrinsic fuzziness or
uncertainty.
A consequence of spatial discreteness is that in any
finite region of space of size L there are only a fi-
nite number of degrees of freedom N ∼ L3. (Hence-
forth we adopt units in which lP = 1.) Although our
universe might be infinite in extent, any experiment
performed by scientists must take place over a finite
period of time. By causality, this implies that the ex-
periment takes place in a region of finite size, which
we take to be L. We therefore assume the existence of
a long-distance (infrared) regulator L in addition to a
short-distance (ultraviolet) regulator lP .
Now consider quantum mechanics in a spatially
discrete universe. The dimensionality of Hilbert space
is itself finite, equal to the number of degrees of free-
dom. Let the space be spanned by a finite set of inde-
pendent basis vectors |n〉 (1 nN ). In conventional
quantum mechanics, we define Hilbert space to consist
of all linear combinations of these basis vectors
(1)|ψ〉 =
N∑
n=1
an|n〉,
modulo rescaling by an arbitrary complex parame-
ter. Since the an are continuous complex parameters,
Hilbert space is continuous even if spacetime is dis-
crete, and the set of possible states |ψ〉 is infinite.
However, in a spatially discrete universe there is no
experiment which can exclude discreteness of the co-
efficients an, if that discreteness is sufficiently small.
We argue as follows. If the number of degrees of free-
dom is finite, so is the set of possible distinct measure-
ment devices one can construct. (By “distinct” devices
we do not mean different in design or construction, but
rather that they measure distinct physical quantities—
in other words, correspond to different operators act-
ing on the Hilbert space. See the qubit example below.)
Equivalently, the number of eigenstates of all possible
distinguishable operators is finite (recall that with ul-
traviolet and infrared regulators present, the spectrum,
and hence the number of eigenstates, of any particular
operator is finite). Thus, the physics of this universecan be described using a Hilbert space with only a
finite number of distinguishable states—that is, a dis-
crete and finite Hilbert space, in which the values of
an are themselves quantized.
As a simple example, consider a single qubit. The
Hilbert space of a spin- 12 particle is simply the set of
all eigenstates of the spin operator. The most general
state |ψ〉 can be written as
(2)|ψ〉 = cos θ |+〉 + eiφ sin θ |−〉,
where θ and φ are continuous parameters. However,
discrete space implies that there are only a finite num-
ber of distinguishable spin operators. A sufficiently
small rotation of the measurement apparatus is in-
distinguishable from no rotation. Hence, one cannot
measure changes in the angular variables smaller than
 ∼ 1/d , where d is the size of the apparatus. This size
is somewhat ill-defined, since by making the apparatus
arbitrarily long it becomes sensitive to very small rota-
tions (neglecting, of course, considerations of rigidity
and causality). One should probably take d to describe
only the part of the apparatus with which the qubit
interacts during a measurement. Alternatively, if the
spin- 12 object described by |ψ〉 has finite size (Comp-
ton wavelength), the set of its possible orientations
might itself be discrete (imagine a vector constrained
to connect two vertices of a lattice). In that case d
might be given by the size of the qubit, rather than
that of the apparatus.
Discrete Hilbert space leads us to a concrete modi-
fication of the linear superposition principle. For ex-
ample, if we were to superpose two states of the
form (2), one with (θ1, φ1) and the other with (θ2, φ2),
then for arbitrary choice of coefficients the resulting
state (θ,φ) will not be in the allowed set. One concrete
proposal would be to replace (θ,φ) by the nearest al-
lowed values (a “snap to nearest lattice site” rule; see
Fig. 1). We imagine that a clever experimentalist could
set a useful bound on this deviation from linear super-
position.
One might be concerned that the SU(2) group
structure of qubit rotations cannot be obtained as the
limit of larger and larger finite discrete subgroups.
However, there exist simple models in which contin-
uous rotational or even Lorentz symmetry is obtained
in the long wavelength limit from underlying dynam-
ics which has only discrete symmetry. For example, in
lattice QCD the symmetries are all discrete, yet contin-
70 R.V. Buniy et al. / Physics Letters B 630 (2005) 68–72Fig. 1. A possible discretization of the Bloch sphere (qubit Hilbert
space). Points on each disc (of size ) are identified. Points between
discs can be assigned to the nearest disc.
uous symmetries emerge in the long wavelength limit.
As another example, in [6] a model of spinless point
particles hopping on a flux lattice gives rise to low-
energy excitations obeying the Dirac equation.
We can deduce similar results concerning discrete-
ness by considering the Schrödinger equation. All the
physics of a universe with discrete spacetime can be
described by a wavefunction ψ(x) whose values are
discrete, rather than continuous. To convince ourselves
of this, we need merely consider simulations of the
Schrödinger equation on a classical digital computer,
in which both the spacetime coordinates and ψ(x) are
discrete. All predictions of quantum mechanics can be
obtained to any desired accuracy using such discrete
simulations; quantum phenomena such as interference
patterns can therefore be reproduced even if quan-
tum mechanics is intrinsically discrete, as suggested
here. Since by assumption we can probe the varia-
tion of ψ(x) only over distances larger than the Planck
length, the magnitude of the required accuracy ψ is
bounded below. (Again, since we have both ultraviolet
and infrared regulators, ψ(x) must be a smooth func-
tion. It cannot vanish identically in an entire region, so
its variation over some finite interval is bounded be-
low.) It is important to note, though, that the required
discreteness might be exponentially small. For exam-
ple, to describe the exponential tail of a wavefunction
might require ψ ∼ e−L, where L is the box size of
the simulation. Nevertheless, for fixed L and lP , the
magnitude of ψ is always bounded below.
A discrete wavefunction ψ(x) implies a discrete
Hilbert space, and vice versa, since the value of the
wavefunction is simply the overlap of a particular
state, |ψ〉, with another, |x〉. In other words, usingEq. (1), ψ(xi) = 〈xi |ψ〉 = ai . If the set of states |ψ〉
and the set of states |x〉 are both finite, then ψ(x)
can take only a finite (discrete) set of values, and vice
versa. As discussed above, to accommodate exponen-
tial fall-off the size of discreteness in the ai must be
exponentially small—potentially of order e−L, where
L is the size of the universe.
However, minimal length seems to imply a stronger
(non-exponential) limitation on the phase information
carried by a quantum state, similar to what we ob-
tained above for a qubit. Suppose information is stored
in a particular quantum state |ψ〉. A Planck-length
uncertainty in the spacetime location of the state (or
of where the measurement of the state takes place)
leads to an uncertainty in the value of the phase, as
seen from the time translation operator U(t) = e−iH t
or the translation operator T (x) = e−ipx . The phase
can be specified only to accuracy E∗ or p∗ in Planck
units, where E∗ and p∗ are roughly the characteris-
tic energy or momentum associated with |ψ〉. To be
explicit, one can expand |ψ〉 in an energy or momen-
tum eigenstate basis, with each term in the expansion
acquiring a phase uncertainty of order E or p. Using
p > 1/L, we obtain a phase uncertainty  > 1/L, sim-
ilar to 1/d in the case of a spin- 12 qubit. If the state |ψ〉
is transported over some path in spacetime, for exam-
ple to be interfered with some other state, we expect a
fundamental limitation on the precision of the relative
phase. This might have some interesting consequences
for Berry’s phase.
We can derive this result another way by consider-
ing a particle of energy E, interacting with an external
probe of energy E′ which measures the phase φ of
its wavefunction. Let the interaction take place over a
time interval t . The phase of the particle necessar-
ily evolves during the time interval, so φ ∼ Et .
Causality requires t > R, where R is the size of the
probe (or the portion of it which interacts with the par-
ticle, which we take to be the entire probe). It must be
the case that R > E′, or the probe would have already
collapsed into a black hole. Finally, using energy-time
uncertainty, E′ > (t)−1, we obtain
(3)φ ∼ Et > EE′ > E2/φ,
which implies φ > E. So, we expect that a phase
discreteness  smaller than ∼ E (recall, we use Planck
units) is undetectable experimentally.
R.V. Buniy et al. / Physics Letters B 630 (2005) 68–72 71The phase uncertainty discussed above is not incon-
sistent with the requirement that ψ might be expo-
nentially small. There is no contradiction between an
exponentially small uncertainty in the magnitude and
larger uncertainty in the phase of a wavefunction ψ .
For example, suppose ψ is expressed as the superposi-
tion of two other states: ψ = a1ψ1 +a2ψ2. Our ability
to measure |ψ | (or |ψi |) to arbitrary accuracy places no
limitation on the precision of the phases in ai or ψi .
We mention some consequences of discrete Hilbert
space:
(1) Only a finite number of classical bits are re-
quired to specify the state of a discrete qubit. Note that,
because we cannot directly measure its state, a single
qubit can be used to transmit or store only a single bit
of classical information. (This is a result of Holevo’s
theorem [7].) Nevertheless, a perfect classical simula-
tion of qubits with continuous Hilbert space requires
an infinite number of classical bits.
(2) There are asymptotic limits to the power of
quantum computation. Consider, for example, Shor’s
algorithm [8] (or similarly the quantum Fourier trans-
form), which requires of order (lnn)3 steps to factor
an integer n. Discreteness of order  limits the pre-
cision of quantum manipulations, and quantum algo-
rithms require a minimum precision of order one over
the number of steps in the algorithm [9]. Thus, factor-
ization of an integer n using Shor’s algorithm requires
manipulations of precision (lnn)−3, and there exists a
largest integer n∗ ∼ exp(−1/3) that can be factored.
In the case of arbitrarily low-energy qubits, we might
have  ∼ 1/L ∼ 1/T , where T is the timescale of
the quantum processor, so n∗ still grows exponentially
with T 1/3. However, for qubits of fixed energy E one
eventually encounters a maximum n∗.
(3) Quantum mechanics is modified at short dis-
tances. It may be that near the Planck scale the Hilbert
space discreteness  is of order unity. For example, a
vector whose length d is of order the Planck length
may have only a few possible orientations (imagine
that the vector must connect two points on a lattice).
According to this analogy, quantum dynamics might
be drastically modified at short distances. It would be
interesting to formulate a superclass of models of this
type which have ordinary quantum mechanics as a lim-
iting case. These might produce a novel approach to
quantum gravity, as current approaches such as stringtheory extrapolate quantum mechanics with continu-
ous Hilbert space all the way to the Planck scale.
One might ask how to evolve a state in a discrete
Hilbert space. There are many possibilities, but one
concrete method would be to write the time evolu-
tion operator e−iH t as a product of discrete evolution
operators e−iHt and apply this product of operators
sequentially to the state, for example as in (2), fol-
lowed by the “snap to” rule after each step. This is
equivalent to taking classical digital computer simula-
tions literally. That is, by accepting the finite precision
of the variable ψ(x) in an ordinary computer program,
one obtains a naive discretization of Hilbert space with
the “snap to” rule implemented by simple numerical
rounding.
In conclusion, it appears that the traditional as-
sumption of continuous Hilbert space is rather strong:
minimal length precludes any experiment showing that
the discreteness parameter  is exactly zero. While we
have motivated a non-zero  using quantum gravity,
we stress that discreteness may appear at a dimen-
sional scale larger than lP , and that experimentalists
should keep an open mind.
Note added
After this work was completed, we were informed
of a related paper by R. Srinkanth [10] in which pos-
sible discreteness of Hilbert space is discussed from a
computational perspective.
Acknowledgements
The authors thank D. Bacon, M. Graesser, B. Mur-
ray and J. Polchinski for useful comments. S.H. and
R.B. are supported by the Department of Energy un-
der DE-FG06-85ER40224. A.Z. was supported in part
by the National Science Foundation under grant num-
ber PHY 99-07949(2004).
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