A Quantum 0−∞ Law
We give conditions under which a sequence of randomly chosen orthogonal subspaces of a separable Hilbert space generates the whole space.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 012, 12 pages
A Quantum 0 − ∞ Law
Michel BAUER ab
a) Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique,
91191 Gif-sur-Yvette, France
b) PSL Research University, CNRS, École normale supérieure,
Département de mathématiques et applications, 75005 Paris, France
E-mail: michel.bauer@ipht.fr
Received October 27, 2021, in final form February 01, 2022; Published online February 10, 2022
https://doi.org/10.3842/SIGMA.2022.012
Abstract. We give conditions under which a sequence of randomly chosen orthogonal
subspaces of a separable Hilbert space generates the whole space.
Key words: random Hamiltonians; random geometry; Markov processes
2020 Mathematics Subject Classification: 60J05; 81-10
Dedicated to Denis Bernard
on his 60 th birthday
1 Introduction
While studying a quantum system, one is often faced with the task of comparing two bases of
a separable Hilbert space. One obvious example is given by the eigenstate bases for a free and
a perturbed Hamiltonian. Another example is when one basis is well-suited to describe local
operators (in this basis they are diagonal, or their matrix elements decay rapidly away from the
diagonal for instance) and the other is the eigenstate basis of the Hamiltonian. This is related
to the eigenstate thermalization hypothesis [1, 3] which can be – at least partly – rephrased as
a physically motivated ansatz for matrix elements of local operators (say, in position space for
instance) in the Hamiltonian eigenstate basis. The question also has some more practical aspects
when some spaces of trial wave functions are used to get approximations of the low-lying energy
eigenstates of a quantum system. In this setting, the dimension of the trial space is larger than
the number of eigenstates which are to be approximated. In one version, all the approximate
eigenstates are looked for in a single large space, but it may also be desirable to look for the
first eigenstate in some trial space, for the second in a larger one and so on, leading to a nested
structure like “Russian dolls”.
Such a Matryoshka picture also paves the way to the construction of random Hamiltoni-
ans, and this is the vantage point we adopt in the following. Suppose we are given a (finite-
dimensional) subspace F1 of the full Hilbert space and we are to choose a ray, i.e., a one-
dimensional subspace, E1 inside it. There is a single unitary-invariant probability measure to
make this choice at random. If one looks for another ray, orthogonal to E1, or, what amounts
to the same, for a two-dimensional subspace E2 ⊃ E1, in a larger (finite-dimensional) subspace
F2 ⊃ F1 of the full Hilbert space, there is again a single unitary-invariant probability measure
to make this choice at random. Given a sequence {0} ⊂ F1 ⊂ F2 ⊂ · · · of trial subspaces, i.e.,
of finite-dimensional subspaces of the full Hilbert space with dimFk ≥ k for k ∈ N such that
∪k∈NFk is dense in the full Hilbert space, this procedure can be continued to produce a random
orthonormal sequence of rays in the Hilbert space, or what amounts to the same a sequence
{0} ⊂ E1 ⊂ E2 ⊂ · · · with Ek ⊂ Fk and dimEk = k for k ∈ N. Statistical properties of matrix
mailto:michel.bauer@ipht.fr
https://doi.org/10.3842/SIGMA.2022.012
2 M. Bauer
elements with respect to this sequence, as compared to those in a fixed basis of the Hilbert
space, can be studied. This is a difficult task. But there is a preliminary question, in fact the
only one we address in this study: does this procedure really construct a basis, or do “holes”
remain in the Hilbert space?
Section 2 makes our framework precise. We introduce the notion of Matryoshka tailored for
our purposes, and state our main result: a necessary and sufficient condition to get a complete
orthonormal system, based on the growth of the dimensions of the trial spaces. Section 3
reinterprets the problem as a quantum version of a classical puzzle/paradox. Section 4 is devoted
to the proofs. Section 5 mention further possible directions.
2 Definitions and main result
Let H denote a complex separable infinite-dimensional Hilbert space. The norm is denoted
by ∥ ∥ and the inner product by ⟨ , ⟩, which we take to be linear in the second argument, as in
quantum mechanics. If S ⊂ H, we set S⊥ := {v ∈ H, ⟨s, v⟩ = 0 for every s ∈ S} as usual.
Let n := (nk)k∈N be an increasing sequence 0 = n0 ≤ n1 ≤ n2 ≤ · · · of integers with
limk→∞ nk = ∞. An n-Matryoshka F := (Fk)k∈N is an increasing sequence {0} = F0 ⊂ F1 ⊂
F2 ⊂ · · · of subspaces of H with dimFk := nk for k ∈ N. An n-Matryoshka F is called total
(in H) if ∪k∈NFk is dense in H.
In the following, until Section 5, we fix once and for all the sequence n and a total n-Mat-
ryoshka F . Note that by the choice of an appropriate orthonormal basis (vn)n≥1 in H we could
reduce the situation to the case when Fk := span(vn)n∈J1,nkK. We shall sometimes exploit this
possibility. In the framework of numerical computation of low-lying eigenstates, Fk would be
the kth trial space, the space in which the first k levels are to be found.
We also fix an increasing sequence m := (mk)k∈N of integers such that mk ≤ nk for k ∈ N
but limk→∞mk = ∞. We set (δm)k := mk+1−mk. An m-Matryoshka in F is an m-Matryoshka
E := (Ek)k∈N such that Ek ⊂ Fk for k ∈ N. In the sequel, if E is an m-Matryoshka in F , E⊥F
k
denotes the orthogonal complement of Ek in Fk (not in H). The case mk = k fits to our original
motivation of finding k low-lying eigenstates in Fk. More general sequences (mk)k∈N would allow
to deal with degenerate energy levels for instance. Though this complicates notations a little
bit, the mathematics is essentially the same.
We denote byMF ,m, or simplyM when there is no risk of confusion, the set ofm-Matryoshkas
in F . We recall that if F is a finite-dimensional Hilbert space1 of dimension n and m ≤ n there
is an unique unitary invariant probability measure µm
F on the set of m-dimensional subspaces
of F (a Grassmannian which we denote by G(F,m)).
An important consequence of the definition of µm
F for our discussion is the following: If v is
a vector in F , E is chosen with µm
F and U is the (orthogonal) projection of v on E (a random
vector) then E
(
∥U∥2
)
= ∥v∥2mn (taken to be 0 if n = 0). Due to the importance of this result,
we give a quick derivation in Appendix A. The vector W := v − U is the projection of v on the
orthogonal complement E⊥ of E in F , and it follows from uniqueness that the image measure
of µm
F by the map from the Grassmannian of m-dimensional subspaces of F to that of n −m-
dimensional subspaces, E 7→ E⊥ or equivalently U 7→ W , is µn−m
F so that E
(
∥W∥2
)
= ∥v∥2 n−m
n ,
which must be because ∥v∥2 = ∥U∥2 + ∥W∥2 identically.
Note in passing that the large n behavior of the formulæ for E
(
∥U∥2
)
and E
(
∥W∥2
)
given
above gives the intuitive reason why, as is well-known, no unitary invariant measure µm
F can
exist when F is infinite-dimensional.
We use the measures µm
F to endow the set M of m-Matryoshkas in F with a probability
measure PF ,m, or simply P , defined recursively as follows:
1In this study, a finite-dimensional subspace of a separable infinite-dimensional reference Hilbert space.
A Quantum 0−∞ Law 3
� E0 = F0 = {0}.
� E0, . . . , Ek being chosen, Ek+1 is chosen as a random (in a unitary invariant way) subspace
of Fk+1 containing Ek. The condition of unitary invariance can be rephrased as follows:
Ek+1 is the orthogonal direct sum of Ek with an (δm)k-dimensional subspace of E⊥F
k
chosen with µ
(δm)k
E
⊥F
k
.
The recursive, Markovian, nature of this definition ensures that the appropriate analog of Kol-
mogorov’s consistency criterion holds automatically, so that it extends to a probability measure
on m-Matryoshkas in F , i.e., to infinite sequences.
More precisely, letting Ek denote the σ-algebra generated by the random subspaces E0, . . . , Ek
and σ
(
∪k∈N Ek
)
the σ-algebra they generate, there is an unique σ
(
∪k∈N Ek
)
-probability measure
on M which has the right finite-dimensional marginals. We complete this probability measure(
with the σ-algebra σ
(
∪k∈N Ek
)
suitably enlarged to a σ-algebra we denote by E to include null
sets
)
, leading to the complete filtered probability space
(
M, E , (Ek)k∈N, P
)
.
It is clear from this construction that only n is relevant. The precise choice of n-Matryoshka F ,
or when this comes into play of an orthonormal basis (vn)n≥1 in H with Fk := span(vn)n∈J1,nkK
is irrelevant.
One basic question is under which conditions E is total
(
recall this means that ∪k∈NEk is
dense in H
)
with P -probability 1. Denoting by E the closure of ∪k∈NEk in H, the question is
whether E
⊥ (
which is also the orthogonal of ∪k∈NEk
)
reduces to {0} (with P -probability 1).
This question has a simple and nice answer.
Theorem 2.1. For k ∈ N∗ define rk :=
mk−mk−1
nk−mk−1
if nk −mk−1 > 0 and rk := 1 otherwise.
If the series
∑
k rk is divergent the space E
⊥
is zero-dimensional (i.e., E is total) with pro-
bability 1.
If the series
∑
k rk is convergent the space E
⊥
is infinite-dimensional with probability 1.
This result is some kind of a paradox: if the series is convergent, even if the sequence n
grows faster that the sequence m so that dimFk − dimEk grows without bounds, the Fks may
succeed in filling the whole of H! As a simple illustration, if mk = k and nk = 2k for k ∈ N then
rk = 1
k+1 and by the divergence of the harmonic series E
⊥
= {0} with probability 1 even if at
each step E⊥F
k has dimension k, which grows without bounds.
This phenomenon is related to a classical paradox that we recall in the next section, explaining
the name “quantum 0−∞ law” on the same occasion
3 The Julie and Jack spending paradox
The Julie and Jack spending paradox is the following. Every month Julie and Jack receive 2
gold coins each and they spend 1. They label the coins gathered on month k as 2k − 1, 2k and
make a pile of coins by putting the new earnings 2k − 1, then 2k on top of what remains of
the previous earnings. However, they chose the coin they spend in a different way. Julie spends
the coin at the top of the pile, while Jack spends the coin at the bottom of the pile. Thus,
by the end of month k, they have received coins labeled 1, 2, . . . , 2k but Julie has spent coins
2, 4, . . . , 2k and kept coins 1, 3, . . . , 2k − 1 while Jack has spent coins 1, 2, . . . , k and kept coins
k + 1, k + 2, . . . , 2k. Of course, each has k gold coins of savings, but letting k go to ∞, we
see that by the end of times Julie has kept every coin with an odd label and is infinitely rich,
while Jack has kept . . . nothing and is ruined! We leave it to he reader to decide whether this
is a true paradox, or whether this simply teaches us something about the way set theory deals
with infinity and limits of sets.
4 M. Bauer
The Julie and Jack spending paradox is classical where classical can be given two meanings.
First it is probably old, it’s origin is obscure (to the author) and the above name is not canonical,
but it is very widely known. Second, we shall exhibit a relation with our original problem which,
we shall argue, is quantum, with a variant of the Julie and Jack spending paradox as a classical
counterpart.
To make contact with our problem, we introduce a third person, John, who earns 2 gold coins
each month and spends 1 just as Jack and Julie do. However, John is a gambler and chooses the
coin he spends at random (uniformly) among the remaining ones. The question arises whether
John is ruined by the end of times. As we shall soon explain, he is with probability 1!
This paradox is related to the so-called Ross–Littlewood paradox (see for instance [2, Exam-
ple 6a], which also deals with the probabilistic side), which is usually presented as a supertask.
A supertask is a task consisting of a denumerable infinity of sub-taks, but which is performed in
a finite amount of time. The name comes from computer science: one postulates the existence
of a computer that performs a task in 1/2 second, the next one in 1/4 seconds, the next in 1/8
seconds and so on, so that after one second a denumerable infinity of tasks, a supertask, has been
accomplished. The possibility of such computers is highly speculative, but their theory is inter-
esting and sheds some light on the limitations of standard computers. In the Ross–Littlewood
paradox, the supertask is the following: an urn contains two balls at time 0, after 1/2 seconds,
one ball is removed and two are added, after another 1/4 seconds one ball is removed and two
are added, after another 1/8 seconds one ball is removed and two are added, and so on. What
remains in the urn at time 1? If at each step the ball that is removed is one that was added
just the step before, one has the analog of Julie’s behavior, and at time 1 the urn contains an
infinite number of balls. However, if at each step the ball that is removed is chosen uniformly
at random among the all the balls that are in the urn, one has an analog of John’s behavior,
and the urn is empty at time 1.
We generalize the question, keeping the notations from Section 2. We assume that by the end
of month k, Julie, Jack and John have earned a total of nk gold coins that they labeled from 1
to nk, and spent mk of them, with the same protocol as above. Each builds a pile of coins by
putting the new earnings of month k, i.e., coins nk−1 + 1 then nk−1 + 2 up to nk on top of the
what remains of the previous ones. The description of the expenses of Julie, who takes the coins
she spends on the top of her pile, is a bit involved but irrelevant for our discussion and we leave
it to the reader. Jack has spent coins labeled from 1 to mk. By hypothesis, limk→∞mk = ∞ so
the fate of Jack is always complete ruin, independently of the growth of the sequence n. For the
expenses of John we give a recursive definition. By the end of month 1, John has spent the coins
labeled by a set S1 of size m1 chosen uniformly at random in J1, n1K. By the end of month 2, he
has spent the coins labeled by a set S2 ⊃ S1 of size m2 chosen uniformly at random in J1, n2K,
and so on. Choosing Sk+1 ⊃ Sk of size mk+1 uniformly at random in J1, nk+1K is the same as
choosing a subset of size (δm)k := mk+1 −mk uniformly at random in J1, nk+1K \ Sk which has
size nk+1 −mk. There are
(
nk+1−mk
mk+1−mk
)
such subsets, so each has probability
(
nk+1−mk
mk+1−mk
)−1
. This
description extends to a probability measure on infinite sequences S := (Sk)k∈N of subsets of N
with ∅ := S0 ⊂ S1 ⊂ S2 ⊂ · · · , where Sk ⊂ J1, nkK and #Sk = mk for k ∈ N. The generalized
question is whether John is ruined or not by the end of times. This is answered by the following
result:
Theorem 3.1. For k ∈ N∗ define rk :=
mk−mk−1
nk−mk−1
if nk −mk−1 > 0 and rk := 1 otherwise.
If the series
∑
k rk is divergent then ∪k∈NSk = N∗ with probability 1.
If the series
∑
k rk is convergent then N∗ \ ∪k∈NSk is infinite with probability 1.
We note the complete analogy with Theorem 2.1. The original paradox is when mk = k and
nk = 2k for k ∈ N, so rk = 1
k+1 and by the end of times, John is ruined with probability 1.
A Quantum 0−∞ Law 5
Note also that John’s fate has some kind of dual relationship with the Birthday paradox and
the Coupon collector problem.
If an orthonormal basis (vn)n≥1 in H is fixed and Fk := span(vn)n∈J1,nkK, there is another
simple way to put a measure, say P classical
F ,m , on m-Matryoshkas in F , namely by choosing S as
above and taking Ek = span(vn)n∈Sk
.
To explain why we consider the measure P classical
F ,m as classical whereas PF ,m
(
which we could
write P quantum
F ,m
)
is of quantum nature, we take the analogy with quantum computing. A classical
bit, or Cbit, can be in only two states, 0 and 1, and there are only two reversible operations
on a Cbit, doing nothing or permuting 0 and 1. A quantum bit, or Qbit, on the other hand,
is a unit vector in a two-dimensional Hilbert space, it can be expressed as a linear combination
of two basis vectors, but every unitary transformation represents a reversible operation. These
considerations generalize immediately to a collection of n > 1 Cbits or Qbits. For Cbits, there
are 2n states and a reversible transformation permutes those states. For Qbits the states span
a Hilbert space of dimension 2n and a reversible transformation is an unitary on that space.
In exactly the same sense, P classical
F ,m involves no superposition of states and randomness involves
uniform choices with respect to permutations, which are classical symmetries, whereas P quantum
F ,m
involves superposition of states, a quantum operation, and randomness involves uniform choices
with respect to unitary transformations, which are quantum symmetries.
However, the outcome is the same: the alternative for the status of ∪k∈NEk within H is the
same under the classical and the quantum constructions.
This classical/quantum distinction can also be rephrased as an illustration of the nuance
between combinatorics and geometric probability (sometimes also called continuous combina-
torics). Cases when the two yield strikingly similar results abound, and the similarity between
Theorems 2.1 and 3.1 is but another illustration.
We shall not give (reproduce?) a detailed proof of Theorem 3.1 because it is simpler than the
proof of Theorem 2.1 but follows closely the same lines.2 The heart of the similarity between the
classical and the quantum case is embodied in the following elementary fact: if n ∈ Jni−1+1, niK
for some i ∈ N∗, and k ≥ i, the probability that n does not belong to Sk is
∏k
i (1−rj). Just as its
quantum counterpart given in Corollary 4.5, this is proven by conditioning and recursion. The
identity says that certain covariances among random unit vectors are the same in the classical
and the quantum case, but this does not survive for higher correlations.
4 Main proofs
Fix an m-Matryoshka E and let v ∈ H. For k ∈ N denote by Uk(v) (or simply Uk when
there is no risk of confusion) the orthogonal projection of v on Ek (a finite-dimensional, hence
closed, subspace of H) and set Wk = Wk(v) := v −Uk(v), the projection of v on the orthogonal
complement of Ek in H. Set V0(v) := 0 and for k ∈ N∗ set Vk = Vk(v) := Uk(v) − Uk−1(v),
which by construction is the orthogonal projection of v on the orthogonal complement of Ek−1
in Ek. To summarize, for k ∈ N∗, v ∈ H splits as a sum of three mutually orthogonal vectors
v = Uk−1(v) + Vk(v) +Wk(v)
corresponding to the orthogonal sum decomposition
H = Ek−1 ⊕
(
E⊥
k−1 ∩ Ek
)
⊕ E⊥
k .
2In fact, the main differences in the proof are really due to the classical/quantum difference: classical discrete
0 − 1 random variables which in terms of subspaces mean orthogonal or included, become continuous, with
subspaces almost orthogonal or close to each other.
6 M. Bauer
Below is a picturesque representation of this decomposition, where each axis represents a vector
space:
Ek−1
E⊥
k−1 ∩ Ek
E⊥
k
Uk−1•
Vk•
Wk •
v
We collect a few simple facts in two lemmas:
Lemma 4.1. Fix an m-Matryoshka E and let v ∈ H.
(i) For k ∈ N∗, v = Wk(v) +
∑k
j=1 Vj(v) is an orthogonal decomposition.
(ii) The series
∑
j Vj(v) converges in H and so does the sequence Wk(v).
(iii) The limit limk→∞Wk(v) is the orthogonal projection of v on E
⊥
.
Proof. The decomposition of v follows immediately from the definitions, and its orthogonality
is a consequence of the fact that (Ek)k∈N is an increasing sequence of subspaces and Vj(v) belongs
by definition to the orthogonal complement of Ej−1 in Ej for j ∈ N∗, while Wk(v) is orthogonal
to Ek, establishing (i).
The convergence of the series
∑
j Vj(v) is an immediate consequence of the orthogonality of
the decomposition in (i) by a standard Hilbert space argument,3 establishing (ii).
Recall that E is the closure of ∪k∈NEk in H. If v ∈ E
⊥
, then Wk(v) = v for every k ∈ N.
If v ∈ ∪k∈NEk, then Wk(v) = 0 for large enough k. If v ∈ E, for every ε > 0 there is a j ∈ N,
and v′, v′′ with v′ ∈ Ej , v
′′ ∈ E⊥
j such that v = v′ + v′′ and ∥v′′∥ ≤ ε. Then ∥Wk(v
′′)∥ ≤ ε
for every k ∈ N while for k ≥ j, Wk(v
′) = 0 so ∥Wk(v)∥ ≤ ε for k ≥ j. This finishes the proof
of (iii). ■
Recall that
(
M, E , (Ek)k∈N, P
)
is the complete filtered probability space we work with, where
M := MF ,m denotes the set of m-Matryoshkas in F , Ek denotes, for k ∈ N, the σ-algebra
generated by k first components of a Matryoshka in M , E contains σ
(
∪k∈N Ek
)
, and P := PF ,m
is a complete E-probability measure defined recursively as explained briefly in Section 2.
Lemma 4.2. Let v ∈ H.
(i) As functions on (M, E) with values in H, Uk(v), Vk(v) and Wk(v) are Ek-measurable for
each k ∈ N.
(ii) The sequence Wk(v) converges almost surely to 0 if and only if
lim
k→∞
E
(
∥Wk(v)∥2
)
= 0.
3For the sake of completeness: as the summands Vj(v) are orthogonal, ∥
∑k
j=1 Vj(v)∥2 =
∑k
j=1∥Vj(v)∥2 =
∥v∥2 − ∥Wk(v)∥2 ≤ ∥v∥2, so that
∑
j≥1∥Vj(v)∥2 < +∞, i.e., the series with general term ∥Vj(v)∥2 is a Cauchy
sequence and then so is the series with general term Vj(v).
A Quantum 0−∞ Law 7
Proof. As Ek is the σ-algebra generated by the random subspaces E0, . . . , Ek of a Matryoshka,
Ek is Ek-measurable by definition, so the projection Uk(v) of v on Ek is Ek-measurable, and then
so are Wk(v) = v − Uk(v) and Vk(v) = Uk(v)− Uk−1(v) (or 0 if k = 0), establishing (i).
By (ii) in Lemma 4.1 and continuity of the norm, Q(v) := limk→∞∥Wk(v)∥2 exists pointwise.
In fact ∥Wk(v)∥2 decreases as a function of k, pointwise, so by dominated convergence (for
instance) limk→∞ E
(
∥Wk∥2
)
exists and equals E (Q). Hence if Wk(v) converges almost surely
to 0, equivalently if ∥Wk(v)∥2 converges almost surely to 0, then limk→∞ E
(
∥Wk(v)∥2
)
= 0,
proving the direct implication of (ii). Conversely, if limk→∞ E
(
∥Wk(v)∥2
)
= 0 then E (Q) = 0
which as Q ≥ 0 implies that Q = 0 almost surely, i.e., limk→∞Wk = 0 almost surely. ■
Recall that by assumption ∪i∈NFi is dense in H. So a subspace of H is dense if and only if
the closure of that subspace contains Fi for every i ∈ N∗. Thus Lemma 4.2 embodies a strategy
of proof of Theorem 2.1 based on the study of E
(
∥Wk(v)∥2
)
at large k for v ∈ ∪i∈NFi.
Suppose that v ∈ Fi for some i ∈ N∗. Then v ∈ Fk for k ≥ i, and then as the Ejs,
j ≤ k are subspaces of Fk, each component in the decompositions v = Uk(v) + Wk(v) =
Uk−1(v) + Vk(v) +Wk(v) = Wk(v) +
∑k
j=1 Vj(v) belong to Fk.
We shall use the following lemma:
Lemma 4.3. Let F be a finite-dimensional Hilbert space and E′ ⊂ F ′ ⊂ F be subspaces, whose
dimensions are denoted by m′, n′, n. Let v ∈ F ′. If m′ ≤ m ≤ n, let E be a random subspace
of F of dimension m containing E′ (chosen in a unitary invariant way) and let E⊥ be its
orthogonal complement in F . Write v = u + V + W as an orthogonal decomposition, where
u ∈ E′, V belongs to the orthogonal complement of E′ in E, and W ∈ E⊥.
Then E
(
∥W∥2
)
= ∥v − u∥2 n−m
n−m′ if n > m′ and 0 otherwise.
Proof. If m′ = n then m′ = m, E′ = E = F ′ = F so v = u, W = 0 and E
(
∥W∥2
)
= 0.
If n > m′, the result is simply the translation of the basic property of the Grassmannian
measure recalled above for the vector v − u, which lays down in the orthogonal complement
of E′ in F (of dimension n −m′) with W its projection on E⊥, a random subspace of dimen-
sion n−m. ■
We can now make use the Markov property of the measure on Matryoshkas. Recall that, for
k ∈ N∗, rk :=
mk−mk−1
nk−mk−1
if nk −mk−1 > 0 and rk := 1 otherwise.
Lemma 4.4. Let i ∈ N∗ and v ∈ Fi. For k ≥ i, E
(
∥Wk(v)∥2|Ek−1
)
= ∥Wk−1(v)∥2(1− rk).
Proof. Write v = Uk−1 + Vk +Wk. Then apriori Uk−1, Vk and Wk are random variables, but
conditionally on Ek−1 we are in position to apply the previous lemma and get E
(
∥Wk∥2|Ek−1
)
=
∥v − Uk−1∥2 nk−mk
nk−mk−1
if nk −mk−1 > 0 and 0 otherwise. By construction, v − Uk−1 = Wk−1 so
using the definition of rk gives the announced formula. ■
There is a useful corollary:
Corollary 4.5. Let i ∈ N∗ and v ∈ Fi. For k ≥ i,
E
(
∥Wk(v)∥2
)
= E
(
∥Wi−1(v)∥2
) k∏
j=i
(1− rj) ≤ ∥v∥2
k∏
j=i
(1− rj),
with equality if v belongs to the orthogonal complement of Fi−1 in Fi.
Proof. For k > 0, we know that v − Uk−1 = Wk−1 the formula in Lemma 4.4 yields
E
(
∥Wk∥2|Ek−1
)
= ∥Wk−1∥2(1 − rk), which can be used recursively by nesting of conditional
expectations to yield E
(
∥Wk∥2
)
= E
(
∥Wi−1∥2
)∏k
j=i(1− rj).
4
8 M. Bauer
The inequality follows from the orthogonality properties in the decompositions v = Ui−1 +
Vi + Wi, equivalently v − Ui−1 = Vi + Wi, which yield ∥v∥2 = ∥Ui−1∥2 + ∥Vi(v)∥2 + ∥Wi∥2 ≥
∥Vi∥2 + ∥Wi∥2 = ∥v − Ui−1∥2 = ∥Wi−1∥2.
Finally, if v belongs to the orthogonal complement of Fi−1 in Fi then v is orthogonal to
Ei−1 ⊂ Fi−1, i.e., Ui−1 = 0 and the inequality turns into an equality. ■
We can now prove the first assertion of Theorem 2.1.
Proof the first assertion of Theorem 2.1. By assumption, ∪i∈NFi is dense inH. So ∪k∈NEk
is dense in H if and only if it is dense in ∪i∈NFi. Equivalently, ∪k∈NEk is dense in H if and
only if any v in some Fi is in the closure of ∪k∈NEk, i.e., if and only if the sequence (Wk(v))k≥i
converges to 0 for any v in some Fi.
Using the linearity of the map v 7→ Wk(v), to prove that limk→∞Wk(v) = 0 for every v ∈
∪i∈NFi, it suffices to check this convergence when v ranges over a basis of the finite-dimensional
space Fi for each i. We fix an orthonormal basis (vn)n≥1 in H such that Fi := span(vn)n∈J1,niK.
Then Wk(vn)) is well-defined for large enough k, i.e., for k such that nk ≥ n: ∪k∈NEk is dense
in H if and only if limk→∞Wk(vn) = 0 for every n ∈ N∗.
By (ii) in Lemma 4.2 we know that limk→∞Wk(v) = 0 holds almost surely for a given v in
some Fi if and only if limk→∞ E
(
∥Wk(v)∥2
)
= 0. By the above remark, it is enough to let v
range over the countable set (vn)n≥1.
By Corollary 4.5, for i ∈ N∗, n ∈ Jni−1+1, niK and k ≥ i, E
(
∥Wk(vn)∥2
)
=
∏k
i (1− rj). Thus
∪k∈NEk is dense in H if and only if the infinite product “diverges” to 0, i.e., if limk→∞
∏k
i (1−rj)
= 0 for every i ∈ N∗. As rj ∈ [0, 1] for each j ∈ N∗, this is equivalent to the divergence of the
series
∑
j rj , concluding the proof. ■
For the second assertion, we shall make use of the following general but elementary Hilbert
space theory standard fact:
Lemma 4.6. Let G, H be two closed subspaces of H and let Φ, Ψ denote the corresponding
orthogonal projectors. Then the norm of Φ restricted to H and the norm of Ψ restricted to G
are equal.
Proof. Recall that ⟨ , ⟩ denotes the inner product in H. For every v ∈ G and w ∈ H, ⟨v,Φw⟩ =
⟨Ψv, w⟩ (because both equal ⟨v, w⟩). By the Cauchy–Schwarz inequality |⟨v,Φw⟩| = |⟨Ψv, w⟩| ≤
∥Ψv∥∥w∥. If δ ≥ 0 is such that ∥Ψv∥ ≤ δ∥v∥ then |⟨v,Φw⟩| ≤ δ∥v∥∥w∥. If ∥Ψv∥ ≤ δ∥v∥ holds
for every v ∈ G, take v = Φw to obtain ∥Φw∥2 = |⟨Φw,Φw⟩| ≤ δ∥Φw∥∥w∥ for every w ∈ H.
If Φw ̸= 0 divide by ∥Φw∥ to get ∥Φw∥ ≤ δ∥w∥, and this inequality is also true if Φw = 0. Thus
the norm of Φ restricted to H is at most the norm of Ψ restricted to G. By symmetry, they are
equal. ■
Lemma 4.7. Let (Hk)k∈N be an increasing sequence of subspaces of H. Then ∪k∈NHk is dense
in H if and only if for every finite-dimensional subspace G of H and every ε > 0 there is a k ∈ N
such that the orthogonal projection on G restricted to the orthogonal complement of Hk in H
has norm ≤ ε.
Proof. For k ∈ N, let H⊥
k denote the orthogonal complement of Hk in H and Ψk the orthogonal
projection on
(
H⊥
k
)⊥
, which is the closure of Hk. Set Ψ
⊥
k := Id−Ψk, the orthogonal projection
on H⊥
k .
The “if” part is the easiest. Fix v in H. We want to approach v arbitrarily closely by vectors
in ∪k∈NHk. The case v = 0 is trivial and we assume v ̸= 0. Let G = span(v). By hypothesis, for
every ε > 0 there is a k such that the orthogonal projection on G restricted H⊥
k has norm ≤ ε
2∥v∥ .
4Note that one cannot go further down because Wi−1 has no reason apriori to belong to Fi−1.
A Quantum 0−∞ Law 9
As Id−Ψk is the orthogonal projector onH⊥
k , by Lemma 4.6, ∥(Id−Ψk)v∥ ≤ ε
2∥v∥∥v∥ = ε
2 . AsHk
is dense in its closure, which contains Ψkv, one can find w ∈ Hk such that ∥Ψkv−w∥ ≤ ε
2 . By the
triangular inequality, ∥v −w∥ ≤ ε. To summarize, for an arbitrary v ̸= 0 in H and an arbitrary
ε > 0 there is a k ∈ N and a w ∈ Hk such that ∥v − w∥ ≤ ε. Hence ∪k∈NHk is dense in H.
For the “only” if part, let G be a finite-dimensional subspace of H, say of dimension d.
We may assume that d > 0. Pick an orthonormal basis (v1, . . . , vd). Let ε > 0. As ∪k∈NHk is
dense in H, for each α ∈ J1, dK there is kα ∈ N and wα ∈ Hkα such that ∥vα − wα∥ ≤ εd−1/2.
Set k := maxα kα. As (Hl)l∈N is an increasing sequence, wα ∈ Hk for each α ∈ J1, dK. The
orthogonal projection is a closest point so ∥Ψ⊥
k vα∥ = ∥(Id − Ψk)vα∥ ≤ ∥vα − wα∥ ≤ εd−1/2.
Write an arbitrary v ∈ G as v :=
∑
α λαvα for a (λα)α∈J1,dK ∈ Cd. Then
∥Ψ⊥
k v∥2 =
∑
α,β∈J1,dK
λαλβ
〈
Ψ⊥
k vα,Ψ
⊥
k vβ
〉
≤
∑
α,β∈J1,dK
|λα| |λβ|∥Ψ⊥
k vα∥|∥Ψ⊥
k vβ∥
≤ ε2d−1
( ∑
α∈J1,dK
|λα|
)2
≤ ε2d−1
(
d
∑
α∈J1,dK
|λα|2
)
= ε2∥v∥2,
using different avatars of the Cauchy–Schwarz inequality when going from the fist line to the
second, and then from the third line to the fourth. We have proven that for every ε > 0 there
is a k ∈ N such that the orthogonal projection on H⊥
k restricted to G has norm ≤ ε. Noting
that G is finite-dimensional, hence closed, we may apply Lemma 4.6 with H := H⊥
k , concluding
the proof. ■
If E is an m-Matryoshka, let GE , or simply G when no confusion is possible, denote the
orthogonal complement of ∪k∈NEk, always a closed subspace of H. Let ΦE , or simply Φ when
no confusion is possible, denote the orthogonal projector on GE .
Set A :=
{
E ∈ MF ,m such that GE is finite-dimensional
}
. Our aim is to show that A is a null
set, and as we work in a complete probability space, it is enough to show that it is contained in
a measurable set of measure 0.
For ε > 0 and k ∈ N, let Ak(ε) :=
{
E ∈ MF ,m such that ∥ΦEv∥ ≤ ε∥v∥ for every v ∈ F⊥
k
}
.
Lemma 4.8. Each Ak(ε) is measurable. For given ε the sets Ak(ε) increase with k ∈ N, and
A ⊂ ∪k∈NAk(ε) = limk→∞Ak(ε) for every ε > 0.
Proof. Fix ε > 0. If E is any m-Matryoshka, recall that, for v ∈ H and k ∈ N, Wk(v) is the
orthogonal projection of w on the orthogonal complement of Ek in H. By (i) in Lemma 4.2,
the map E ∈ (M, E) 7→ Wk(v) is Ek-measurable, hence E-measurable, for each k ∈ N and each
v ∈ H. The content of (iii) in Lemma 4.1 is that ΦEv = limk→∞Wk(v), so E ∈ (M, E) 7→ ΦE(v)
is measurable as a limit of measurable maps. Thus, for given v ∈ H, the set Av :=
{
E ∈
MF ,m such that ∥ΦEv∥ ≤ ε∥v∥
}
is measurable. The orthogonal projection ΦE is continuous
and H is separable. Hence, if H is any closed subspace of H, and (vn)n ∈ N a dense sequence
of vectors in H, the set
{
E ∈ MF ,m such that ∥ΦEv∥ ≤ ε∥v∥for every v ∈ H
}
is measurable
because it can be written as the countable intersection ∩n∈NAvn . Applying this result toH = F⊥
k
gives the measurability of each Ak(ε).
As (Fk)k∈N is an increasing sequence of subspaces of H, (F⊥
k )k∈N is a decreasing sequence of
subspaces of H, so the sets Ak(ε) increase with k ∈ N for fixed ε.
Recall that ∪k∈NFk is dense in H. Applying Lemma 4.7 to Hk = Fk for k ∈ N yields that A
is contained in ∪k∈NAk(ε), which is also limk→∞Ak(ε) by the previous argument. ■
10 M. Bauer
We can now turn to the heart of the matter.
Proof the second assertion of Theorem 2.1. Our assumption now is that
∑
k rk is conver-
gent. Hence, for large enough i, si :=
∏
k≥i(1− rk) > 0, and si ↗ 1 when i → ∞.
By assumption, limi→∞ ni = ∞. Thus S := {i ∈ N∗, ni −ni−1 > 0} is infinite. For each i ∈ S
we choose a non-zero vector vi ∈ F⊥
i−1∩Fi. By Corollary 4.5, E
(
∥Wk(vi)∥2
)
= ∥vi∥2
∏k
j=i(1−rj)
for k ≥ i. Taking the large k limit, E
(
∥Φvi∥2
)
= ∥vi∥2si for i ∈ S.
Fix ε > 0, k ∈ N and v ∈ F⊥
k . Write
E
(
∥Φv∥2
)
= E
(
∥Φv∥21Ak(ε)
)
+ E
(
∥Φv∥21Ak(ε)c
)
.
On Ak(ε), ∥Φv∥2 ≤ ε2∥v∥2 so
E
(
∥Φv∥21Ak(ε)
)
≤ ε2∥v∥2P (Ak(ε)).
As ∥Φv∥2 ≤ ∥v∥2 (Φ is an orthogonal projector!)
E
(
∥Φv∥21Ak(ε)c
)
≤ ∥v∥2P
(
Ak(ε)
c
)
.
Thus we have proven that, for ε > 0, k ∈ N and v ∈ F⊥
k ,
E
(
∥Φv∥2
)
≤ ε2∥v∥2P (Ak(ε)) + ∥v∥2(1− P (Ak(ε)).
For i ∈ S, i > k, it holds that vi ∈ F⊥
i−1 ∩ Fi ⊂ F⊥
k , so putting things together
∥vi∥2si = E
(
∥Φvi∥2
)
≤
(
ε2 − 1
)
∥vi∥2P (Ak(ε)) + ∥vi∥2.
Dividing by ∥vi∥2 (which is > 0 by construction) and rearranging yields(
1− ε2
)
P (Ak(ε)) ≤ 1− si,
and letting i → ∞ along S we obtain that P (Ak(ε)) = 0 for every ε < 1, so ∪k∈NAk(ε) =
limk→∞Ak(ε) is a null set for every ε < 1.
Thus A :=
{
E ∈ MF ,m such that GE is finite-dimensional
}
, a subset of ∪k∈NAk(ε) is also
a null set. To summarize, if
∑
k rk is convergent, the orthogonal GE of ∪k∈NEk is infinite-
dimensional with probability 1, concluding the proof. ■
5 Conclusions
The conclusions of Theorem 2.1 give another example in probability theory when in fact the
outcome of randomness is certitude, and depending on a condition (the convergence or divergence
of a series) only two extreme alternatives are possible.
However, the above mathematical considerations, neat as they are, should be seen as no more
than a preliminary to more interesting physical questions: once one is guaranteed that the full
Hilbert space is “covered”, some generic features of interest can in principle be studied.
To fix the ideas, let us assume that mk = k for k ∈ N, i.e., at each step a new ray in H is
added, constructing step by step a random orthonormal basis (modulo phases) (uk)k≥1, which is
to be compared to the original orthonormal basis (vl)l≥1 in H such that Fk := span(vl)l∈J1,nkK.
More precisely, we would like to understand the behavior of matrix elements in the u-basis
of operators which are local in the v-basis. To really make contact with, say, the eigenstate
thermalization hypothesis, we would also need a Hamiltonian, providing an increasing sequence
(ωk)k≥1 of energies such that such uk has energy ωk.
A Quantum 0−∞ Law 11
Up to now, we have said nothing neither about the choice of sequence of dimensions of the trial
spaces Fk, nor about the energies. As a conclusion, we propose a model of random Hamiltonians
in which both sequences are proportional.
We thus choose a random integer sequence nk =: dimFk, an energy scale ω and set ωk =
ω dimFk. The energy scale itself could be deterministic or random. It would be nice to have
some physical intuition for such a proportionality relation, which is admittedly artificial.5
Our proposal is based on the use of renewal processes. We fix a sequence (pn)n∈N∗ , where
each pn is ≥ 0 and
∑
n∈N∗ pn = 1, and use it to define a renewal process (Dk)k∈N: D0 = 0 and
the differences Dk−Dk−1, k ∈ N∗, are integer valued independent random variables taking value
n ∈ N∗ with probability pn. Taking dimFk := Dk for k ∈ N and carrying out the construction,
the strong laws of large numbers lead to criteria to ensure that (uk)k≥1 is indeed total (with
probability 1 with respect to the renewal process). This is the case obviously if (but not only if)
the distribution (pn)n∈N∗ has a mean i.e., if
∑
n∈N∗ npn < +∞: the criterion in Theorem 2.1
simply bowls down to the divergence of the harmonic series. We make the finite mean assumption
to proceed.
The main virtue of this construction is that it provides a complete statistical model which we
summarize: we take a renewal sequence (Dk)k∈N∗ , set Fk := span(vl)l∈J1,DkK, choose a random
Matryoshka in F , (Ek)k∈N∗ with dimEk = k and take as Hamiltonian
Ĥ =
∑
l∈N∗
ωl|l⟩⟨l|,
where |l⟩ = ul, i.e.,
∑
l∈J1,kK |l⟩⟨l| is the orthogonal projector on Ek and ωk = ωDk.
This model neglects all correlations between levels but for those which come from nearest level
(the gap) correlation, as described by the common distribution of the dimFk−dimFk−1, k ∈ N∗.
An interesting limit occurs when ω goes to 0 but the gap itself has a limiting distribution.
We have insisted enough on the artificial features of this construction, so let us conclude on
a more optimistic note. First the model at least exhibits correlations between the statistical
properties of the eigenvalues and of the eigenstates. Second, as the distribution of the gap is our
choice, one can in principle study the influence of level repulsion, expected in generic quantum
system, and compare it with Poissonian statistics expected in integrable systems for instance.
Third, the renewal structure ensures that the density of eigenvalues is asymptotically flat, so
there is no need to straighten it; we could even take the first gap using the stationary measure,
leading to a flat density everywhere.
We plan to return to the study of the statistical properties of this model in forthcoming
works.
A The measure on subspaces
In this appendix we recall in an informal way a few salient features of µm
F , the unitary invariant
probability measure on the set of m-dimensional subspaces of F , a finite-dimensional Hilbert
space of dimension n. We do not address uniqueness of µm
F .
Our starting point is the Haar probability measure, µF on U(F ),6 the group of unitary
transformations of F . We fix an orthonormal basis (v1, . . . , vn) of F .
For m ∈ J1, nK we consider the map Πm : U(F ) → G(F,m) (recall this is the set of m-
dimensional subspaces of F ) defined by Πm(g) := span(gvl)l∈J1,mK. If U(F ) and G(F,m) are
5The relationship with the more physical feature that the Hamiltonian should be somewhat local (though not
strictly diagonal) in the original orthonormal basis (vl)l≥1 is obscure to say the least. Interpreted in terms of
trial wave functions, the proportionality suggests that the larger the gap between two states, the more one should
enlarge the trial space, a principle which looks questionable as well.
6We apologize for a clash in notation with the main text, where U denotes a random variable.
12 M. Bauer
endowed with there usual smooth manifold structure and the associated Borel σ-algebra, this
map is smooth (hence measurable). So it induces a probability measure µF ◦ Π−1
m on G(F,m):
the measure of a measurable subset B is of G(F,m) is defined to be µF
(
Π−1
m (B)
)
. This measure
on G(F,m) is unitary invariant because the Haar measure on U(F ) is, and we denote if by µm
F .
The group U(F ) contains the (linear extension of the) permutation group acting on the basis
(v1, . . . , vn). So even without knowing about uniqueness, it is clear by symmetry that using an
arbitrary m-subset of the basis (v1, . . . , vn) instead of (v1, . . . , vm) to define Πm would lead to
the same measure.
If f : G(F,m) → R is µm
F -integrable, the definition of µm
F leads to∫
E∈G(F,m)
f(E) dµm
F (E) :=
∫
g∈U(F )
f(Πm(g)) dµF (g),
and we apply this formula for special classes of functions f . Fix v ∈ F and let hv(E) be the
orthogonal projection of v on the m-dimensional subspace E of F . For g ∈ U(F ) we compute
that
∥hv(Πm(g))∥2 = |⟨gv1, v⟩|2 + · · ·+ |⟨gvm, v⟩|2.
Thus ∫
E∈G(F,m)
∥hv(E)∥2 dµm
F (E) :=
∫
g∈U(F )
∥hv(Πm(g))∥2 dµF (g)
=
∑
l∈J1,mK
∫
g∈U(F )
|⟨gvl, v⟩|2 dµF (g).
By symmetry, it is plain that the m summands on the right-hand side are equal, and for m = n
they sum to ∥v∥2. Hence, for each l ∈ J1, nK,∫
g∈U(F )
|⟨gvl, v⟩|2 dµF (g) = ∥v∥2 1
n
.
Consequently∫
E∈G(F,m)
∥hv(E)∥2 dµm
F (E) = ∥v∥2m
n
.
Now
∫
E∈G(F,m) f(E) dµm
F (E) is simply the expectation of (the random variable) f under the
probability measure µm
F and hv(E) is what we denoted by U in the main text, so we retrieve
the formula E
(
∥U∥2
)
= ∥v∥2mn .
Acknowledgments
I thank Denis Bernard for discussions about the eigenstate thermalization hypothesis that finally
led me to this study, and Philippe Biane for discussions on some technical points.
References
[1] Deutsch J.M., Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991), 2046–2049.
[2] Ross S., A first course in probability, 10th ed., Pearson, 2019.
[3] Srednicki M., Chaos and quantum thermalization, Phys. Rev. E 50 (1994), 888–901, arXiv:cond-
mat/9403051.
https://doi.org/10.1103/PhysRevA.43.2046
https://doi.org/10.1103/PhysRevE.50.888
https://arxiv.org/abs/cond-mat/9403051
https://arxiv.org/abs/cond-mat/9403051
1 Introduction
2 Definitions and main result
3 The Julie and Jack spending paradox
4 Main proofs
5 Conclusions
A The measure on subspaces
References
|
| id | nasplib_isofts_kiev_ua-123456789-211533 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T15:20:52Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bauer, Michel 2026-01-05T12:28:31Z 2022 A Quantum 0−∞ Law. Michel Bauer. SIGMA 18 (2022), 012, 12 pages 1815-0659 2020 Mathematics Subject Classification: 60J05; 81-10 arXiv:2110.07456 https://nasplib.isofts.kiev.ua/handle/123456789/211533 https://doi.org/10.3842/SIGMA.2022.012 We give conditions under which a sequence of randomly chosen orthogonal subspaces of a separable Hilbert space generates the whole space. I thank Denis Bernard for discussions about the eigenstate thermalization hypothesis that finally led me to this study, and Philippe Biane for discussions on some technical points. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Quantum 0−∞ Law Article published earlier |
| spellingShingle | A Quantum 0−∞ Law Bauer, Michel |
| title | A Quantum 0−∞ Law |
| title_full | A Quantum 0−∞ Law |
| title_fullStr | A Quantum 0−∞ Law |
| title_full_unstemmed | A Quantum 0−∞ Law |
| title_short | A Quantum 0−∞ Law |
| title_sort | quantum 0−∞ law |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211533 |
| work_keys_str_mv | AT bauermichel aquantum0law AT bauermichel quantum0law |