1. The Outline
In a stark contrast to its classical counterpart, br quantum mechanics builds the element of uncertainty into its notion of state. Indeed, while a definite entity per se, becomes chancy upon probing. How do we usefully characterize in terms of its uncertainties?
Questions of this type are as old as quantum mechanics and its Copenhagen interpretation [
1]. To address the issue in that vein, consider a prototypical quantum measurement experiment on a system with states in
N-dimensional Hilbert space. In particular, the system is repeatedly prepared in state
and the observable associated with a single non-degenerate Hermitian operator
is measured, producing a sequence
Here is the eigensystem of , and the outcome of the ℓ-th trial, namely the state into which collapsed and the measured value.
The uncertainty of
with respect to its probing by
refers to the indeterminacy associated with the stochastic nature of the sequence
. This feature is commonly characterized by some form of statistical spread in the encountered eigenvalues, where “spread” has the meaning of separation/distance on the spectrum. We will refer to such characteristics of
as
metric uncertainties or
-uncertainties. (In addition to these “spectral”
-uncertainties, one can also use the metric on the Hilbert space to define “state”
-uncertainties. If the inner-product metric is used, the latter formulas are simple and elegant due to the fact that all pairs of distinct orthonormal states are equidistant.) Standard deviation is a popular quantifier of this type since it enters the Heisenberg relations [
1,
2].
In contrast, our aim here is to characterize quantum uncertainty by its amount/measure. More precisely, we seek characteristics conveying how many distinct effectively appear in . The larger such effective number, the larger the uncertainty. We will call such characteristics measure uncertainties or -uncertainties.
The sequence (1) encodes probabilities
of encountering
in a trial. These then determine the value of a given
-uncertainty. According to quantum mechanics, the experimental analysis will yield
. With this being independent of the eigenvalues,
-uncertainties are functions of the measurement basis only. On the other hand,
-uncertainties depend on the entire eigensystem
since any metric quantifier is a function of
. Denoting the two types of quantifiers as
and
, respectively, we have
where
fully represents
. In other words,
-uncertainties characterize
in relation to measurement operators while
-uncertainties in relation to measurement bases.
To achieve its intended meaning, a properly constructed
-uncertainty has to be realized by a function that admits the interpretation as a number of states from
contained in
. The theory of such objects has been developed in Ref. [
3]. In fact, the identity-counting functions
constructed there directly correspond to
-uncertainties.
Theoretical structure leading to these quantifiers formalizes the notion of effective number
assigned to a collection of
N objects endowed with probability weights
. The associated analysis simplifies when
is equivalently treated as a function of counting weights
where
. The concept is then represented by the set
of all effective number functions (ENFs)
, each realizing one consistent effective counting scheme. The effective number theory of Ref. [
3] defines
axiomatically and then finds it explicitly (Theorem A1). Thus, all ENFs, and hence all valid
-uncertainties, are known. A short overview of the ideas underlying the effective number theory and of its main results is given in
Appendix A.
A consequential structural feature of
(Theorem A2) implies the existence of the minimal ENF which is central to our present purposes. More precisely, the function
belongs to
and
for all
C and all
from
. (It is easy to see that if the function with this property exists, it has to be unique. Note also that the function
is universal in that it does not depend on
N.) In other words, there exists a sharp notion of “minimal amount” for collections of objects with probability weights, realized by
. Hence, there is a sharp notion of minimal quantum
-uncertainty. In explicit terms,
[U] Let , be the counting weights associated with quantum state and the Hilbert space basis . The μ-uncertainty of with respect to is at least states.
Given [U], we will refer to as the intrinsic -uncertainty of with respect to . Indeed, this “uncertainty amount” is inherent to the state since it cannot be lowered or removed by the optional change of a quantifier. Its existence reflects the innate nature of uncertainty in quantum mechanics.
The novelty of the above arises largely due to the inclusion of additivity among the defining properties of ENFs [
3]. This step is dictated by the intended measure-like nature of effective numbers. In fact, each
extends the counting measure from ordinary sets to those endowed with probability measures. In
Section 3, we will construct quantifiers that play this role for the Jordan content (Lebesgue measure of “regular domains” in
). Together with the discrete case, this will cover most situations arising in quantum physics. As an elementary example,
-uncertainties of a Schrödinger particle with respect to the position basis are effective volumes. In the spinless case, our analysis implies the intrinsic value
Here is the wave function of a particle contained in the region of finite volume V. The existence of intrinsic -uncertainty implies in this case that quantum particle cannot be associated with the effective volume smaller than .
Quantifying the indeterminacy is sometimes approached via entropy. It is thus of theoretical interest to understand the relations between the measure-like and the entropy-like angles on the concept. Here, we start such discussion by conveying
-uncertainty in an entropy-like manner, which may find uses in the context of field-theoretic and many-body systems. We proceed in analogy with the original Boltzmann approach in classical statistical mechanics [
4], where
N accessible states of a priori equal probability generate the entropy
. In our case,
N quantum states with arbitrary probabilities effectively represent
“accessible” ones, leading to
as a Boltzmann-like characteristic we refer to as the
-entropy. The effective number theory then implies the existence of minimal
-entropy associated with state
and basis
, namely
where
C has the meaning specified in [U
]. The motivation for
is to express
-uncertainty as the number of degrees of freedom effectively “active” in the measurement (
Section 4).
It is natural to ask in this context whether our measure approach can be applied to quantum entanglement. This is indeed the case and the relevant construction is given in
Section 5. It is based on a new elementary notion of
quantum effective number (Definition 3), which is a basis-independent characteristic of a density matrix, expressing the number of states effectively comprising a mixture. In the effective number methodology this exemplifies a context in which it is necessary to take into account that counted objects may “share content”, or be generally correlated in a way affecting the total. The resulting measure-based notion of entanglement (
-
entanglement) may provide a useful alternative characterization of entangled states. In addition, we use quantum effective numbers to obtain the quantum version of
-entropy, which is the analog of von Neumann entropy [
5].
Before presenting the details of the above outline, we remark that the results of Ref. [
3], extended here, may also find fruitful applications in the general area of localization, both in the original Anderson [
6,
7] and many-body guises [
8]. Characterizing states by their intrinsic
-uncertainty with respect to the position basis invokes a somewhat unusual perspective on this vast topic. This and other applications of
-uncertainty and quantum effective numbers will be discussed in dedicated forthcoming publications. In
Appendix B, we provide tutorial examples of
-uncertainty in simple situations, both in the discrete and continuum case.
2. -Uncertainty
In this and the next section we develop the theory of
-uncertainty in detail. To that end, we emphasize at the outset that our aim here is not to question the merits of standard metric approach in the analyses of quantum experiments. Rather, our intent is to point out that there exists a complementary, measure outlook on quantum uncertainty that offers new conceptual insights and a different type of practical use. The example of the former is a surprising existence of uniquely-defined intrinsic uncertainty. The latter can be illustrated by the utility of
-uncertainty in quantum computation. Indeed, the cost of realizing a quantum algorithm is proportional to the effective number of possible collapsed states in its measurement step [
3]. Hence,
-uncertainty can be used in the associated efficiency analysis.
We start by analyzing quantum uncertainty in a general setting. In fact, the discussion of
Section 1 needs to be extended in two ways. The first one involves the inclusion of probing by multiple and possibly degenerate commuting operators. The second one is concerned with the form of
-uncertainty in situations that require taking the dimension of Hilbert space to infinity, e.g., when removing the regularization cutoffs.
Thus, rather than the prototypical situation of
Section 1, consider the experiment involving
D commuting operators assembled into a
D-tuple
. It is implicitly understood that the eigensubspace decompositions associated with individual operators are distinct so that the redundant setups, such as
, are avoided. Since
does not necessarily represent a complete system, each combination
of measured individual eigenvalues specifies the subspace
of the underlying
N-dimensional Hilbert space
. Collectively, this leads to a decomposition of
into
M orthogonal subspaces
The set
of effective number functions [
3] specifies all consistent
-uncertainties associated with the above experimental setup. Specifically, we have the following definition.
Definition 1. Let and let be its (non-normalized) projection into subspace from orthogonal decomposition (6) specified by . Let further , , be the collection of associated counting weights, and . We refer to as the μ-uncertainty of with respect to and the effective number function .
If
is a complete set of commuting operators, then
and the description in terms of a basis (
), utilized in
Section 1 becomes convenient. The arguments resulting in [U
] also lead to the intrinsic
-uncertainty limits in this general setting. In particular,
Albeit starting from the experiment specified by probing operators, measure uncertainty only depends on the associated orthogonal decomposition of the Hilbert space. On the other hand,
-uncertainties are fully
-dependent. To highlight this, consider
involving individual operators of the same physical dimension. Let
be the
D-tuple of eigenvalues associated with subspace
, and
the probability of
collapsing into it upon probing. Expressing the
-uncertainty as a standard deviation leads to
where
is a metric of choice on
. Thus, while
for any
-uncertainty, we have
in case of
-uncertainties.
The above makes it clear that -uncertainties can be viewed as abstract entities which, given a wide variety of possible decompositions , define a rich collection of characteristics describing . They reflect an inherently quantum aspect of the state and have a sharp physical interpretation in terms of quantum experiments. The effective number theory, and [U] in particular, imply that it is meaningful to view with varying as a complete description of in terms of its -uncertainties. It is not known at this time whether a similarly definite structure exists in case of -uncertainties as well.
The above native setup for the theory of
-uncertainty (finite-dimensional Hilbert space) affords direct applications to many interesting systems, such as those of qbits realizing a quantum computer. However, a transition to infinite case is frequently necessary. Since
generically diverges in the process, we will work with the ratio of the effective number to its nominal counterpart, namely the relative
-uncertainty. More explicitly, consider a regularization procedure involving a sequence of Hilbert spaces
of growing dimension
. At the
k-th step of the process, the target state
is represented by the vector
, and the target Hilbert space decomposition
by the collection
of
subspaces. The relative
-uncertainty of
with respect to
and
is
where
is the counting vector associated with
and
. Unlike
, the number of subspaces
does not necessarily grow unbounded in the
limit. In fact, the virtue of
is that it can be used universally: it is applicable to quantum state of arbitrary nature as long as it can be defined via a regularization involving finite-dimensional Hilbert spaces.
3. Continuous Spectra and Effective Uncertainty Volumes
For the purposes of this section, it is convenient to label the subspaces of the Hilbert space decomposition by eigenvalue D-tuples of some fixed generating them as its eigenspaces. Thus, is the subspace of represented by , namely a point in the “spectrum” of . The decomposition itself will be denoted as .
Upon measurements entailed by the operators in , state undergoes a collapse described by the pair . While we associated -uncertainty with the abundance of distinct in repeated experiments, it is also the abundance of and individually because their pairing is one to one. Focusing on , if the spectra turn continuous upon regularization removal, -uncertainty of the target state should thus be expressible in terms of a measure on . In this section, such general expression will be derived.
We use the regularization setup described in connection with the relative
-uncertainty formula (8), and assume that the spectra of all operators involved in
become continuous in their target
. Consider arbitrary
specified by its counting function
, so that
. The corresponding relative
-uncertainty at
k-th regularization step involves the expression
where
with
the projection of
into subspace
, i.e., the probability associated with eigenvalue
D-tuple
. On the RHS, we introduced a hypercubic grid in
with spacing
, and grouped individual counts by the elementary hypercube the associated
falls into (
is a hypercube centered at
). Note that the
j-sum receives non-zero contributions only from hypercubes containing
.
The target relative
-uncertainty for continuous spectra corresponds to taking
followed by
limit of expression (9). Given that each counting function
is continuous, and assuming that
is chosen so that the association between
and
in target
becomes expressible via probability density
(see below), this limiting procedure is equivalently carried out with
where
is the number of
contained in
. To cast this into a continuous form, we introduce the probability density
of encountering
in the experiment involving
and
, as well as the probability density
of
-eigenvalue
D-tuples
Since the sum in the numerator of the latter is
we have from (10) that
where the spectral support
of
is defined by
. The integrand vanishes at
since each
is bounded, leading to the restriction of the integral to
. Note that we have distinguished the generic variable
parametrizing entire
from the spectral variable
labeling the actual continuum of subspaces. Via standard manipulations, one can (formally) write
with
the projection of
into
.
Several comments regarding the formula (12) are important to make.
- (i)
Recall that in discrete case we have identified
-uncertainties with effective number functions
. However, in the continuum, where effective number generically loses its direct meaning (diverges), this correspondence becomes facilitated by counting functions
of Theorem 1 in Ref. [
3]. Thus, in full detail we have
but the last dependence will remain implicit in what follows.
- (ii)
Since relative
-uncertainty depends on the Hilbert space decomposition
but not on a particular
associated with it, formula (12) should reflect this invariance. To see it, consider relabeling the subspaces
as
, where
is a one-to-one differentiable map. This defines
D-tuple of new operators
, and the associated transformed probability densities
and
. The change of variables then confirms
- (iii)
How does the additivity, carefully enforced in the regularization process, explicitly translate into Equation (12)? Consider the partition of the spectral support
into subregions
and
, thus specifying both the decomposition
of the underlying Hilbert space, as well as the operators
,
acting on them, i.e.,
. Moreover, spectral probability densities
on
descend from
via
Extending the concatenation notation of Ref [
3] to this continuous case, we have equivalently
From (12) it then directly follows that
where ⊞ was also extended to the elements of mutually orthogonal Hilbert spaces in an obvious manner (
), and
is the decomposition of
associated with
. Relation (16) is precisely the one for composing two fractions of distinct amounts into that of a combined amount (
), and is an equivalent representation of additivity. In terms of probability distributions involved, this reads
- (iv)
[U] and (12) lead to the notion of minimal -uncertainty in the context of continuous spectra. In particular, with the above definitions and notation in place, we have
[U
]
Let be a D-tuple of Hermitean operators on with continuous spectra, and , , the associated spectral characteristics. There exists a minimal relative μ-uncertainty of states from with respect to , assigned bywhere is the probability density of obtaining in -measurements of .
- (v)
Important special case of formula (12) arises for uniform
. Among other things, this setting applies to several relevant operators, such as those of position and momentum in quantum mechanics. Thus, let
occupy a finite volume
in
. A unique feature of uniform
is that the effective fraction of states, quantified by
, also expresses the effective fraction of spectral volume in this case. Indeed, uniformity at the regularized level implies that distinct subspaces represent non-overlapping elementary volumes, and the ratio
becomes the effective volume fraction in the continuum limit. Thus, it is meaningful in this case to define
-uncertainty (rather than relative
-uncertainty) and interpret it as the
effective spectral volume. In particular, from (12) we obtain
Note that the -uncertainty of Schrödinger particle with respect to the position basis Equation (4) is a special case of this general relationship.
- (vi)
The results of this section entail a notable mathematical corollary. Thus, leaving the realm of quantum mechanics for the moment, consider
with well-defined non-zero Jordan content (ordinary volume), i.e.,
. (Speaking of Jordan content simply means that ∫ is understood to denote the Riemann integral.) Can we extend the meaning of Jordan content so that, in addition to
itself, the volume is assigned to any pair
, where
is a continuous probability distribution on
? The effective number theory [
3] provides a positive answer to this question, and Equation (12) the corresponding prescription. Indeed, introducing a Riemann partition of
and the associated discrete probability distribution descended from
, effective volume fraction associated with counting function
can be evaluated. Adopting any sequence of Riemann refinements producing
V, one obtains a result that can be read off directly from Equation (12). The conversion from
to effective volume
then leads to the analogue of (19), namely
Here, the first equality specifies all consistent effective volume assignments (labeled by
). The inequality, valid for all
P and all
, expresses the existence of
minimal effective volume quantifier specified by
and guaranteed to play this role by Theorem 2 of Ref. [
3].
- (vii)
Finally, consider the case involving both continuous and discrete operators. Thus, let the
D-tuple
contain
operators
with continuous spectra upon regularization removal. Expression (12) for relative
-uncertainty then generalizes into
Here, and , are associated with whose components are discrete target eigenvalues. Note that , and similarly for .
5. Quantum Effective Numbers, Quantum -Entropy and -Entanglement
Similarly to naturals, effective numbers were constructed to characterize collections of objects acting as autonomous wholes, i.e., not sharing “parts” with one other. This aspect is generic in situations where counting is normally considered to make sense. Thus, we were justified to use effective numbers to count the states of orthonormal basis, or the subspaces from the orthogonal decomposition of the Hilbert space. Incidentally, these autonomous objects play a crucial role in quantum measurement process, and thus the uncertainty.
When the boundaries between objects become fuzzy and/or their contents can be shared in some manner, counting has to be modified, if at all possible, to accommodate the commonality. In the quantum context, situations of this type arise when inquiring about the state content of a density matrix. Here, we do not mean the abundance of elements from arbitrary fixed basis. (The answer to that question, namely
where
, represents the
-uncertainty of
with respect to basis
, and involves only a direct application of effective counting). Rather, we are interested in a basis-independent characteristic specifying the number of independent states effectively participating in the mixture. Thus, consider a density matrix
, namely
where the number
J of distinct states
from
N-dimensional Hilbert space is arbitrary. Recalling that each effective number function
is uniquely associated with its counting function
so that
(Theorem 1 of Ref. [
3]), we define
quantum effective numbers associated with
as follows.
Definition 3. Let be density matrix and a counting function. Thenwill be referred to as the quantum effective number of with respect to . The rationale for the above construct is quite clear. States
in definition (29) cannot be directly counted since they are not necessarily orthogonal. However, equivalently expressing
in terms of its eigenstates gives the latter the role of autonomous components to which effective counting applies. From the mathematical standpoint, the connection between effective numbers and their quantum counterparts is analogous to that of Shannon [
9] and von Neumann entropies [
5]. To avoid confusion, we emphasize that
is not a
-uncertainty and we do not refer to it such. Rather, it is an useful object that allows us to define quantum
-entropy and
-entanglement (see below).
Several comments regarding are important to make.
- (i)
Quantum effective numbers can be introduced as a well-motivated extension of ordinary effective numbers, as done here, or as an axiomatic construct of its own. Without going into details, we note that the key property of exact additivity, required to be satisfied by , concerns combining density matrices defined in mutually orthogonal Hilbert subspaces. Definition 3 manifestly accommodates this feature.
- (ii)
The notion of minimal effective number applies also to its quantum version. In particular, it follows from Theorem 2 of Ref. [
3] that
Hence, the same reasons that give its absolute meaning in case of ordinary effective counting, apply to in the quantum case.
- (iii)
Quantum effective numbers allow us to express a degree of entanglement between parts of the system as an effective number of states. Thus, given a bipartite system specified by
, state
, and the associated density matrix
, we define
and refer to
as
-entanglement of
with respect to partition specified by
A and the counting function
. Note that
by virtue of the Schmidt decomposition argument. The notion of minimal
-entanglement
follows.
- (iv)
The quantum
-entropy, namely the
-entropy associated with a density matrix, is
where
is the minimal entropy quantifier. Similarly to its classical counterpart, the utility of
is mainly envisioned in many body and field theory applications. The concept of
-entanglement can be equivalently based on quantum
-entropy in analogy with the standard quantum information approach to entanglement using von Neumann entropy. In the same way, the general entanglement-related construct of quantum mutual information has a counterpart in the measure-based notion of mutual “state content”, which can also be equivalently treated in terms of quantum
-entropy (33).
6. The Summary
In this work, we proposed and analyzed the approach to quantum uncertainty that characterizes it as an effective total of possible measurement outcomes (
-uncertainty). Unlike in the case of conventional spectral metric approach (
-uncertainty), the mathematical theory governing
-uncertainties exists. It is the effective number theory of Ref. [
3], which implies that there is an amount
of
-uncertainty, associated with each quantum state and type of measurement, that cannot be reduced by using a different
-uncertainty quantifier. Hence, this minimal amount is intrinsic to a quantum situation at hand. Statements [U
], [U] and [U
] convey this in various generic contexts of interest.
The conclusion that uncertainty is encoded by quantum formalism at such a basic level via the universal quantifier
is interesting conceptually. Moreover, its unique explicit form
is useful from a practical standpoint. In that regard, it is also useful to recall the proposal to characterize state
by all
, i.e., by the number of basis states from
that
effectively resides in, for all bases
[
3]. The present discussion casts that into describing
by all of its intrinsic
-uncertainties. This viewpoint gives uncertainty a privileged role in the description of quantum state indeed.
While obvious from our discussion, it may be worth pointing out that Heisenberg relations and statements of minimal -uncertainty ([U], [U] and [U]) offer very different kinds of insight into the nature of quantum uncertainty. Indeed, while Heisenberg relation infers certain minimum which is associated with a pair of incompatible operators and universal with respect to the state involved in simplest cases, the intrinsic -uncertainties are the minima associated with each state individually and universal with respect to the operators sharing the same basis. Clearly, more can be said along these lines, both qualitatively and quantitatively, once -uncertainties become utilized more fully.
A significant portion of the present work entailed deriving -uncertainty expressions in situations where the measurement setup entails an orthogonal decomposition of the Hilbert space labeled by continuous spectral parameters. In particular, Formulas (12) and (21) are the results of the regularization cutoff removals, needed in such cases. The latter represents the most general form of -uncertainty, applicable to arbitrary Hilbert space and any of its decompositions specified by a set of commuting Hermitean operators.
It is worth emphasizing that the treatment of uncertainty as a measure became possible by virtue of extending ordinary counting (counting measure) into effective counting (effective counting measure) [
3]. Our treatment of continuous spectra here similarly corresponds to extending the notion of Jordan content in
(ordinary volume) to effective Jordan content (effective volume), as expressed by Equation (20). The resulting approach may have uses in applied mathematics, e.g., as a suitable way to define the effective support of a function.
The concept of effective numbers naturally leads to the auxiliary notion of
-entropy. In the context of quantum states, its motivation mainly relates to convenience in dealing with exponentially growing Hilbert spaces of many-body physics. Working with entropy translates into considering the equivalent number of degrees of freedom and their density, Equations (26) and (27). This approach may be useful in the analysis of thermalization (see, e.g., [
8] for a relevant review).
In order to construct quantum -entropy and a measure-based approach to quantum entanglement, we have shown how to use effective numbers to analyze the state content of density matrices (Definition 3). A suitable extension is necessary since the states specifying the matrix may not be independent (mutually orthogonal). As is obvious from its intended meaning and the resulting formula (30), this quantum effective number is a basis-independent concept. Among other things, it allows us to express quantum entanglement as the effective number of states “generated” in the Hilbert space of one bipartite component due to the influence of the other (32). Substantially more can be said about the ensuing approach to entanglement and to quantum entropy (33), with a dedicated account forthcoming.