# Are superpositions of an infinite number of states realizable?

I first encounted this idea in Constructing finite dimensional codes with optical continuous variables where it mentions "superpositions of an infinite number of infinitely squeezed states" in the introduction.

I was able to find this question which states in regards to free fermions that:

you can argue that the wavefunctions are superpositions of an infinite number of space "basis states"

I was also able to find references to the idea in Quantum Cosmology And Baby Universes where it is stated:

One can therefore regard the singular $$K$$ eigenstates as being superpostions of an infinite number of regular harmonic oscillator solutions.

Additionally, in this paper (PDF) on Quantum Theory of the Electric Field in the Coherent States section (page 6) it states:

Coherent states are superpositions of an infinite number of Fock states

Are objects that are in a superpostion of an infinite number of states used in practice or is it strictly a theoretical and/or mathematical concept?

Related:

• Are you looking for quantum computing experiments that run continuous-variables algorithms (i.e. that make use of the infinite superposition for some purpose)? Or just some systems that can be described by an infinite superposition of states? Because if it's the later, every particles can be described by an infinite superposition (for instance of positions and momentum states). If you're looking for a precise experiment, a laser produces coherent states, which are an infinite superposition of photon numbers. In qubit-based QC, we're usually interested only in a small part of the Hilbert space – Arthur Pesah May 16 '20 at 18:45
• @ArthurPesah I am interested in both, however it seems the "real question" is the former. – user820789 May 16 '20 at 20:18

First things first, theoretically infinite dimensional Hilbert space is natural in quantum mechanics due to the relation $$[x,p]=ih$$. This produces the Heisenberg algebra. This algebra is solvable and then invoking the Lie-Kolchin theorem, if the dimension of representation is finite, then it is necessarily one dimensional which is not useful as it would make everything commute and nothing would happen. So we consider the other alternative which is that our Hilbert space is infinite dimensional.

Fair enough, the mathematical ground is solid. But in reality, since we use only a finite space and work for finitely small time scales, we generally approximate the operators in our infinite Hilbert space to a finite subspace of it. Classic examples are this

The spectrum and eigenstates of any field quadrature operator restricted to a finite number N of photons are studied, in terms of the Hermite polynomials. By (naturally) defining approximate eigenstates, which represent highly localized wavefunctions with up to N photons, one can arrive at an appropriate notion of limit for the spectrum of the quadrature as N goes to infinity, in the sense that the limit coincides with the spectrum of the infinite-dimensional quadrature operator. In particular, this notion allows the spectra of truncated phase operators to tend to the complete unit circle, as one would expect. A regular structure for the zeros of the Christoffel-Darboux kernel is also shown.

and this

We present several new techniques for approximating spectra of linear operators (not necessarily bounded) on an infinite-dimensional, separable Hilbert space. Our approach is to take well-known techniques from finite-dimensional matrix analysis and show how they can be generalized to an infinite-dimensional setting to provide approximations of spectra of elements in a large class of operators. We conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n-pseudospectrum and argue how that can be used as an approximation to the spectrum. .

But does that mean that infinite Hilbert spaces are relegated to the corners of theoretical physics? The answer is, not surprisingly, no. The experimenters did find application of this, and is technically known as continuous-variable quantum information.

Quoting Wikipedia :

One approach to implementing continuous-variable quantum information protocols in the laboratory is through the techniques of quantum optics. By modeling each mode of the electromagnetic field as a quantum harmonic oscillator with its associated creation and annihilation operators, one defines a canonically conjugate pair of variables for each mode, the so-called "quadratures", which play the role of position and momentum observables. These observables establish a phase space on which Wigner quasiprobability distributions can be defined. Quantum measurements on such a system can be performed using homodyne and heterodyne detectors.

But remember what we had mentioned in the beginning, about why theoretically our Hilbert space needs to be infinite dimensional? Well, even that exact formulation has been proposed as the basis of a quantum computer.

Another proposal is to modify the ion-trap quantum computer: instead of storing a single qubit in the internal energy levels of an ion, one could in principle use the position and momentum of the ion as continuous quantum variables.

And all these have been formalised. For example, one can look here:

The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

and here:

Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometr y of phase space to protect against errors that shift the value s of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.

and finally here:

We consider the quantum processor based on a chain of trapped ions to propose an architecture wherein the motional degrees of freedom of trapped ions (position and momentum) could be exploited as the computational Hilbert space. We adopt a continuous-variables approach to develop a toolbox of quantum operations to manipulate one or two vibrational modes at a time. Together with the intrinsic non-linearity of the qubit degree of freedom, employed to mediate the interaction between modes, arbitrary manipulation and readout of the ionic wave function could be achieved.

Thus, there are several proposals for using infinite dimensional Hilbert spaces, and the same is used in optics sometimes. These are just a few ways in which this can be utilised, and more are expected to open as research progresses.

• I think CV quantum computing is about the weakest example where infinite dimensional spaces are really required -- after all, these are all separable Hilbert spaces and can thus be well approximated by finite dimensional spaces (though this might not be a very nice description). Generally, QFTs should make a much stronger case for the requirement for infinite dimensional spaces. – Norbert Schuch May 26 '20 at 19:37
• By QFT in this context , I take you mean lattice gauge theory or something to that effort? And yes, I agree that all of these spaces are separable, but OP really wanted to see if there was any use, and I just pointed out that there are. That they can be finitely approximated is a separate issue. – Yuzuriha Inori May 26 '20 at 20:20
• Thanks for just an incredibly thorough answer! – user820789 May 26 '20 at 20:34
• @YuzurihaInori Well, no lattice, in particular. -- Again, I think it depends what you (or the OP) mean by "in practice". I think that there is a significant number of theoretical physicists who think that anything can be done with finite dimensional spaces. And CVs for a finite number of modes is a prime example: If you truncate to a finite number of bosons, you pretty much get everything what you want. Another one of these examples are spin chains: Most things can be captured very well by sequences of finite lattices for $N\to\infty$. – Norbert Schuch May 26 '20 at 20:50

For any problem which is described by an infinite dimensional Hilbert space, you can regard any state as a superposition of an infinite number of states. The only real question is thus whether infinite-dimensional Hilbert space are "used in practice".

• Are they used in practice? – user820789 May 24 '20 at 2:35
• What do you even mean by "use in practice"? – Norbert Schuch May 24 '20 at 12:23