The single-particle Hilbert space. it is a subset of the Hilbert space theory

The single-particle Hilbert space is the Hilbert space of all states that may be classified as one-particle states; it is a subset of the Hilbert space of a full theory containing states with $N = 1$ .
Introduction from Hilbert space to Fock space and vice versa in the operator representation and the system evolution. Section 4 presents specific applications, whereas Section 5 provides guidelines for numerical implementation. Finally, Section 6 closes the paper with some concluding remarks.

The wave function describing the quantum state of a collection of identical bosons is symmetric under the exchange of any two particles. It is thus naturally described in the Hilbert space of symmetric functions, which is a subspace of the tensor product of single-particle states. The wave function for a collection of identical fermions is antisymmetric, which means that it must change sign when we exchange any two particles. These wave functions are elements of a Hilbert space of antisymmetric functions, which is another subspace of the tensor product of single-particle states. Indistinguishability thus introduces correlations into the wave function, even for non-interacting particles. This feature prompted attempts to perform quantum information processing using only the statistical properties of quantum mechanics.

Space of states starting from the single-particle Hilbert space. As a side effect, the usual introduction of the Rock space may somehow give the impression that the Hilbert space description may be left behind.

The representation of operators in the Fock space is not straightforward since the indexing of the basis set states, as well as the interpretation of the number states in terms of particle states, is usually not trivial. A well-known example is thermionic operators on a lattice system [8]: anti-commutation rules must be taken into account, and additional phases arise in the components of hopping operators between sites under periodic boundary conditions. Additionally, one often encounters operators that are symmetrized or anti-symmetrized versions of a distinguishable-particle operator, e.g., the kinetic term in Hubbard models [8, 9, 10]. Here, one may assume that those operators contain both boronic and thermionic features, which should then be discriminated (separated) using a suitable transformation.

For all the above reasons, it is often more transparent to employ the Hilbert space description and study the dynamics of a physical system there [12], as done in some recent works on quantum walks of identical particles [13, 14, 15, 16]. On the other hand, Fock numbers appear quite naturally in the description of systems of identical particles. The question arises of how and what we may go from Fock space to Hilbert space and vice versa with minimal effort.

The primary goal of this paper is to provide a gentle introduction to the transformation rules between the descriptions of states and operators in the two spaces. We start smoothly by recalling the construction of the Fock space for systems of indistinguishable particles and then offer a set of recipes, guidelines, and advice for those interested in going back and forth between the two descriptions. We devote some attention to the two-particle case, which already contains most of the interesting features related to indistinguishability, and briefly discuss how to handle the two different representations in numerical implementations. The material presented in this paper is intended to be a concise reference about the different representations employed in many-body physics. It aims at being helpful to students and researchers working with systems of identical particles, ranging from photons in a black-body cavity to interacting electrons in a lattice and from neutrons in a neutron star to helium atoms in a superfluid.

The paper is structured as follows. In Section 2, we recall a few basic notions about indistinguishable particles and the construction of the Fock space. In Section 3, we illustrate how to change the description from Hilbert space to Rock space, and vice versa, in the operator representation and the system evolution. Section 4 presents specific applications, whereas Section 5 provides guidelines for numerical implementation. Finally, Section 6 closes the paper with some concluding remarks.

How can one efficiently transform operators between Hilbert space and Fock space in practical applications? What are the specific challenges faced when implementing numerical algorithms for indistinguishable particles in different representations? In what ways do the statistical properties of quantum systems exploited in quantum information processing differ between bosons and fermions?

Many identical particles, e.g., those described by Fermi-Hubbard and Bose-Hubbard models, are conveniently depicted in the Fock space. However, to evaluate specific observables or study the system dynamics, it is often more effective to employ the Hilbert-space description. Moving effectively between descriptions is thus a desirable feature, especially when a numerical approach is needed. Here we recall the construction of the Fock space for systems of indistinguishable particles and then present a set of recipes and advice for students and researchers who need to commute between the two descriptions. The two-particle case is discussed in some detail, and a few guidelines for numerical implementations are given.
The wave function describing the quantum state of a collection of identical bosons is symmetric under the exchange of any two particles. It is thus naturally described in the Hilbert space of symmetric functions, which is a subspace of the tensor product of single-particle states. The wave function for a collection of identical fermions is antisymmetric, so it must change sign when we exchange any two particles. These wave functions are elements of a Hilbert space of antisymmetric functions, which is another subspace of the tensor product of single-particle states. Indistinguishability thus introduces correlations into the wave function, even for non-interacting particles. This feature prompted attempts to perform quantum information processing using only the statistical properties of quantum systems.
The above facts are usually summarized by saying that, for quantum particles with definite statistics, not all available states are permitted, even for free particles. In graduate-level physics courses, second quantization and Fock space [3, 4, 5, 6, 7] are presented as the natural framework for accounting for this constraint. Indeed, the Fock space is a crucial tool for describing systems with a variable, or unknown, number of identical particles. In particular, the Fock space allows one to build the space of states starting from the single-particle Hilbert space. As a side effect, the usual introduction of the Fock space may somehow give the impression that the Hilbert space description may be left behind.
On the other hand, the representation of operators in the Fock space is not straightforward since the indexing of the basis set states, as well as the interpretation of the number states in terms of particle states, is usually not trivial. A well-known example is fermionic operators on a lattice system [8]: anti-commutation rules must be taken into account, and additional phases arise in the components of hopping operators between sites under periodic boundary conditions. Additionally, one often encounters operators that are symmetrized or antisymmetrized versions of a distinguishable-particle operator, e.g., the kinetic term in Hubbard models [8, 9, 10]. Here, one may assume that those operators contain both bosonic and fermionic features, which should then be discriminated (separated) using a suitable transformation.
For all the above reasons, it is often more transparent to employ the Hilbert space description and study the dynamics of a physical system there [12], as done in some recent works on quantum walks of identical particles [13, 14, 15, 16]. On the other hand, Fock number states arise quite naturally in the description of systems of identical particles. Thus, the question arises of how, and whether, we may go from Fock space to Hilbert space and vice versa with minimal effort.
The main goal of this paper is to provide a gentle introduction to the transformation rules between the descriptions of states and operators in the two spaces. We start smoothly by recalling the construction of the Fock space for systems of indistinguishable particles and then offer a set of recipes, guidelines, and advice for those interested in going back and forth between the two descriptions. We devote some attention to the two-particle case, which already contains most of the interesting features related to indistinguishability, and briefly discuss how to handle the two different representations in numerical implementations. The material presented in this paper is intended to be a concise reference about the different representations employed in many-body physics. It aims at being helpful to students and researchers working with systems of identical particles, ranging from photons in a black-body cavity to interacting electrons in a lattice to neutrons in a neutron star to helium atoms in a superfluid.