Entangled Phase States via Quantum Beam Splitter

Abstract: We study the entanglement effect of beam splitter on the temporally stable phase states. Specifically, we consider the eigenstates (phase states) of an unitary phase operator resulting from the polar decomposition of ladder operators of generalized Weyl--Heisenberg algebras possessing finite dimensional representation space. The linear entropy that measures the degree of entanglement at the output of the beam splitter is analytically obtained. We find that the entanglement is not only strongly dependent on the Hilbert space dimension but also quite related to strength the parameter ensuring the temporal stability of the phase states. Finally, we discuss the evolution of the entangled phase states.

• The paper establishes a method for constructing temporally stable phase states as eigenstates of a unitary phase operator in a finite-dimensional Hilbert space.
• It demonstrates that a quantum beam splitter transforms these states into entangled two-mode outputs, with linear entropy quantifying entanglement based on both the system dimension and a phase stability parameter.
• Numerical analysis reveals that maximum entanglement occurs at a 50:50 beam splitter, highlighting tunable entanglement dynamics through temporal evolution.

Entanglement of Temporally Stable Phase States via Quantum Beam Splitter

Overview

This paper provides a rigorous analysis of the entanglement properties of temporally stable phase states constructed as eigenstates of a unitary phase operator. These states are defined within the context of a finite-dimensional generalized Weyl--Heisenberg algebra. By analyzing the action of a beam splitter on these phase states, the study delineates the dependence of output entanglement, measured via linear entropy, on both the Hilbert space dimension and an intrinsic phase stability parameter $\varphi$. Numerical explorations highlight distinctive entanglement behaviors across varying system dimensions and parameter regimes.

Generalized Weyl--Heisenberg Algebra: Finite-Dimensional Structure

The foundation of the study is a generalized Weyl--Heisenberg algebra generated by $\{a^+, a^-, N\}$ with commutation relations $[N, a^\pm] = \pm a^\pm$ and $[a^-, a^+] = G(N)$, where $G(N)$ is Hermitian. When $G(N) = \mathbb{I}$, this reduces to the usual oscillator algebra. To guarantee a finite-dimensional ($2s+1$) representation, the authors enforce a structure function $F(N)$ such that $F(2s+1) = 0$. This leads to nilpotency conditions $(a^-)^{2s+1} = (a^+)^{2s+1} = 0$, ensuring a compact Hilbert space essential for defining the unitary phase operator.

Different instantiations of this algebra capture standard, nonlinear, and truncated oscillator dynamics, generalizing prior constructions such as the Pegg-Barnett formalism. Explicit Hamiltonians, of the form $\{a^+, a^-, N\}$0, include as limits the truncated harmonic oscillator and higher-order finite oscillators.

Construction of Temporally Stable Phase States

Central to the analysis is the constructive definition of phase states as eigenstates of a unitary operator $\{a^+, a^-, N\}$1, obtained via polar decomposition $\{a^+, a^-, N\}$2. These "phase states" are

$\{a^+, a^-, N\}$3

where $\{a^+, a^-, N\}$4, $\{a^+, a^-, N\}$5, and $\{a^+, a^-, N\}$6 is a phase parameter ensuring temporal stability. The evolution is given by $\{a^+, a^-, N\}$7, demonstrating a one-parameter family of phase-stable states.

These phase states exhibit key properties: orthonormality for fixed $\{a^+, a^-, N\}$8, closure, and equiprobability across the Fock occupation number basis. The parameter $\{a^+, a^-, N\}$9 imparts both a record of the algebraic structure and controls the temporal behavior and mutual unbiasedness with computational basis states. Notably, for $[N, a^\pm] = \pm a^\pm$0, these states devolve to Pegg-Barnett states, eradicating the dynamical distinction between different algebraic settings.

Beam Splitter Transformation and Entanglement Quantification

The paper conducts a detailed analysis of how a quantum beam splitter transforms these finite-dimensional phase states. For a beam splitter with reflection and transmission coefficients $[N, a^\pm] = \pm a^\pm$1 and $[N, a^\pm] = \pm a^\pm$2, the transformation applied to input $[N, a^\pm] = \pm a^\pm$3 yields an entangled two-mode state. The output density matrix is constructed, and the reduced single-mode density matrix is explicitly computed. The degree of entanglement is quantified using the linear entropy $[N, a^\pm] = \pm a^\pm$4.

The paper presents analytic expressions for the linear entropy in terms of $[N, a^\pm] = \pm a^\pm$5, $[N, a^\pm] = \pm a^\pm$6, and $[N, a^\pm] = \pm a^\pm$7. One notable result is the independence of entropy from the index $[N, a^\pm] = \pm a^\pm$8. The dependence of $[N, a^\pm] = \pm a^\pm$9 on $[a^-, a^+] = G(N)$0 is nontrivial for $[a^-, a^+] = G(N)$1, and the symmetry properties within the expressions for linear entropy are systematically delineated.

Numerical Results and Entanglement Behaviour

Systematic numerical evaluation, focusing on the structure function $[a^-, a^+] = G(N)$2, reveals:

• Maximum Entanglement at 50:50 Beam Splitter: Across all finite-dimensional Hilbert spaces analyzed, maximal output entanglement occurs when $[a^-, a^+] = G(N)$3.
• Dimension-Dependent Entanglement: For two-level systems ($[a^-, a^+] = G(N)$4), entanglement after the beam splitter is independent of $[a^-, a^+] = G(N)$5. For higher dimensions ($[a^-, a^+] = G(N)$6), $[a^-, a^+] = G(N)$7 displays nontrivial, often Gaussian-like, dependence on $[a^-, a^+] = G(N)$8, with the location and width of maximum entanglement shifting with $[a^-, a^+] = G(N)$9.
• Enhanced Entanglement for Larger Hilbert Spaces: The degree of entanglement at fixed $G(N)$0 and $G(N)$1 increases strictly with the system dimension $G(N)$2, but saturates more slowly for larger $G(N)$3.
• Temporal Evolution as Entanglement Oscillation: Since $G(N)$4 parametrizes time evolution, the $G(N)$5-dependence of $G(N)$6 can be interpreted as a temporal dynamics of entanglement for phase states—showing regimes of enhancement, suppression, and recurrences as a function of this parameter.

Implications and Prospects

The study establishes that the generalized phase states built from finite-dimensional algebras exhibit enhanced and tunable entanglement properties under standard optical beam splitting operations. This is not only a function of system size but also intricately dependent on the phase stability parameter $G(N)$7, which encodes both spectral and dynamical details of the underlying algebra.

From a practical standpoint, these results highlight the role of engineered algebraic structures and state preparation in controlling entanglement generation—suggesting new classes of nonclassical, finite-dimensional states for quantum information and quantum optics protocols. Theoretically, this framework extends the landscape of mutually unbiased bases and temporally robust quantum states.

Of particular note is the limitation for infinite-dimensional (canonical oscillator) Hilbert spaces, where unitary phase operators are not well-defined, precluding the direct construction of these temporally stable phase states. As such, the work motivates further exploration of truncated or algebraically deformed oscillator structures for practical entanglement engineering.

Conclusion

The paper delivers a comprehensive and explicit framework for analyzing the generation of entanglement in temporally stable phase states via beam splitter operations. By generalizing the Weyl--Heisenberg algebra to finite dimensions and constructing intrinsic phase-stable states, the work shows that entanglement can be maximized and manipulated through both system dimension and phase parameters. These insights may inform the design of quantum optical systems and generalized coherent state constructions for quantum information processing applications.