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A Bargmann representation of the quantum absorption refrigerator

M. W. AlMasri, Mohamed Ridza Wahiddin

Published May 6, 2021• 0 citations
OpenAlex

Abstract

Instead of representing quantum states by vectors in the Hilbert space one could represent them as a convergent monomial in the space of holomorphic functions with Gaussian measure using the Bargmann representation of the lowering and raising operators. This strategy simplifies the calculations since for a given quantum system the coordinate representation of any quantum state is always more complicated than its corresponding expression in the Bargmann space. We solve the quantum absorption refrigerator analytically in term of the phase space coordinates $z_{i}$ and their partial derivatives. We finally discuss the computational complexity of the holomorphic representation and compare it with the computational complexity of the standard operator method and prove the efficiency of the holomorphic representation in computing some tasks. Our treatment is applicable in all quantum heat engines and refrigerators.

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Instead of representing quantum states by vectors in the Hilbert space one could represent them as a convergent monomial in the space of holomorphic functions with Gaussian measure using the Bargmann representation of the lowering and raising operators. This strategy simplifies the calculations since for a given quantum system the coordinate representation of any quantum state is always more complicated than its corresponding expression in the Bargmann space. We solve the quantum absorption refrigerator analytically in term of the phase space coordinates ziz_{i}zi​ and their partial derivatives. We finally discuss the computational complexity of the holomorphic representation and compare it with the computational complexity of the standard operator method and prove the efficiency of the holomorphic representation in computing some tasks. Our treatment is applicable in all quantum heat engines and refrigerators.
We study bosonic quantum computations using the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which not only gives a canonical description of bosonic quantum computing using basic elements of complex analysis but also provides a unifying picture which delineates the boundary between discrete- and continuous-variable quantum information theory. Using this representation, we show that the evolution of a single bosonic mode under a Gaussian Hamiltonian can be described as an integrable dynamical system of classical Calogero-Moser particles corresponding to the zeros of the holomorphic function, together with a conformal evolution of Gaussian parameters. We explain that the Calogero-Moser dynamics is due to unique features of bosonic Hilbert spaces such as squeezing. We then generalize the properties of this holomorphic representation to the multimode case, deriving a non-Gaussian hierarchy of quantum states and relating entanglement to factorization properties of holomorphic functions. Finally, we apply this formalism to discrete- and continuous- variable quantum measurements and obtain a classification of subuniversal models that are generalizations of Boson Sampling and Gaussian quantum computing.
We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since {znn!}\{\frac{z^{n}}{\sqrt{n!}}\}{n!​zn​} form an orthonormal basis, one could in principle expand any complex function f(z)f(z)f(z) in powers of zzz and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of zzz carry no physical information only their non-negative exponents nnn are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.
We introduce a model of computing on holomorphic functions, which captures bosonic quantum computing through the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which not only gives a canonical definition of bosonic quantum computing using basic elements of complex analysis but also provides a unifying picture which delineates the boundary between discrete- and continuous-variable quantum information theory. Using this representation, we show that the evolution of a single bosonic mode under a Gaussian Hamiltonian can be described as an integrable dynamical system of classical Calogero-Moser particles corresponding to the zeros of the holomorphic function, together with a conformal evolution of Gaussian parameters. We explain that the Calogero-Moser dynamics is due to unique features of bosonic Hilbert spaces such as squeezing. We then generalize the properties of this holomorphic representation to the multimode case, deriving a non-Gaussian hierarchy of quantum states and relating entanglement to factorization properties of holomorphic functions. Finally, we apply this formalism to discrete- and continuous-variable quantum measurements and introduce formally the model of Holomorphic Quantum Computing. Non-adaptive circuits in this model allow for a classification of subuniversal models that are generalisations of Boson Sampling and Gaussian quantum computing, while adaptive circuits capture existing universal bosonic quantum computing models. Our results leave open the possibility that continuous-variable quantum computations may outperform their discrete-variable counterpart.
Spin chain Hamiltonians can be written in terms of complex differential operators using the Bargmann representation of the Jordan-Schwinger map. In this case, the eigenfunctions are expressed as the product of orthonormal monomials of the phase-space coordinates zi=Qi+iPiz_{i}=Q_{i}+i P_{i}zi​=Qi​+iPi​ in the complex plane. Furthermore, the series constructed from each phase-space coordinate converges uniformly in any compact domain of the complex plane. Formulating spin chains with respect to the phase-space coordinates helps in discussing their classical limit and in the calculations of quasi-probability distributions.
In the Bargmann representation of quantum mechanics, physical states are mapped into entire functions of a complex variable z *, whereas the creation and annihilation operators and play the role of multiplication and differentiation with respect to z *, respectively. In this paper we propose an alternative representation of quantum states, conjugate to the Bargmann representation, where the roles of and are reversed, much like the roles of the position and momentum operators in their respective representations. We derive expressions for the inner product that maintain the usual notion of distance between states in the Hilbert space. Applications to simple systems and to the calculation of semiclassical propagators are presented.
In the Bargmann representation of quantum mechanics, physical states are mapped into entire functions of a complex variable z∗, whereas the creation and annihilation operators â † and a ̂ play the role of multiplication and differentiation with respect to z∗, respectively. In this paper we propose an alternative representation of quantum states, conjugate to the Bargmann representation, where the roles of â † and a ̂ are reversed, much like the roles of the position and momentum operators in their respective representations. We derive expressions for the inner product that maintain the usual notion of distance between states in the Hilbert space. Applications to simple systems and to the calculation of semiclassical propagators are presented. 1.