Some quantum-to-classical asymptotics

We study asymptotic expansions of Gaussian integrals of analytic functionals on infinite-dimensional spaces (Hilbert and nuclear Frechet). We obtain an asymptotic equality coupling the Gaussian integral and the trace of the composition of scaling of the covariation operator of a Gaussian measure and the second (Frechet) derivative of a functional. In this way we couple classical average (given by an infinite-dimensional Gaussian integral) and quantum average (given by the von Neumann trace formula). We can interpret this mathematical construction as a procedure of “dequantization” of quantum mechanics. We represent quantum mechanics as an asymptotic projection of classical statistical mechanics with infinite-dimensional phase space. This space can be represented as the space of classical fields, so quantum mechanics is represented as a projection of “prequantum classical statistical field theory.”We study asymptotic expansions of Gaussian integrals of analytic functionals on infinite-dimensional spaces (Hilbert and nuclear Frechet). We obtain an asymptotic equality coupling the Gaussian integral and the trace of the composition of scaling of the covariation operator of a Gaussian measure and the second (Frechet) derivative of a functional. In this way we couple classical average (given by an infinite-dimensional Gaussian integral) and quantum average (given by the von Neumann trace formula). We can interpret this mathematical construction as a procedure of “dequantization” of quantum mechanics. We represent quantum mechanics as an asymptotic projection of classical statistical mechanics with infinite-dimensional phase space. This space can be represented as the space of classical fields, so quantum mechanics is represented as a projection of “prequantum classical statistical field theory.”

Journal of Mathematical PhysicsJan 1, 200721 citations

Stein’s Lemma for Classical-Quantum Channels

Mario Berta, Christoph Hirche, Eneet Kaur +1 more

It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We show that, for the discrimination of classical-quantum channels, adaptive strategies do not lead to an asymptotic advantage. As our main result, this establishes Stein's lemma for classical-quantum channels. Our proofs are based on the concept of amortized distinguishability of channels, which we analyse using entropy inequalities.It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We show that, for the discrimination of classical-quantum channels, adaptive strategies do not lead to an asymptotic advantage. As our main result, this establishes Stein's lemma for classical-quantum channels. Our proofs are based on the concept of amortized distinguishability of channels, which we analyse using entropy inequalities.

IEEE International Symposium on Information Theory (ISIT)Jul 1, 20195 citations

Two‐State Quantum Asymptotics

Michael Berry

Annals of the New York Academy of SciencesApr 1, 199543 citations

Classical Nonclassical Asymptotics

Evert Jan Post

Boston studies in the philosophy of scienceJan 1, 19950 citations

Asymptotic localization of symbol correspondences for spin systems

P. A. S. Alcântara, Pedro de M. Rios

Quantum or classical mechanical systems symmetric under $SU(2)$ are called spin systems. A $SU(2)$ -equivariant map from $(n+1)$ -square matrices to functions on the $2$ -sphere, satisfying some basic properties, is called a spin- $j$ symbol correspondence ( $n=2j\in\mathbb N$ ). Given a spin- $j$ symbol correspondence, the matrix algebra induces a twisted $j$ -algebra of symbols. In this paper, we establish a new, more intuitive criterion for when the Poisson algebra of smooth functions on the $2$ -sphere emerges asymptotically ( $n\to\infty$ ) from the sequence of twisted $j$ -algebras of symbols. This new, more geometric criterion, which in many cases is equivalent to the numerical criterion obtained in [Rios&Straume], is now given in terms of a classical (asymptotic) localization of the symbols of projectors (quantum pure states). For some important kinds of symbol correspondence sequences, classical localization of all projector-symbols is equivalent to asymptotic emergence of the Poisson algebra. But in general, such a classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. Finally, we obtain some relations between asymptotic localization of a symbol correspondence sequence and its quantizations of the classical spin system.Quantum or classical mechanical systems symmetric under $SU(2)$ are called spin systems. A $SU(2)$-equivariant map from $(n+1)$-square matrices to functions on the $2$-sphere, satisfying some basic properties, is called a spin-$j$ symbol correspondence ($n=2j\in\mathbb N$). Given a spin-$j$ symbol correspondence, the matrix algebra induces a twisted $j$-algebra of symbols. In this paper, we establish a new, more intuitive criterion for when the Poisson algebra of smooth functions on the $2$-sphere emerges asymptotically ($n\to\infty$) from the sequence of twisted $j$-algebras of symbols. This new, more geometric criterion, which in many cases is equivalent to the numerical criterion obtained in [Rios&Straume], is now given in terms of a classical (asymptotic) localization of the symbols of projectors (quantum pure states). For some important kinds of symbol correspondence sequences, classical localization of all projector-symbols is equivalent to asymptotic emergence of the Poisson algebra. But in general, such a classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. Finally, we obtain some relations between asymptotic localization of a symbol correspondence sequence and its quantizations of the classical spin system.

Apr 8, 20200 citations

Quantum or classical mechanical systems symmetric under

SU(2)

are called spin systems. A

SU(2)

-equivariant map from

(n+1)

-square matrices to functions on the

2

-sphere, satisfying some basic properties, is called a spin-

j

symbol correspondence (

n=2j\in\mathbb N

). Given a spin-

j

symbol correspondence, the matrix algebra induces a twisted

j

-algebra of symbols. In this paper, we establish a new, more intuitive criterion for when the Poisson algebra of smooth functions on the

2

-sphere emerges asymptotically (

n\to\infty

) from the sequence of twisted

j

-algebras of symbols. This new, more geometric criterion, which in many cases is equivalent to the numerical criterion obtained in [Rios&Straume], is now given in terms of a classical (asymptotic) localization of the symbols of projectors (quantum pure states). For some important kinds of symbol correspondence sequences, classical localization of all projector-symbols is equivalent to asymptotic emergence of the Poisson algebra. But in general, such a classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. Finally, we obtain some relations between asymptotic localization of a symbol correspondence sequence and its quantizations of the classical spin system.