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arXiv:2509.07882

Effective approach to open systems with probability currents and the Grothendieck formalism

Apostol Vourdas

Published Sep 9, 2025• 0 citations
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Abstract

An effective approach to open systems and irreversible phenomena is presented, where an open system $Σ(d)$ with $d$-dimensional Hilbert space, is a subsystem of a larger isolated system $Σ(2d)$ (the `full universe') with $2d$-dimensional Hilbert space. A family of Bargmann-like representations (called $z$-Bargmann representations) introduces naturally the larger space. The $z$-Bargmann representations are defined through semi-unitary matrices (which are a coherent states formalism in disguise). The `openness' of the system is quantified with the probability current that flows from the system to the external world. The Grothendieck quantity ${\cal Q}$ is shown to be related to the probability current, and is used as a figure of merit for the `openness' of a system. ${\cal Q}$ is expressed in terms of `rescaling transformations' which change not only the phase but also the absolute value of the wavefunction, and are intimately linked to irreversible phenomena (e.g., damping/amplification). It is shown that unitary transformations in the isolated system $Σ(2d)$ (full universe), reduce to rescaling transformations when projected to its open subsystem $Σ(d)$. The values of the Grothendieck ${\cal Q}$ for various quantum states in an open system, are compared with those for their counterpart states in an isolated system.

DOI
10.48550/arxiv.2509.07882

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An effective approach to open systems and irreversible phenomena is presented, where an open system Σ(d)Σ(d)Σ(d) with ddd-dimensional Hilbert space, is a subsystem of a larger isolated system Σ(2d)Σ(2d)Σ(2d) (the full universe') with $2d$-dimensional Hilbert space. A family of Bargmann-like representations (called $z$-Bargmann representations) introduces naturally the larger space. The $z$-Bargmann representations are defined through semi-unitary matrices (which are a coherent states formalism in disguise). The openness' of the system is quantified with the probability current that flows from the system to the external world. The Grothendieck quantity Q{\cal Q}Q is shown to be related to the probability current, and is used as a figure of merit for the openness' of a system. ${\cal Q}$ is expressed in terms of rescaling transformations' which change not only the phase but also the absolute value of the wavefunction, and are intimately linked to irreversible phenomena (e.g., damping/amplification). It is shown that unitary transformations in the isolated system Σ(2d)Σ(2d)Σ(2d) (full universe), reduce to rescaling transformations when projected to its open subsystem Σ(d)Σ(d)Σ(d). The values of the Grothendieck Q{\cal Q}Q for various quantum states in an open system, are compared with those for their counterpart states in an isolated system.
An effective approach to open systems and irreversible phenomena is presented, where an open system Σ ( d ) with d -dimensional Hilbert space, is a subsystem of a larger isolated system Σ ( 2 d ) (the ‘full universe’) with 2 d -dimensional Hilbert space. A family of Bargmann-like representations (called z -Bargmann representations) introduces naturally the larger space. The z -Bargmann representations are defined through semi-unitary matrices (which are a coherent states formalism in disguise). The ‘openness’ of the system is quantified with the probability current that flows from the system to the external world. The Grothendieck quantity Q is shown to be related to the probability current, and is used as a figure of merit for the ‘openness’ of a system. Q is expressed in terms of ‘rescaling transformations’ which change not only the phase but also the absolute value of the wavefunction, and are intimately linked to irreversible phenomena (e.g., damping/amplification). It is shown that unitary transformations in the isolated system Σ ( 2 d ) (full universe), reduce to rescaling transformations when projected to its open subsystem Σ ( d ) . The values of the Grothendieck Q for various quantum states in an open system, are compared with those for their counterpart states in an isolated system.
The Grothedieck bound formalism is studied using “rescaling transformations,” in the context of a single quantum system. The rescaling transformations enlarge the set of unitary transformations (which apply to isolated systems), with transformations that change not only the phase but also the absolute value of the wavefunction, and can be linked to irreversible phenomena (e.g., quantum tunneling, damping and amplification, etc). A special case of rescaling transformations are the dequantisation transformations, which map a Hilbert space formalism into a formalism of scalars. The Grothendieck formalism considers a “classical” quadratic form C(θ) which takes values less than 1, and the corresponding “quantum” quadratic form Q(θ) which takes values greater than 1, up to the complex Grothendieck constant kG. It is shown that Q(θ) can be expressed as the trace of the product of θ with two rescaling matrices, and C(θ) can be expressed as the trace of the product of θ with two dequantisation matrices. Values of Q(θ) in the “ultra-quantum” region (1, kG) are very important, because this region is classically forbidden [C(θ) cannot take values in it]. An example with Q(θ)∈(1,kG) is given, which is related to phenomena where classically isolated by high potentials regions of space, communicate through quantum tunneling. Other examples show that “ultra-quantumness” according to the Grothendieck formalism (Q(θ)∈(1,kG)), is different from quantumness according to other criteria (like quantum interference or the uncertainty principle).
The transition probability of an isolated system for a time-dependent unitary evolution is invariant under the reversal of protocols. In this paper, we generalize the expression of microscopic reversibility to externally perturbed large quantum open systems, which provides a model-independent equality between time forward and reversed joint transition probabilities. A time-dependent external perturbation acts on the subsystem during a transient duration, and subsequently the perturbation is switched off so that the total system would thermalize. We concern ourselves with the net transition probability for the subsystem from the initial to final states after a time evolution during which the energy is irreversibly exchanged between the subsystem and reservoir. The time-reversed probability is given by the reversal of the forcing protocol and the initial ensemble. Microscopic reversibility equates the time forward and reversed probabilities, and therefore appears as a thermodynamic symmetry for open quantum systems.
Abstract A set of n coherent states is introduced in a quantum system with d-dimensional Hilbert space H(d). It is shown that they resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits. A n-tuple representation of arbitrary states in H(d), analogous to the Bargmann representation, is defined. There are two other important properties of these coherent states which make them ‘ultra-quantum’. The first property is related to the Grothendieck formalism which studies the ‘edge’ of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form C that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form Q that uses vectors in the unit ball of the Hilbert space. It shows that if C ⩽ 1 , the corresponding Q might take values greater than 1, up to the complex Grothendieck constant k G . Q related to these coherent states is shown to take values in the ‘Grothendieck region’ ( 1 , k G ) , which is classically forbidden in the sense that C does not take values in it. The second property complements this, showing that these coherent states violate logical Bell-like inequalities (which for a single quantum system are quantum versions of the Frechet probabilistic inequalities). In this sense also, our coherent states are deep into the quantum region.
Hilbert's sixth problem placed probability and mechanics at the center of the axiomatization of physics. It asks how irreversible statistical probability and thermodynamics can follow from the reversible, deterministic dynamics of a system. Building on an exact open quantum system theory developed in the past two decades, we construct a dynamical route from the reversible quantum dynamics to reduced statistics. For quadratic bosonic and fermionic systems bilinearly coupled to arbitrary environments, we derive the exact reduced density operator determined completely by a dissipative propagator u and a fluctuation correlation function v of the system. For continuous spectral densities with no bound-state pole, u vanishes in the long-time limit, information about the system initial states is lost completely, and the reduced density operator of systems approaches a thermal Gibbs state. Gapped spectral densities with localized bound states instead preserve long-time memory and prevent thermalization. We then remove the usual assumption of an initially thermal, mixed environment, start from a product of pure states for the system and environment, unitary evolution first generates system-environment entanglement and a periodic quasi-Gibbs reduced state for a finite environment. When the number of environmental degrees of freedom tends to infinity, the spectrum becomes continuous, the recurrence disappears, the exact reduced density operator converges to a thermal Gibbs state. The total entropy remains zero, whereas the system entropy becomes entanglement entropy and reaches its maximum. Probability is therefore not an additional random postulate imposed on unitary dynamics, it is the objective statistical structure of an open system in a composite. These results provide an exact, testable dynamical solution of the determinism to statistics.