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arXiv:2409.07270 · Journal of Mathematical Physics

Rescaling transformations and the Grothendieck bound formalism in a single quantum system

Apostol Vourdas

Published Sep 1, 2024• 3 citations
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Abstract

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).

DOI
10.1063/5.0201690

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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).
Abstract The Grothendieck formalism considers a classical quadratic form C(θ), and a quantum quadratic form Q(θ). It shows that if C(θ) ≤ 1, the Q(θ) might take values greater than 1, up to the complex Grothendieck constant k G . The region (1, k G ) is ‘ultra-quantum’ in the sense that Q can take values in it, but C cannot take values in it. The mathematical formalism is applied to various physical examples. It is shown that many physically interesting density matrices give Q(θ) ≤ 1. In some examples Q is the Wigner and Weyl functions, and in that case also Q(θ) ≤ 1. In many examples Fourier dual density matrices both give Q(θ) ≤ 1, and in this sense Fourier duality does not play an important role within the Grothendieck formalism. Therefore quantumness according to Grothendieck (Q ε (1, k G )) is different from quantumness according to other criteria.
Abstract The Grothendieck theorem considers a ‘classical’ quadratic form <?CDATA C{\mathcal{C}}C?> C that uses complex scalars in the unit disc, and the corresponding ‘quantum’ quadratic form <?CDATA Q{\mathcal{Q}}Q?> Q that replaces the scalars with vectors in the unit ball of a Hilbert space. It shows that when <?CDATA C≤1{\mathcal{C}}\le 1C≤1?> C ≤ 1 then <?CDATA Q{\mathcal{Q}}Q?> Q might take values greater than 1, up to the complex Grothendieck constant kG . Previous work in a quantum context, used Grothendieck’s theorem with multipartite entangled systems, in contrast to the present work which uses it for a single quantum system. The emphasis in the paper is in examples with <?CDATA Q∈(1,kG){\mathcal{Q}}\in (1,{k}_{G})Q∈(1,kG​)?> Q ∈ ( 1 , k G ) , which is a classically forbidden region in the sense that <?CDATA C{\mathcal{C}}C?> C cannot take values in it.
Abstract Grothendieck’s bound is used in the context of a single quantum system, in contrast to previous work which used it for multipartite entangled systems and the violation of Bell-like inequalities. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form C that uses complex numbers in the unit disc, and takes values less than 1. It then proves that if the complex numbers are replaced with vectors in the unit ball of the Hilbert space, then the ‘quantum’ quadratic form Q might take values greater than 1, up to the complex Grothendieck constant k G . The Grothendieck theorem is reformulated here in terms of arbitrary matrices (which are multiplied with appropriate normalisation prefactors), so that it is directly applicable to quantum quantities. The emphasis in the paper is in the ‘Grothendieck region’ ( 1 , k G ) , which is a classically forbidden region in the sense that C cannot take values in it. Necessary (but not sufficient) conditions for Q taking values in the Grothendieck region are given. Two examples that involve physical quantities in systems with six and 12-dimensional Hilbert space, are shown to lead to Q in the Grothendieck region ( 1 , k G ) . They involve projectors of the overlaps of novel generalised coherent states that resolve the identity and have a discrete isotropy.
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.
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.
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 2d2d2d-dimensional Hilbert space. A family of Bargmann-like representations (called zzz-Bargmann representations) introduces naturally the larger space. The zzz-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. Q{\cal Q}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.