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.
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.
Abstract The Grothendieck theorem considers a ‘classical’ quadratic form <?CDATA C?> C that uses complex scalars in the unit disc, and the corresponding ‘quantum’ quadratic form <?CDATA Q?> Q that replaces the scalars with vectors in the unit ball of a Hilbert space. It shows that when <?CDATA C≤1?> C ≤ 1 then <?CDATA 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)?> Q ∈ ( 1 , k G ) , which is a classically forbidden region in the sense that <?CDATA C?> C cannot take values in it.
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.
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 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.
Grothendieck constants KG(d) bound the advantage of d-dimensional strategies over 1-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from d=2, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For d∈{3,4,5}, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for d⩽9. For d∈{4,7,8}, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants KG(d↦2) as the advantage of complex d-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on KG(d↦2).
Abstract Known results for multi-particle entangled systems that link the Grothendieck formalism with the violation of Bell-like inequalities, are generalised to a single quantum system. The Frechet probabilistic inequalities in a single quantum system, are the analogue of Bell-like inequalities in an entangled multipartite system. Kolmogorov probabilities obey Boole’s inequality and the Frechet probabilistic inequalities, but quantum probabilities violate them. Through examples, a link is made between the violation of these inequalities and the Grothendieck formalism in a single quantum system.