On the resolution to Tsirelson's problem
(2025) FYSM34 20251Particle and nuclear physics
Department of Physics
- Abstract
- Correlations between two quantum agents, Alice and Bob, are typically modeled by assigning each a separate Hilbert space and representing their joint system as a tensor product of these spaces. Alternatively, one can let Alice and Bob act on a single Hilbert space, and require that their measurement operators commute. Tsirelson's problem asks whether any such commuting correlation can be approximated arbitrarily well by a tensor product correlation. A result in computational complexity theory has shown that this is false. In this thesis, we present a self-contained, accessible proof of this negative solution to Tsirelson's problem. We then use the framework of operator systems to prove that numerous models in physics cannot distinguish... (More)
- Correlations between two quantum agents, Alice and Bob, are typically modeled by assigning each a separate Hilbert space and representing their joint system as a tensor product of these spaces. Alternatively, one can let Alice and Bob act on a single Hilbert space, and require that their measurement operators commute. Tsirelson's problem asks whether any such commuting correlation can be approximated arbitrarily well by a tensor product correlation. A result in computational complexity theory has shown that this is false. In this thesis, we present a self-contained, accessible proof of this negative solution to Tsirelson's problem. We then use the framework of operator systems to prove that numerous models in physics cannot distinguish between these two descriptions of correlations. Most notably, this includes all physical systems that exhibit some form of a finite-dimensional approximation. We discuss how as a consequence of this, an observation of a correlation which separates the commuting and tensor product model would experimentally verify that the underlying system genuinely has infinite degrees of freedom. Finally, we use the theory of C$^*$-algebras to argue that quantum networks with bipartite sources of quantum states do not provide any new insights into Tsirelson's problem, and provide a precise conjecture that would show this conclusively. (Less)
- Popular Abstract
- Imagine two physicists, Alice and Bob, who each have a coin in a little box. They don't know whether their coin is showing heads or tails. Alice and Bob are far apart, and Alice peeks in her box to see that her coin is showing heads. Instantly, she knows that Bob's coin is showing tails, even if Bob hasn't opened his box yet. This phenomenon, known as quantum entanglement, is one of the strangest predictions of quantum mechanics. When we prepare two entangled particles, measuring one particle instantly tells us something about the other particle, no matter how far apart they are.
This thesis explores the setup where Alice and Bob share a (potentially entangled) quantum system, on which they conduct some measurements. The standard... (More) - Imagine two physicists, Alice and Bob, who each have a coin in a little box. They don't know whether their coin is showing heads or tails. Alice and Bob are far apart, and Alice peeks in her box to see that her coin is showing heads. Instantly, she knows that Bob's coin is showing tails, even if Bob hasn't opened his box yet. This phenomenon, known as quantum entanglement, is one of the strangest predictions of quantum mechanics. When we prepare two entangled particles, measuring one particle instantly tells us something about the other particle, no matter how far apart they are.
This thesis explores the setup where Alice and Bob share a (potentially entangled) quantum system, on which they conduct some measurements. The standard mathematical picture used to describe this configuration is to assign Alice and Bob each their own separate quantum lab. Another way to describe this is to lump them together in a single, large quantum lab.
Tsirelson's problem asks whether these two descriptions lead to the same predictions. For a long time, it was believed that this was the case. However, a recent discovery in a completely different field of science has shown that this is not true! The two descriptions are in fact different. This follows from a result in computational complexity theory, which deals with how efficiently modern computers can solve computational problems. In this thesis, we walk through this negative solution to Tsirelson's problem.
The negative solution states that with our current understanding of physics, an experiment that demonstrates the difference between these two descriptions can exist. However, no such experiment is currently known. We show that due to the underlying mathematical structure, most physical systems cannot distinguish between these two descriptions. For example, if the underlying quantum system on which Alice and Bob conduct their measurements only has a finite amount of possible configurations, then the two descriptions are identical.
From this, we discuss that if an experiment which demonstrates the negative solution to Tsirelson's problem is found, it could verify that the underlying physical system has infinite possible configurations. This could experimentally prove that our physical world has an infinite characteristic. If no such experiment can be found, then the negative solution to Tsirelson's problem could teach us something new about the principles that physical theories must obey. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9192556
- author
- Nieuwenhuis, Rutger LU
- supervisor
- organization
- course
- FYSM34 20251
- year
- 2025
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Tsirelson's problem, quantum correlations, MIP*, C*-algebras, operator systems, finite-dimensional approximations, quantum networks, quantum entanglement
- language
- English
- id
- 9192556
- date added to LUP
- 2025-06-05 08:31:31
- date last changed
- 2025-06-05 12:45:44
@misc{9192556,
abstract = {{Correlations between two quantum agents, Alice and Bob, are typically modeled by assigning each a separate Hilbert space and representing their joint system as a tensor product of these spaces. Alternatively, one can let Alice and Bob act on a single Hilbert space, and require that their measurement operators commute. Tsirelson's problem asks whether any such commuting correlation can be approximated arbitrarily well by a tensor product correlation. A result in computational complexity theory has shown that this is false. In this thesis, we present a self-contained, accessible proof of this negative solution to Tsirelson's problem. We then use the framework of operator systems to prove that numerous models in physics cannot distinguish between these two descriptions of correlations. Most notably, this includes all physical systems that exhibit some form of a finite-dimensional approximation. We discuss how as a consequence of this, an observation of a correlation which separates the commuting and tensor product model would experimentally verify that the underlying system genuinely has infinite degrees of freedom. Finally, we use the theory of C$^*$-algebras to argue that quantum networks with bipartite sources of quantum states do not provide any new insights into Tsirelson's problem, and provide a precise conjecture that would show this conclusively.}},
author = {{Nieuwenhuis, Rutger}},
language = {{eng}},
note = {{Student Paper}},
title = {{On the resolution to Tsirelson's problem}},
year = {{2025}},
}