We prove a theorem about positive-operator-valued measures (POVMs) that is an analog of the Kolmogorov extension theorem, a standard theorem of probability theory. According to our theorem, if a sequence of POVMs G n on ${\mathbb{R}}^n$ satisfies the consistency (or projectivity) condition $G_{n+1}(A\times {\mathbb{R}}) = G_n(A)$ then there is a POVM G on the space ${\mathbb{R}}^{\mathbb{N}}$ of infinite sequences that has G n as its marginal for the first n entries of the sequence. We also describe an application in quantum theory.We prove a theorem about positive-operator-valued measures (POVMs) that is an analog of the Kolmogorov extension theorem, a standard theorem of probability theory. According to our theorem, if a sequence of POVMs G n on $${\mathbb{R}}^n$$ satisfies the consistency (or projectivity) condition $$G_{n+1}(A\times {\mathbb{R}}) = G_n(A)$$ then there is a POVM G on the space $${\mathbb{R}}^{\mathbb{N}}$$ of infinite sequences that has G n as its marginal for the first n entries of the sequence. We also describe an application in quantum theory.

Jan 1, 200712 citations

Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories

Simon Milz, Fattah Sakuldee, Felix A. Pollock +1 more

In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.

arXiv:1712.02589QuantumApr 20, 202087 citations

Uniform Continuity of POVMs

Roberto Beneduci

arXiv:1307.5733International Journal of Theoretical PhysicsNov 5, 20131 citation

Two Simple Proofs of Müller's Theorem

Samuel Epstein

Due to Müller's theorem, the Kolmogorov complexity of a string was shown to be equal to its quantum Kolmogorov complexity. Thus there are no benefits to using quantum mechanics to compress classical information. The quantitative amount of information in classical sources is invariant to the physical model used. These consequences make this theorem arguably the most important result in the intersection of algorithmic information theory and physics. The original proof is quite extensive. This paper contains two simple proofs of this theorem. This paper also contains new bounds for quantum Kolmogorov complexity with error.Due to Müller's theorem, the Kolmogorov complexity of a string was shown to be equal to its quantum Kolmogorov complexity. Thus there are no benefits to using quantum mechanics to compress classical information. The quantitative amount of information in classical sources is invariant to the physical model used. These consequences make this theorem arguably the most important result in the intersection of algorithmic information theory and physics. The original proof is quite extensive. This paper contains two simple proofs of this theorem. This paper also contains new bounds for quantum Kolmogorov complexity with error.

arXiv:2402.05328Feb 8, 20240 citations

{\mathbb{R}}^n

satisfies the consistency (or projectivity) condition

G_{n+1}(A\times {\mathbb{R}}) = G_n(A)

then there is a POVM G on the space

{\mathbb{R}}^{\mathbb{N}}

of infinite sequences that has G n as its marginal for the first n entries of the sequence. We also describe an application in quantum theory.