Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.
Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.
We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs). This formulation provides a direct interpretation of density matrices as quasi-moment matrices. Using QEOs, we will provide a series of representation theorems, a' la de Finetti, relating a classical probability mass function (satisfying certain symmetries) to a quasi-expectation operator. We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. Although particles indistinguishability is considered a truly "weird" quantum phenomenon, it is not special. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT. Using this connection, we will rederive the first and second quantisation in QT for bosons through the classical statistical concept of exchangeable random variables. Using this approach, we will show how classical reality emerges in QT as the number of identical bosons increases (similar to what happens for finitely exchangeable sequences of rolls of a classical dice).
De Finetti theorems tell us that if we expect the likelihood of outcomes to be independent of their order, then these sequences of outcomes could be equivalently generated by drawing an experiment at random from a distribution, and repeating it over and over. In particular, the quantum de Finetti theorem says that exchangeable sequences of quantum states are always represented by distributions over a single state produced over and over. The main result of this paper is that this quantum de Finetti construction has a universal property as a categorical limit. This allows us to pass canonically between categorical treatments of finite dimensional quantum theory and the infinite dimensional. The treatment here is through understanding properties of (co)limits with respect to the contravariant functor which takes a C*-algebra describing a physical system to its convex, compact space of states, and through discussion of the Radon probability monad. We also show that the same categorical analysis also justifies a continuous de Finetti theorem for classical probability.
The quantum versions of de Finetti's theorem derived so far express the convergence of n-partite symmetric states, i.e., states that are invariant under permutations of their n parties, toward probabilistic mixtures of independent and identically distributed (IID) states of the form ${\ensuremath{\sigma}}^{\ensuremath{\bigotimes}n}$. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a quantum de Finetti theorem particularly relevant to continuous-variable systems. Specifically, an n-mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge toward a probabilistic mixture of IID Gaussian states (actually, n identical thermal states).
We prove a de Finetti theorem for exchangeable sequences of states on test spaces , where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptions may fail, leading to probabilistic models that do not have a de Finetti theorem.
Standard finite quantum de Finetti theorems approximate the k-system marginals of permutation-invariant states of n-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/n error bound in max-relative entropy improves on the previously best known k2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗n, where dA=dimA, we prove that the k-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of r defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2 and decays exponentially in n for a suitable choice of parameters r and k.
What does it mean for a causal structure to be `unknown'? Can we even talk about `repetitions' of an experiment without prior knowledge of causal relations? And under what conditions can we say that a set of processes with arbitrary, possibly indefinite, causal structure are independent and identically distributed? Similar questions for classical probabilities, quantum states, and quantum channels are beautifully answered by so-called "de Finetti theorems", which connect a simple and easy-to-justify condition – symmetry under exchange – with a very particular multipartite structure: a mixture of identical states/channels. Here we extend the result to processes with arbitrary causal structure, including indefinite causal order and multi-time, non-Markovian processes applicable to noisy quantum devices. The result also implies a new class of de Finetti theorems for quantum states subject to a large class of linear constraints, which can be of independent interest.