Oliver Rudolph, J. D. M. Wright
It is shown here that a strengthening of Wallach's Unentangled Gleason Theorem can be obtained by applying results of the present authors on generalised Gleason theorems for quantum multi-measures arising from investigations of quantum decoherence functionals.
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The purpose of this note is to give a generalization of Gleason's theorem inspired by recent work in quantum information theory on "nonlocality without entanglement." For multipartite quantum systems, each of dimension three or greater, the only nonnegative frame functions over the set of unentangled states are those given by the standard Born probability rule. However, if one system is of dimension 2 this is not necessarily the case.
Gleason's theorem asserts the equivalence of von Neumann's density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third author's Unentangled Gleason's Theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of …
Gleason's theorem [A. Gleason, J. Math. Mech., \textbf{6}, 885 (1957)] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice $\PH$ extend to positive linear functionals on the algebra of bounded operators $\BH$. Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann alg…
Consider a finite dimensional complex Hilbert space $\cH$, with $dim(\cH) \geq 3$, define $\bS(\cH):= \{x\in \cH \:|\: ||x||=1\}$, and let $ν_\cH$ be the unique regular Borel positive measure invariant under the action of the unitary operators in $\cH$, with $ν_\cH(\bS(\cH))=1$. We prove that if a complex frame function $f : \bS(\cH)\to \bC$ satisfies $f \in \cL^2(\bS(\cH), ν_\cH)$, then it verifies Gleason's statement: There is a unique linear operator $A: \cH \to \cH$ such that $f(u) = < u| A u>$ for every $u \in \bS(\cH)$. $A$ is Hermitean when $f$ is real. No boundedness requirement is thu…
We present a reply to the objections raised by M. J. W. Hall against our extension of Gleason's theorem.
Kochen and Specker's theorem can be seen as a consequence of Gleason's theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason's theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated.
We discuss concrete examples for frame functions and their associated density operators, as well as for non-Gleason type probability measures.
We derive Born's rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason's theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.
Gleason's theorem is a fundamental 60 year old result in the foundations of quantum mechanix, setting up and laying out the surprisingly minimal assumptions required to deduce the existence of quantum density matrices and the Born rule. Now Gleason's theorem and its proof have been continuously analyzed, simplified, and revised over the last 60 years, and we will have very little to say about the theorem and proof themselves. Instead, we find it useful, (and hopefully interesting), to make some clarifying comments concerning the explicit construction of the quantum density matrix that Gleason'…
Gleason-type theorems derive the density operator and the Born rule formalism of quantum theory from the measurement postulate, by considering additive functions which assign probabilities to measurement outcomes. Additivity is also the defining property of solutions to Cauchy's functional equation. This observation suggests an alternative proof of the strongest known Gleason-type theorem, based on techniques used to solve functional equations.
On the connection between the theorems of Gleason and of Kochen and Specker
本文通过函数 m 在单位球面上取值的线性方程，给出 Gleason 定理与 Kochen-Specker 定理简化版本的初等证明，揭示二者分别依赖整个球面与有限点集的联系。
The quantum-mechanical rule for probabilities, in its most general form of positive-operator valued measure (POVM), is shown to be a consequence of the environment-assisted invariance (envariance) idea suggested by Zurek [Phys. Rev. Lett. 90, 120404 (2003)], being completed by Gleason's theorem. This provides also a method for derivation of the Born rule.
Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment. We identify the class of general probabilistic theories which also admit Gleason-type theorems. It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes. Our result also implies that the standard no-restriction hypothes…
Gleason's Theorem for a Qubit as Part of a Composite System
通过复合系统公理将Gleason定理扩展到二维希尔伯特空间，从概率独立性要求导出密度矩阵和Born规则。
As it is known, Gleason's theorem is not applicable for a two-dimensional Hilbert space since in this situation Gleason's axioms are not strong enough to imply Born's rule thus leaving room for a dispersion-free probability measure i.e., one that has only values 0 and 1. To strengthen Gleason's axioms one must add at least one more assumption. But, as it is argued in the present paper, alternatively one can give up the lattice condition lying in the foundation of Gleason's theorem. Particularly, the lattice structure based on the closed linear subspaces in the Hilbert space could be weakened b…
We present a derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. Combined to Gleason's theorem, this approach naturally leads to the usual quantum formalism, within a new conceptual framework that is discussed heuristically in details. The structure of Quantum Mechanics, from its probabilistic nature to its mathematical expression, appears as a result of the interplay between the quantized number of "modalities" accessible to a quantum system, and the continuum of "contexts" that are required to…
We introduce a novel notion of probability within quantum history theories and give a Gleasonesque proof for these assignments. This involves introducing a tentative novel axiom of probability. We also discuss how we are to interpret these generalised probabilities as partially ordered notions of preference and we introduce a tentative generalised notion of Shannon entropy. A Bayesian approach to probability theory is adopted throughout, thus the axioms we use will be minimal criteria of rationality rather than ad hoc mathematical axioms.
The Kochen--Specker (KS) theorem reveals the nonclassicality of single quantum systems. In contrast, Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. Accordingly, unlike incompatibility, entanglement and Bell non-locality are not necessary to demonstrate KS-contextuality. However, here we find that for multiqubit systems, entanglement and non-locality are both essential to proofs of the Kochen--Specker theorem. Firstly, we show that unentangled measurements (a strict superset of local measurements) can never yield a logical (state-independent) proof of …
A Gleason-type theorem is proved for two restricted classes of informationally complete POVMs in the qubit case. A particular (incomplete) Kochen-Specker colouring, suggested by Appleby in dimension three, is generalized to arbitrary dimension. We investigate its effectivity as a function of dimension, using two different measures of this. In particular, we will derive a limit for the fraction of the sphere that can be satisfactorily coloured using the generalized Appleby construction as the number of dimensions approaches infinity. The second, and physically more relevant measure of effectivi…
Quantum processes cannot be reduced, in a nontrivial way, to classical processes without specifying the context in the description of a measurement procedure. This requirement is implied by the Kochen-Specker theorem in the outcome-deterministic case and, more generally, by the Gleason theorem. The latter establishes that there is only one non-contextual classical model compatible with quantum theory, the one that trivially identifies the quantum state with the classical state. However, this model requires a breaking of the unitary evolution to account for macroscopic realism. Thus, a causal c…
A quantum state can be understood in a loose sense as a map that assigns a value to every observable. Formalizing this characterization of states in terms of generalized probability distributions on the set of effects, we obtain a simple proof of the result, analogous to Gleason's theorem, that any quantum state is given by a density operator. As a corollary we obtain a von Neumann-type argument against non-contextual hidden variables. It follows that on an individual interpretation of quantum mechanics, the values of effects are appropriately understood as propensities.
Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to sequential measurement processes is the aim of this paper. Given only a simple set of postulates, a probability measure is derived utilising the concept of Liouville space and the most general permissible quantum channel arises in the same manner. Super-Liouville space is constructed and a Bayesian interpretation of this object is provided. An important applica…
Busch's theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleason's theorem. Here we show that a further generalisation is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Ou…
It has recently been claimed by De Zela that Gleason's theorem, for probability measures on the lattice of projection operators, can be extended to qubits by adding assumptions related to continuity and the existence of 'eigenstates'. This amounts to a claim of the derivation of Born's rule for Hermitian qubit observables. I point out a simple counterexample, and the flaw in De Zela's derivation (these are equally applicable to the repetition of the derivation given in a recent Reply). I also briefly discuss a valid extension to qubits given by Busch.
From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Solér's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $σ$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-…
Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for n = 2. The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should be excluded from consideration.
In a previous article [1] we presented an argument to obtain (or rather infer) Born's rule, based on a simple set of axioms named "Contexts, Systems and Modalities" (CSM). In this approach there is no "emergence", but the structure of quantum mechanics can be attributed to an interplay between the quantized number of modalities that are accessible to a quantum system, and the continuum of contexts that are required to define these modalities. The strong link of this derivation with Gleason's theorem was emphasized, with the argument that CSM provides a physical justification for Gleason's hypo…
I review the two theorems referred to in the title, and then suggest that it would be interesting to know how much of Hilbert space one can use without forcing the proof of these theorems. It would also be interesting to know what parts of Hilbert space that are essential for the proofs. I go on to discuss cubes, graphs, and pentagrams.
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for…
We develop a synthesis of Turing's paradigm of computation and von Neumann's quantum logic to serve as a model for quantum computation with recursion, such that potentially non-terminating computation can take place, as in a quantum Turing machine. This model is based on the extension of von Neumann's quantum logic to partial states, defined here as sub-probability measures on the Hilbert space, equipped with the natural pointwise partial ordering. The sub-probability measures allow a certain probability for the non-termination of the computation. We then derive an extension of Gleason's theor…
Why qubits are exceptional in Gleason's theorem: a Bloch-space perspective
本文通过布洛赫空间视角简化格莱森定理，揭示二维系统中非Born概率规则的存在及高维系统中其不可能性，强调Born规则在高维下的不可避免性及量子比特的特殊性。
We pose and solve a problem concerning consistent assignment of quantum probabilities to a set of bases associated with maximal projective measurements. We show that our solution is optimal. We also consider some consequences of the main theorem in the paper in conjunction with Gleason's theorem. Some potential applications to state tomography and probabilistic quantum secret-sharing scheme are discussed.
Probability-based approach to hybrid classical-quantum systems of any size: Generalized Gleason and Kraus theorems
本文提出基于概率的四条公理框架描述任意规模混合经典-量子系统，证明广义Gleason定理和广义Kraus定理，给出多种情形下演化变换的显式表达，并讨论经典-量子关联量化与量子操作推广。
This paper reports three almost trivial theorems that nevertheless appear to have significant import for quantum foundations studies. 1) A Gleason-like derivation of the quantum probability law, but based on the positive operator-valued measures as the basic notion of measurement (see also Busch, quant-ph/9909073). Of note, this theorem also works for 2-dimensional vector spaces and for vector spaces over the rational numbers, where the standard Gleason theorem fails. 2) A way of rewriting the quantum collapse rule so that it looks almost precisely identical to Bayes rule for updating probabil…
It is sometimes stated that Gleason's theorem prevents the construction of hidden-variable models for quantum entities described in a more than two-dimensional Hilbert space. In this paper however we explicitly construct a classical (macroscopical) system that can be represented in a three-dimensional real Hilbert space, the probability structure appearing as the result of a lack of knowledge about the measurement context. We briefly discuss Gleason's theorem from this point of view.
Contextuality and the fundamental theorems of quantum mechanics
本研究拓展拓扑斯量子力学视角，证明Wigner定理、Gleason定理等四大量子力学基本定理均与语境性相关，并用语境预层语言重述定理，给出适用于冯·诺依曼代数的通用版本，揭示语境性对量子理论结构的核心编码作用。
The purpose of this note is to complete the interesting review on quantum contextuality [1] that appeared recently. In particular we will introduce and discuss the ideas of extracontextuality and extravalence, that allow one to relate Kochen-Specker's and Gleason's theorems, and also to shift the emphasis from the first to the second one. We will also argue that whereas Kochen-Specker's is essentially a negative result (a no-go theorem), Gleason's is a positive one since it provides a mathematical justification of Born's rule. The link between these issues is provided by a specific quantum fea…
We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision measurements on defining aspects of quantum mechanics such as contextuality and the Born rule. While foundational results such as the Kochen-Specker and Gleason theorems are valid in the context of infinite precision, they fail to hold in general in a world with limited resources. Here we employ an interval-valued framework to establish bounds on the validity of those theorems in realistic experimental environments. In this way, not on…
Based on the Bell theorem, it has been believed that a theoretical computation of the Bell correlation requires explicit use of an entangled state. Such a physical superposition of light waves occurs in the downconverter sources used in Bell experiments. However, this physical superposition is eliminated by wave propagation to spatially separated detectors. Bell correlations must therefore result from local waves, and the source boundary conditions of their previously entangled state. In the present model, Bell correlations are computed from disentangled separated waves, boundary conditions of…
Generalised Kochen-Specker Theorem for Finite Non-Deterministic Outcome Assignments
本文提出了一种广义 Kochen-Specker 定理，排除了结果分配集合为 {0, p, 1-p, 1} 的隐变量理论，并构建了双人非局部博弈展示了其设备无关应用。
Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory
研究排除量子理论离散非确定性替代方案，通过构建有限测量配置，采用状态依赖哈迪型测试和广义KS定理两种方式，扩展层理论框架引入强\(O\)值语境性，探讨对多值逻辑及相关协议的影响。
The probabilistic rule that links the formalism of Quantum Mechanics (QM) to the real world was stated by Born in 1926. Since then, there were many attempts to derive the Born postulate as a theorem, Gleason's being the most prominent. The Gleason derivation, however, is generally considered rather intricate and its physical meaning, in particular in relation with the noncontextuality of probability (NP), is not quite evident. More recently, we are witnessing a revival of interest in possible demonstrations of the Born rule, like Zurek's and Deutsch's based on the decoherence and on the theory…
We discuss the problem of hidden variables and the motivation for introducting them in quantum mechanics. These include determinism, and the problem of meassurement and incompleteness. We first discuss Von-Neumann's imposisbility proof and then analyse it's weakness in terms of Bell's explicit hidden variable model of spin one-half particles. We next discuss Gleason's theorem and Kochen-Specker theorem and bring out the troublems with non contextual hidden variable theories. An important role is played by Einstein locality in the discussion of hidden variable theories as was first brought out …
This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by A.M. Gleason to prove his celebrated theorem. In particular, the problem of associating quantum state with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical observables representing quantum one…
Revisiting Quantum Contextuality in an Algebraic Framework
本文在代数框架下重新审视量子语境性，通过超语境性与超价性思想联系Kochen-Specker定理和Gleason定理，并借助无限张量积与扇区化机制，构建统一量子与经典物理的数学模型。
Summing to Uncertainty: On the Necessity of Additivity in Deriving the Born Rule
本文探讨了在推导本规则中加法性假设的必要性，指出其在多个关键推导中的核心作用，并揭示了缺乏该假设可能导致的漏洞。
In this paper, we show that Erwin Schroedinger's generalization of the Einstein Podolsky Rosen argument can be connected to certain mathematical theorems - Gleason's and also Kochen and Specker's - in a manner analogous to the relation of EPR itself with Bell's theorem. In both cases, the conclusion is quantum nonlocality, as we discuss. The "Schroedinger nonlocality" proofs share some features with the Greenberger, Horne, and Zeilinger quantum-nonlocality work, yet also differ in significant ways. For clarity and completeness, we begin with a detailed discussion of the topic of hidden variabl…
Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs
本文针对非负无纠缠量子证明类QMA⁺(2)，证明了近最优间隙放大结果，揭示其复杂度相变，结合对称子空间投影与德芬蒂定理完成证明，为量子复杂度理论提供关键进展。
We obtain sufficient conditions for the efficient simulation of a continuous variable quantum algorithm or process on a classical computer. The resulting theorem is an extension of the Gottesman-Knill theorem to continuous variable quantum information. For a collection of harmonic oscillators, any quantum process that begins with unentangled Gaussian states, performs only transformations generated by Hamiltonians that are quadratic in the canonical operators, and involves only measurements of canonical operators (including finite losses) and suitable operations conditioned on these measurement…
We prove that the correlations present in a multipartite quantum state have an \emph{operational} quantum character as soon as the state does not simply encode a multipartite classical probability distribution, i.e. does not describe the joint state of many classical registers. Even unentangled states may exhibit such \emph{quantumness}, that is pointed out by the new task of \emph{local broadcasting}, i.e. of locally sharing pre-established correlations: this task is feasible if and only if correlations are classical and derive a no-local-broadcasting theorem for quantum correlations. Thus, l…
In this thesis we explore the questions of what should be considered a "classical" theory, and which aspects of quantum theory cannot be captured by any theory that respects our intuition of classicality. This exploration is divided in two parts: in the first we review classical results of the literature, such as the Kochen-Specker theorem, von Neumann's theorem, Gleason's theorem, as well as more recent ideas, such as the distinction between $ψ$-ontic and $ψ$-epistemic ontological models, Spekkens' definition of contextuality, Hardy's ontological excess baggage theorem and the PBR theorem. Th…
Simplicial quantum contextuality
本文提出基于单纯集的量子上下文性新框架，将测量场景扩展至空间并推广非信号分布为单纯分布，利用上同调见证将拓扑构造扩展至概率分布层面，并自然表达Fine、Gleason及Kochen-Specker等基础定理。
Algebraic Kolmogorov--Arnold representation theorem for quantum measurement
提出代数有界多项式版本的Kolmogorov-Arnold表示定理，证明无纠缠多量子比特乘积态的物理性质可精确分解为局部内观测量和浅层单变量多项式架构，并证明该表示对对抗性扰动具有稳定性。
The celebrated Bell-Kochen-Specker no-go theorem asserts that quantum mechanics does not present the property of realism, the essence of the theorem is the lack of a joint probability distributions for some experiment settings. In this work, we exploit the information theoretic form of the theorem using information measure instead of probabilistic measure and indicate that quantum mechanics does not present such entropic realism neither. The entropic form of Gleason's no-disturbance principle is developed and it turns out to be characterized by the intersection of several entropic cones. Entro…
This paper is concerned with the nature of probability in physics, and in quantum mechanics in particular. It starts with a brief discussion of the evolution of Itamar Pitowsky's thinking about probability in quantum theory from 1994 to 2008, and the role of Gleason's 1957 theorem in his derivation of the Born Rule. Pitowsky's defence of probability therein as a logic of partial belief leads us into a broader discussion of probability in physics, in which the existence of objective "chances" is questioned, and the status of David Lewis' influential Principal Principle is critically examined. T…
Quantum mechanics dramatically differs from classical physics, allowing for a wide range of genuinely quantum phenomena. The goal of quantum information is to understand information processing from a quantum perspective. In this mindset, it is thus natural to focus on tasks where quantum resources provide an advantage over classical ones, and to overlook tasks where quantum mechanics provides no advantage. But are the latter tasks really useless from a more general perspective? Here we discuss a simple information-theoretic game called 'guess your neighbour's input', for which classical and qu…
This Chapter develops a realist information-theoretic interpretation of the nonclassical features of quantum probabilities. On this view, what is fundamental in the transition from classical to quantum physics is the recognition that \emph{information in the physical sense has new structural features}, just as the transition from classical to relativistic physics rests on the recognition that space-time is structurally different than we thought. Hilbert space, the event space of quantum systems, is interpreted as a kinematic (i.e., pre-dynamic) framework for an indeterministic physics, in the …
De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of i.i.d. states. Recently, it was shown that in many quantum information applications a more relaxed de Finetti reduction (i.e. only a matrix inequality between the symmetric state and one of de Finetti form) is enough, and that it leads to more concise and elegant arguments. Here we show several uses and general flexible applicability of a constrained de Finetti reduction in quantum information theory, which was recently discovered by Duan, Severini and Winter. In particular we show tha…
Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $ω(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. cond…
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