J. Finkelstein
This note is a somewhat-lighthearted comment on a recent paper by David Wallace, arXiv:0906.2718[quant-ph] entitled "A formal proof of the Born rule from decision-theoretic assumptions".
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I comment briefly on derivations of the Born rule presented by Masanes et al. and by Hossenfelder.
The Born rule, which is one of foundational axioms of quantum theory, states that the probability of obtain outcome $a$ for the quantum state $|ψ\rangle$ is determined by $P(a)=|\langle a|ψ\rangle|^{2}$. Despite its great success in predicting the experimental outcomes, there still lacks a direct way to verify the Born rule. Here, we show that the weak value based direct measurement of superposition wavefunction is feasible, which can be used to verify the Born rule directly. The plausible experiment is suggested.
The paper is withdrawn due to an error in Section 2.
We have shown that the Born rule $(ρ=|ψ|^{2})$ is inconsistent with Lorentz symmetry of the Salpeter equation (square root Klein-Gordon equation). So we find relativistic modification of the Born rule as $ρ=\frac{1}{2mc^{2}}(|\sqrt{\widehat{H}-mc^{2}}ψ|^{2}+|\sqrt{\widehat{H}+mc^{2}}ψ|^{2})$ , which is consistent with Lorentz symmetry of this equation
Considerable effort has been devoted to deriving the Born rule (e.g. that $|ψ(x)|^2 dx$ is the probability of finding a system, described by $ψ$, between $x$ and $x + dx$) in quantum mechanics. Here we show that the Born rule is not solely quantum mechanical; rather, it arises naturally in the Hilbert space formulation of {\it classical} mechanics as well. These results provide new insights into the nature of the Born rule, and impact on its understanding in the framework of quantum mechanics.
In this article we discuss few new derivations of the so called Born's rule for quantum probability in the context of the pilot wave theory proposed by de Broglie in 1927.
As a fundamental postulate of quantum mechanics, Born's rule assigns probabilities to the measurement outcomes of quantum systems and excludes multi-order quantum interference. Here we report an experiment on a single spin in diamond to test Born's rule by inspecting the third-order quantum interference. The ratio of the third-order quantum interference to the second-order in our experiment is ceiled at the scale of $10^{-3}$, which provides a stringent constraint on the potential breakdown of Born's rule.
Character of observables in classical physics and quantum theory is reflected upon. Born rule in the context of measurement being an interaction between two quantum systems is discussed. A pedagogical introduction to Finkelstein-Hartle frequency operator is presented.
It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born Rule is postulated rather than derived from the first principles. In this work the Born Rule is derived from the time-reversal symmetry of quantum equations of motion. The derivation is based on a simple functional equation that takes into account properties of probability, as well as the linearity and time-reversal symmetry of quantum equations of motion. The derivation presented in this work also allows to determine certain limits to applicability of the Born Rule.
We present a reply to the objections raised by M. J. W. Hall against our extension of Gleason's theorem.
We present a new experimental approach using a three-path interferometer and find a tighter empirical upper bound on possible violations of Born's Rule. A deviation from Born's rule would result in multi-order interference. Among the potential systematic errors that could lead to an apparent violation we specifically study the nonlinear response of our detectors and present ways to calibrate this error in order to obtain an even better bound.
The argument of environment-assisted invariance (known as envariance) implying Born's rule is widely used in models for quantum measurement to reason that they must yield the correct statistics, specifically for linear models. However, it has recently been shown that linear collapse models can never give rise to Born's rule. Here, we address this apparent contradiction and point out an inconsistency in the assumptions underlying the arguments based on envariance. We use a construction in which the role of the measurement machine is made explicit and show that the presence of envariance does no…
Where does the Born Rule come from? Superposition
该研究针对玻恩定则的来源争议，通过在有限概率论中加入叠加事件的概念，证明玻恩定则可自然出现，指出其是叠加数学的固有特性，无需物理学推导。
Despite the tremendous empirical success of quantum theory there is still widespread disagreement about what it can tell us about the nature of the world. A central question is whether the theory is about our knowledge of reality, or a direct statement about reality itself. Regardless of their stance on this question, current interpretations of quantum theory regard the Born rule as fundamental and add an independent state-update (or "collapse") rule to describe how quantum states change upon measurement. In this paper we present an alternative perspective and derive a probability rule that su…
I argue that various derivations of the Born probability rule within strictly unitary quantum mechanics implicitly use anthropic constraints on the properties of the probabilities, but without proposing an ensemble from which those constraints limit a selection. An argument by Mallah obtains the Born probabilities explicitly for standard unitary dynamics but via postulating a hypothetical extra noise-like component of the quantum state. I argue that the implicit anthropic constraints required for well-behaved probability rules would select for a state of the sort Mallah proposes.
This paper presents a novel explanation of the cause of quantum probabilities and the Born rule based on the intuitionistic interpretation of quantum mechanics where propositions obey constructive (intuitionistic) logic. The use of constructive logic makes it possible (through a replacement of the concept of truth with the concept of constructive probability) to abandon the law of excluded middle in the intuitionistic interpretation.
The absolute value squared of the probability amplitude corresponding to the overlap of an initial state with a continuum wave solution to the Schrödinger equation of the problem, has the physical interpretation provided by the Born rule. Here, it is shown that for an open quantum system, the above probability may be written in an exact analytical fashion as an expansion in terms of the non-Hermitian resonance (quasinormal) states and complex poles to the problem which provides an underlying non-Hermitian character of the Born rule.
Born's rule deviations from temporal non-local effects
本研究针对量子平衡假设不成立的系统，借助量子流体动力学框架揭示渡越干涉与本征记忆效应诱导的有限时间非局域关联会导致玻恩定则偏离，其表现为不稳定态生存概率的非指数衰减，挑战传统量子概率解释并为时间非局域性实验提供新动力学指导。
Having analyzed the formal aspects of Wallace's proof of the Born rule, we now discuss the concepts and axioms upon which it is built. Justification for most axioms is shown to be problematic, and at times contradictory. Some of the problems are caused by ambiguities in the concepts used. We conclude the axioms are not reasonable enough to be taken as mandates of rationality in Everettian Quantum Mechanics. This invalidates the interpretation of Wallace's result as meaning it would be rational for Everettian agents to decide using the Born rule.
Is Born-Jordan really the universal Path Integral Quantization Rule?
本文探讨了路径积分量子化规则中Born-Jordan规则的有效性，指出其仅适用于二次动量的哈密顿量，且Weyl量化方案同样可得相同结果。
The Born rule is at the foundation of quantum mechanics and transforms our classical way of understanding probabilities by predicting that interference occurs between pairs of independent paths of a single object. One consequence of the Born rule is that three way (or three paths) quantum interference does not exist. In order to test the consistency of the Born rule, we examine detection probabilities in three path intereference using an ensemble of spin-1/2 quantum registers in liquid state nuclear magnetic resonance (LSNMR). As a measure of the consistency, we evaluate the ratio of three way…
Many physicists believe that the EPR experiment exhibits instantaneous non-local effects. I argue below that an application of Born's Rule to EPR shows no such instantaneous effects and that EPR is consistent with full Lorentz invariance. I then discuss how it is possible to understand the EPR correlations for non-commuting spin components without recourse to non-local effects. Finally, I use these results to discuss the Free Will Theorem, which I formulate in a new way.
A non-collapse scenario for ``conscious'' selection of a term from a superposition was proposed in quant-ph/0309166: thermally assisted tunneling of neuronal pore molecules. But ``observers'' consisting of only two neurons appear to be at odds with Born's rule. In the present paper, an observer is assumed to possess a large number of auxilliary properties irrelevant for the result of the measurement. Born's rule then reduces to postulating that, prior to the result becoming conscious, irrelevant properties are in an entangled state with maximum likelihood, in the sense that phase-equivalent en…
A physical experiment comprises along the time trajectory a start, a time evolution (duration), and an end, which is the measurement. In non relativistic quantum mechanics the start of the experiment is defined by the wave function at time 0 taking into account the starting conditions, the evolution is described by the wave function following the Schrödinger equation and the measurement by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that makes quantum mechanics statistical with all its consequences. The nature of the Born rule is thereby unknown albeit n…
Born's rule, one of the cornerstones of quantum mechanics, relates detection probabilities to the modulus square of the wave function. Single-particle interference is accordingly limited to pairs of quantum paths and higher-order interferences are prohibited. Deviations from Born's law have been quantified via the Sorkin parameter which is proportional to the third-order term. We here extend this formalism to many-particle interferences and find that they exhibit a much richer structure. We demonstrate, in particular, that all interference terms of order $(2M+1)$ and greater vanish for $M$ par…
A concise review of the derivation of the Born rule and Schrödinger equation from first principles is provided. The starting point is a formalization of fundamental notions of measurement and composition, leading to a general framework for physical theories known as the positive formalism. Consecutively adding notions of spacetime, locality, absolute time and causality recovers the well established convex operational framework for classical and quantum theory. Requiring the partially ordered vector space of states to be an anti-lattice one obtains quantum theory in its standard formulation. Th…
In which-way double-slit experiments with perfect detectors, it is assumed that having a second detector at the slits is redundant, as it will not change the interference pattern. We however show that if higher-order or non-classical paths are accounted for, the presence of the second detector will have an effect on the interference pattern. Accounting for these non-classical paths also means that the Sorkin parameter in triple-slit experiments is only an approximate measure of Born rule violation. Using the difference between single and double which-way detectors, we give an alternative param…
We deduce the Born rule. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics--a linear, not Hilbert', vector space--and empirical notion of the statistical length of a state. Its statistical nature comes from the experimental detector-clicks being formalized into the abstract quantum micro-events. We also comment on that it is not only that the use has not been made of some quantum axioms when deriving the rule but, in a sense, their invoking would be inconsistent.
I develop the decision-theoretic approach to quantum probability, originally proposed by David Deutsch, into a mathematically rigorous proof of the Born rule in (Everett-interpreted) quantum mechanics. I sketch the argument informally, then prove it formally, and lastly consider a number of proposed ``counter-examples'' to show exactly which premises of the argument they violate.
Recently, W. H. Zurek presented a novel derivation of the Born rule based on a mechanism termed environment-assisted invariance, or "envariance" [W. H. Zurek, Phys. Rev. Lett. 90(2), 120404 (2003)]. We review this approach and identify fundamental assumptions that have implicitly entered into it, emphasizing issues that any such derivation is likely to face.
Improving the proof of the Born rule using a physical requirement on the dynamics of quantum particles
本文通过引入“短时间内粒子性质保持不变”的动力学公设，改进已有结果并给出 Born 规则的完整证明，同时指出该公设被所有实验满足却无法由标准量子力学公设推出。
Zurek's derivation of the Born rule from envariance (environment-assisted invariance) is tightened up, somewhat generalized, and extended to encompass all possibilities. By this, besides Zurek's most important work, also the works of 5 other commentators of the derivation are taken into account, and selected excerpts commented upon. All this is done after a detailed theory of twin unitaries, which are the other face of envariance.
Testing Born's rule via photoionization of helium
本研究提出利用先进阿秒光电离实验通过Sorkin检验验证玻恩定则，模拟显示考虑实验噪声和采集效率后精度可达现有最优Sorkin检验水平，并探讨了更多量子力学基础检验的实现方案。
It is often claimed that the collapse of the wave function and Born's rule to interpret the square of the norm as a probability, have to be introduced as separate axioms in quantum mechanics besides the Schroedinger equation. Here we show that this is not true in certain models where quantum behavior can be attributed to underlying deterministic equations. It is argued that indeed the apparent spontaneous collapse of wave functions and Born's rule are features that strongly point towards determinism underlying quantum mechanics.
In this work we derive Born's rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to a environement made of "qubits" (i.e., Bohmian pointers) we show that entanglement together with deterministic chaos lead to a fast relaxation from any statistitical distribution $ρ(x)$ (of finding a particle at point $x$) to the Born probability law $|Ψ(x)|^2$. Our model is discussed in the context of Boltzmann's kinetic theory and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.
The spatiotemporal Born rule is quasiprobabilistic
本文提出一种准概率描述，将量子理论中的时空本规则扩展到时间领域，并结合量子贝叶斯规则，为开放量子系统的时间反转对称性提供操作性定义。
The auxiliary rules of quantum mechanics have always included the Born rule that connects probability with square modulus. This need not be the case, for it is possible to introduce probability into the theory through probability current alone. When this is done, other rules can provide for stochastically triggered measurements within a system of any size, microscopic or macroscopic; and solutions to the Schrodinger equation can be consistently applied to individual trials, not just to ensembles of trials. Other advantages appear. The rules can then resolve the paradox associated with the Schr…
Zurek's derivation of Born's rule using envariance (invariance due to entanglement) is considered to capture the probability in full generality, but only as applied to measurement of a quantum observable. Contrariwise, textbook formulations of Born's rule begin with a pure state of a closed, undivided system. The task of this study is to show that a rearrangement of the Zurek approach is possible in which the latter is viewed as giving the probabilities for Schmidt states of an arbitrary composite state vector, and afterwards it is extended to probabilities in a closed, undivided system. This …
We formalize the hidden measurement approach within the very general notion of an interactive probability model. We narrow down the model by assuming the state space of a physical entity is a complex Hilbert space and introduce the principle of consistent interaction which effectively partitions the space of apparatus states. The normalized measure of the set of apparatus states that interact with a pure state giving rise to a fixed outcome is shown to be in accordance with the probability obtained using the Born rule.
The possibility to recover the which-way information, for example in the two slit experiment, is based on a natural but implicit assumption about the position of a particle {\it before} a position measurement is performed on it. This assumption cannot be deduced from the standard postulates of quantum mechanics. In the present paper this assumption is made explicit and formally postulated as a new rule, the {\it quantum typicality rule}. This rule correlates the positions of the particles at two different times, thus defining their trajectories. Unexpectedly, this rule is also equivalent to th…
We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr's doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C*-algebra of observables is empirically accessible only through associated commutative C*-algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of single-case probabilities (which will be the subject of a sequel pa…
We study in detail the trajectories, ordered and chaotic, of two entangled Bohmian qubits when their initial preparation satisfies (or not) Born's rule for various amounts of quantum entanglement. For any non zero value of entanglement ordered and chaotic trajectories coexist and the proportion of ordered trajectories increases with the decrease of the entanglement. In the extreme cases of zero and maximum entanglement we have only ordered and chaotic trajectories correspondingly. The chaotic trajectories of this model are ergodic, for any given value of entanglement, namely the limiting distr…
The Born rule is derived from operational assumptions, independent of the normalization of the state. Unlike Gleason's theorem, the argument applies even if probabilities are defined for only a single resolution of the identity, so it applies to all the major foundational approaches to quantum mechanics. There are important points of contact with Deutsch's program for deducing the probabilistic interpretation of quantum mechanics from decision thoery, as recently completed by Wallace. Decision theory can be used to supplement the present derivation, in application to the Everett interpretation…
Relaxation to quantum equilibrium and the Born rule in Nelson's stochastic dynamics
该研究通过数值模拟Nelson随机动力学的三种量子系统，发现引力场中自由下落粒子可在量子弛豫完成前出现干涉，为区分标准量子力学与Nelson理论提供实验方向。
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…
Unified Quantum Dynamics: The Emergence of the Born Rule
本文提出含“小信号截断”的修正Schrödinger方程，证明Born规则可自然涌现，并将多世界分支及其统计性质统一于单一运动定律。
The Born rule is part of the collapse axiom in the standard version of quantum theory, as presented by standard textbooks on the subject. We show here that its signature quadratic dependence follows from a single additional physical assumption beyond the other axioms - namely, that the probability of a particular measurement outcome (the state $ϕ_k$, say) is independent of the choice of observable to be measured, so long as one of its eigenstates corresponds to that outcome. We call this assumption ``observable independence.'' As a consequence, the Born rule cannot be completely eliminated fro…
To understand the dynamical origin of the measurement in quantum mechanics, several models have been put forward which have a quantum system coupled to an apparatus. The system and the apparatus evolve in time and the Born rule for the system to be in various eigenstates of the observable is naturally obtained. In this work, we show that the effect of the drive-induced dissipation in such a system can lead to the Born rule, even if there is no separate apparatus. The applied drive needs to be much stronger than the system-environment coupling. In this condition, we show that the dynamics of a …
The Born rule, a foundational axiom used to deduce probabilities of events from wavefunctions, is indispensable in the everyday practice of quantum physics. It is also key in the quest to reconcile the ostensibly inconsistent laws of the quantum and classical realms, as it confers physical significance to reduced density matrices, the essential tools of decoherence theory. Following Bohr's Copenhagen interpretation, textbooks postulate the Born rule outright. However, recent attempts to derive it from other quantum principles have been successful, holding promise for simplifying and clarifying…
Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. In this paper, assuming that our usual inference procedure makes sense for every set of logical propositions represented in terms of commuting projectors on a given Hilbert space, we extend the logical interpretation to quantum mechanics and derive the Born rule. Our result implies that, from the epistemological viewpoints, we can regard quantum mechanics as a natur…
During the last ten years or so, derivations of the Born rule based on decision theory have been proposed and developed, and it is claimed that these are valid in the context of the Everett interpretation. This claim is critically assessed and it is shown that one of its key assumptions, although natural in the context of the Copenhagen interpretation, is not consistent with that of Everett. It is further argued that any interpretation that relates outcome likelihood to the expansion coefficients connecting the wavefunction with the eigenfunctions of the measurement operator must be inconsiste…
The Born rule -- 100 years ago and today
本文梳理玻恩定则从1926年提出至今的发展历程，涵盖早期历史、POVM推广及现代推导，同时指出其具有有限适用域，超出范围应用会产生问题。
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…
This work is an attempt to justify Born's rule within the framework of the many-minds interpretation seen as a development of the many-worlds interpretation of Everett. More precisely, here we develop a unitary model of many-minds based on the work of Albert and Loewer (Synthese \textbf{77}, 195 (1988)). At the difference of previous models ours is not genuinely stochastic and dualistic and also involves some classical-like randomness concerning the initial conditions of the Universe. We also compare the present method for recovering Born's rule with previous works based on decision theory \em…
Born's Rule from Quantum Frequentism
本文提出玻恩规则的广义形式，证明任意可分解态测量中不存在违反玻恩规则的历史，从而说明纯幺正演化可在更宽松条件下与玻恩规则共存，并将其应用于量子电路保真度，提出基于自信息率的单次基准测试替代方案。
In this paper I present a thought experiment that gives different results depending on whether or not the wavefunction collapses. Since the wavefunction does not obey the Schrodinger equation during the collapse, conservation laws are violated. This is the reason why the results are different. Quantities that are conserved if the wavefunction does not collapse might change if it does. I also show that using the Born Rule to derive probabilities of states before a measurement given the state after it (rather than the other way round as it is usually used) leads to the conclusion that the memori…
The Born rule provides a fundamental connection between theory and observation in quantum mechanics, yet its origin remains a mystery. We consider this problem within the context of quantum optics using only classical physics and the assumption of a quantum electrodynamic vacuum that is real rather than virtual. The connection to observation is made via classical intensity threshold detectors that are used as a simple, deterministic model of photon detection. By following standard experimental conventions of data analysis on discrete detection events, we show that this model is capable of repr…
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。
Does Born Rule Imply Unitarity of Time Evolution in Quantum Mechanics?
探讨量子力学中玻恩规则与时间演化幺正性的关系，指出认为玻恩规则暗示幺正性的观点有缺陷，即便动力学非幺正，玻恩规则也能确保总概率守恒，适用于多种量子系统系综，还讨论了相关守恒及反对观点。
Synthesizing the Born rule with reinforcement learning
本文通过强化学习算法模拟非理想智能体对SIC测量结果下注，量化其决策偏离玻恩规则的程度并分析限制因素，同时提出单光子实验实现方案。
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