It is part of information theory folklore that, while quantum theory prohibits the generic cloning of states, such cloning is allowed by classical information theory. Indeed, many take the phenomenon of no-cloning to be one of the features that distinguishes quantum mechanics from classical mechanics. In this paper, we use symplectic geometry to argue that pace conventional wisdom, in the case where one does not include a machine system, there is an analog of the no-cloning (...) class='Hi'>theorem for classical systems. However, upon adjoining a non-trivial machine system one finds that, pace the quantum case, the obstruction to cloning disappears for pure states. We then discuss the difference between this result and the quantum case, and show that it can be explained in terms of the rigidity of the theories' respective geometries. Finally, we discuss the relationship between this result and classical no-cloning arguments in the context of symmetric monoidal categories and statistical classical mechanics. (shrink)

Our conscious minds exist in the Universe, therefore they should be identified with physical states that are subject to physical laws. In classical theories of mind, the mental states are identified with brain states that satisfy the deterministic laws of classical mechanics. This approach, however, leads to insurmountable paradoxes such as epiphenomenal minds and illusionary free will. Alternatively, one may identify mental states with quantum states realized within the brain and try to resolve the above paradoxes using the standard Hilbert (...) space formalism of quantum mechanics. In this essay, we first show that identification of mind states with quantum states within the brain is biologically feasible, and then elaborating on the mathematical proofs of two quantum mechanical no-go theorems, we explain why quantum theory might have profound implications for the scientific understanding of one's mental states, self identity, beliefs and free will. (shrink)

The aim of the paper is to investigate the role played by causality, and more specifically the no-signaling condition, in the assessment of the quantum theory. To this end, we discuss why it is important that even a non-relativistic theory such as Quantum Mechanics doesn’t imply a violation of this condition. Then, we use this argument to prove an original result stating that the destructive behaviour of the measurement process on the entanglement properties of quantum systems is a necessary and (...) unavoidable feature of the quantum theory. Finally, we critically review the no-cloning theorem. The original formulation of the theorem states that a linear quantum cloning machine, designed in order to successfully clone states that coincide with appropriate basis vectors, fails to copy states that are a non-trivial superposition of those basis vectors; we will furthermore prove that such a linear cloning device, even with the hypothesis that it can only clone basis vectors successfully, may provide a violation of the no-signaling condition and therefore cannot exist. (shrink)

No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal \ -imperfect cloning operation which tolerates a finite loss \ of fidelity in the cloned state, and show that an individual system’s algebra of observables is abelian if and only if there is a universal (...) \ -imperfect cloning operation in the case where the loss of fidelity is less than \ . Therefore in this case no universal \ -imperfect cloning operation is possible in algebraic quantum theory. (shrink)

The 'no-supervenience' theorem (Reason, 2019; Reason and Shah, 2021) is a proof that no fully self-aware system can entirely supervene on any objectively observable system. I here present a simple, non-technical summary of the proof and demonstrate its implications for four separate theories of consciousness: the 'property dualism' theory of David Chalmers; the 'reflexive monism' of Max Velmans; Galen Strawson's 'realistic monism'; and the 'illusionism' of Keith Frankish. It is shown that all are ruled out in their current form (...) by the no-supervenience theorem, except for Chalmers' theory, which I show requires humans to behave irrationally. (shrink)

It seems to me that it is among the most sure-footed of quantum physicists, those who have it in their bones, that one finds the greatest impatience with the idea that the ‘foundations of quantum mechanics’ might need some attention. Knowing what is right by instinct, they can become a little impatient with nitpicking distinctions between theorems and assumptions. —John Stewart Bell [4, p. 33].

In this article I raise a new problem for quantum mechanics, which I call the control problem. Like the measurement problem, the control problem places a fundamental constraint on quantum theories. The characteristic feature of the problem is its focus on state preparation. In particular, whereas the measurement problem turns on a premise about the completeness of the quantum state ('no hidden variables'), the control problem turns on a premise about our ability to prepare or control quantum states. After raising (...) the problem, I discuss some applications. I suggest that it provides a useful new lens through which to view existing theories or interpretations, in part because it draws attention to aspects of those theories that the measurement problem does not. I suggest that it also helps clarify the physical significance of the well-known no-go result—the no-cloning theorem—on which it is based. (shrink)

We prove a no-triplets theorem for spin 1 particles, which implies a strengthened form of the no-cloning theorem.

In his classic work The Mind and its Place in Nature published in 1925 at the height of the development of quantum mechanics but several years after the chemists Lewis and Langmuir had already laid the foundations of the modern theory of valence with the introduction of the covalent bond, the analytic philosopher C. D. Broad argued for the emancipation of chemistry from the crass physicalism that led physicists then and later—with support from a rabblement of philosophers who knew as (...) much about chemistry as etymologists—to believe that chemistry reduced to physics. Here Broad’s thesis is recast in terms more familiar to chemists. In the hard sell of particle physics, several prominent figures in chemistry—Hoffmann, Primas, and Pauling—have had their views interpreted to imply that they were sympathetic to greedy reductionism when in fact they were not. Indeed, being chemists without physicists as alter egos, they could not but side with Broad’s contention that chemistry, as a science that deals primarily in emergent phenomena which are beyond the purview of physicalism, owes no acquiescence to particle physics and its ethereal wares. Historically, among the most widely used expediencies in chemistry and materials science are additivity or mixture rules and their cohort transferability, all of which are devised and used under the mantle of naive reductionism. Here it is argued that while the transfer of functional groups between molecules works empirically to an extent, it is strictly outlawed by the no-cloning theorem of quantum mechanics. Several illustrative examples related to chemistry’s irreducibility to physics are presented and discussed. The failure of naive reductionism exhibited by the deep-inelastic scattering of leptons by A > 2 nuclei is traced to the same flawed reasoning that was the original basis of Moffitt’s ‘atoms in molecules’ hypothesis, the neglect of context, nuclei in the case of high-energy physics and molecules in the case of chemistry. A non-exhaustive list of other contexts from physics, chemistry, and molecular biology evidencing similar departures from the ideal of additivity or reductionism is provided for the perusal of philosophers. Had the call by the mathematician J. T. Schwartz for developments in mathematical linguistics possessed of a less single, less literal, and less simple-minded nature been met, perhaps it might have persuaded scientists to abandon their regressive fixation with unphysical reductionism and to adapt to new methodologies that engender a more nuanced handling of ubiquitous emergent phenomena as they arise in Nature than is the case today. (shrink)

The no-free-lunch theorems promote a skeptical conclusion that all possible machine learning algorithms equally lack justification. But how could this leave room for a learning theory, that shows that some algorithms are better than others? Drawing parallels to the philosophy of induction, we point out that the no-free-lunch results presuppose a conception of learning algorithms as purely data-driven. On this conception, every algorithm must have an inherent inductive bias, that wants justification. We argue that many standard learning algorithms should rather (...) be understood as model-dependent: in each application they also require for input a model, representing a bias. Generic algorithms themselves, they can be given a model-relative justification. (shrink)

A recent claim by Meehan that quantum mechanics has a new “control problem” that puts limits on our ability to prepare quantum states and revises our understanding of the no-cloning theorem is examined. We identify flaws in Meehan’s analysis and argue that such a problem does not exist.

White proposes an a priori justification of the reliability of inductive prediction methods based on his thesis of induction-friendliness. It asserts that there are by far more induction-friendly event sequences than induction-unfriendly event sequences. In this paper I contrast White's thesis with the famous no free lunch theorem. I explain two versions of this theorem, the strong NFL theorem applying to binary and the weak NFL theorem applying to real-valued predictions. I show that both versions refute (...) the thesis of induction-friendliness. In the conclusion I argue that an a priori justification of the reliability of induction based on a uniform probability distribution over possible event sequences is impossible. In the outlook I consider two alternative approaches: justification externalism and optimality justifications. (shrink)

In the history of quantum physics several no-go theorems have been proved, and many of them have played a central role in the development of the theory, such as Bell’s or the Kochen–Specker theorem. A recent paper by F. Laudisa has raised reasonable doubts concerning the strategy followed in proving some of these results, since they rely on the standard framework of quantum mechanics, a theory that presents several ontological problems. The aim of this paper is twofold: on the (...) one hand, I intend to reinforce Laudisa’s methodological point by critically discussing Malament’s theorem in the context of the philosophical foundation of quantum field theory; secondly, I rehabilitate Gisin’s theorem showing that Laudisa’s concerns do not apply to it. (shrink)

The important recent book by Schurz ( 2019 ) appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only concern the case where there is a uniform prior—they prove that there are “as many priors” (loosely speaking) for which any induction algorithm _A_ out-generalizes some induction algorithm _B_ as vice-versa. Importantly though, in addition to the NFL theorems, there are many _free lunch_ theorems. (...) In particular, the NFL theorems can only be used to compare the expected performance of an induction algorithm _A_, considered in isolation, with the expected performance of an induction algorithm _B_, considered in isolation. There is a rich set of free lunches which instead concern the statistical _correlations_ among the generalization errors of induction algorithms. As I describe, the meta-induction algorithms that Schurz advocates as a “solution to Hume’s problem” are simply examples of such a free lunch based on correlations among the generalization errors of induction algorithms. I end by pointing out that the prior that Schurz advocates, which is uniform over bit frequencies rather than bit patterns, is contradicted by thousands of experiments in statistical physics and by the great success of the maximum entropy procedure in inductive inference. (shrink)

No-free-lunch theorems are important theoretical result in the fields of machine learning and artificial intelligence. Researchers in this fields often claim that the theorems are based on Hume’s argument about induction and represent a formalisation of the argument. This paper argues that this is erroneous but that the theorems correspond to and formalise Goodman’s new riddle of induction. To demonstrate the correspondence among the theorems and Goodman’s argument, a formalisation of the latter in the spirit of the former is sketched.

Recent experiments have shown that certain fluid-mechanical systems, namely oil droplets bouncing on oil films, can mimic a wide range of quantum phenomena, including double-slit interference, quantization of angular momentum and Zeeman splitting. Here I investigate what can be learned from these systems concerning no-go theorems as those of Bell and Kochen-Specker. In particular, a model for the Bell experiment is proposed that includes variables describing a ‘background’ field or medium. This field mimics the surface wave that accompanies the droplets (...) in the fluid-mechanical experiments. It appears that quite generally such a model can violate the Bell inequality and reproduce the quantum statistics, even if it is based on local dynamics only. The reason is that measurement independence is not valid in such models. This opens the door for local ‘background-based’ theories, describing the interaction of particles and analyzers with a background field, to complete quantum mechanics. Experiments to test these ideas are also proposed. (shrink)

We discuss the role of the notion of information in the description of physical reality. We consider theories for which dynamics is linear with respect to stochastic mixing. We point out that the no-cloning and no-deleting principles emerge in any such theory, if law of conservation of information is valid, and two copies contain more information than one copy. We then describe the quantum case from this point of view.

The debate over the question whether quantum mechanics should be considered as a complete account of microphenomena has a long and deeply involved history, a turning point in which has been certainly the Einstein-Bohr debate, with the ensuing charge of incompleteness raised by the Einstein-Podolsky-Rosen argument. In quantum mechanics, physical systems can be prepared in pure states that nevertheless have in general positive dispersion for most physical quantities; hence in the EPR argument, the attention is focused on the question whether (...) the account of the microphysical phenomena provided by quantum mechanics is to be regarded as an exhaustive description of the physical reality to which those phenomena are supposed to refer, a question to which Einstein himself answered in the negative. However, there is a mathematical side of the completeness issue in quantum mechanics, namely the question whether the kind of states with positive dispersion can be represented as a different, dispersion-free kind of states in a way consistent with the mathematical constraints of the quantum mechanical formalism. From this point of view, the other source of the completeness issue in quantum mechanics is the no hidden variables theorem formulated by John von Neumann in his celebrated book on the mathematical foundations of quantum mechanics, the preface of which already anticipates the program and the conclusion concerning the possibility of ‘neutralizing’ the statistical character of quantum mechanics. (shrink)

We present a discrete model theory similar in structure to ordinary quantum mechanics, but based on a finite field instead of complex amplitudes. The interpretation of this theory involves only the “modal” concepts of possibility and necessity rather than quantitative probability measures. Despite its simplicity, our model theory includes entangled states and has versions of both Bell’s theorem and the no cloning theorem.

In this paper we show that ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}-ontic models, as defined by Harrigan and Spekkens (HS), cannot reproduce quantum theory. Instead of focusing on probability, we use information theoretic considerations to show that all pure states of ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}-ontic models must be orthogonal to each other, in clear violation of quantum mechanics. Given that (i) Pusey, Barrett and Rudolph (PBR) previously showed that ψ\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}-epistemic models, as defined by HS, also contradict quantum mechanics, and (ii) the HS categorization is exhausted by these two types of models, we conclude that the HS categorization itself is problematic as it leaves no space for models that can reproduce quantum theory. (shrink)

I argue that our judgements regarding the locally causal models that are compatible with a given constraint implicitly depend, in part, on the context of inquiry. It follows from this that certain quantum no-go theorems, which are particularly striking in the traditional foundational context, have no force when the context switches to a discussion of the physical systems we are capable of building with the aim of classically reproducing quantum statistics. I close with a general discussion of the possible implications (...) of this for our understanding of the limits of classical description, and for our understanding of the fundamental aim of physical investigation. _1_ Introduction _2_ No-Go Results _2.1_ The CHSH inequality _2.2_ The GHZ equality _3_ Classically Simulating Quantum Statistics _3.1_ GHZ statistics _3.2_ Singlet statistics _4_ What Is a Classical Computer Simulation? _5_ Comparing the All-or-Nothing GHZ with Statistical equalities _6_ General Discussion _7_ Conclusion. (shrink)

A recent no-go theorem gives an extension of the Wigner’s Friend argument that purports to prove the “Quantum theory cannot consistently describe the use of itself.” The argument is complex and thought provoking, but fails in a straightforward way if one treats QM as a statistical theory in the most fundamental sense, i.e. if one applies the so-called ensemble interpretation. This explanation is given here at an undergraduate level, which can be edifying for experts and students alike. A recent (...) paper has already shown that the no-go theorem is incorrect with regard to the de Broglie Bohm theory and misguided in some of its general claims. This paper’s contribution is three fold. It shows how the extended Wigner’s Friend argument fails in the ensemble interpretation. It also makes more evident how natural a consistent statistical treatment of the wave function is. In this way, the refutation of the argument is useful for bringing out the core statistical nature of QM. It, in addition, manifests the unnecessary complications and problems introduced by the collapse mechanism that is part of the Copenhagen interpretation. The paper uses the straightforwardness of the ensemble interpretation to make the no-go argument and its refutation more accessible. (shrink)

In this article I raise a new problem for quantum mechanics, which I call the control problem. Like the measurement problem, the control problem places a fundamental constraint on quantum theories. The characteristic feature of the problem is its focus on state preparation. In particular, whereas the measurement problem turns on a premise about the completeness of the quantum state (‘no hidden variables’), the control problem turns on a premise about our ability to prepare or control quantum states. After raising (...) the problem, I discuss some applications. I suggest that it provides a useful new lens through which to view existing theories or interpretations, in part because it draws attention to aspects of those theories that the measurement problem does not (such as the role of conditional and relative states). I suggest that it also helps clarify the physical significance of the well-known no-go result—the no-cloning theorem—on which it is based. (shrink)

Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis $\alpha _{i}$ chosen by Alice, irrespective of Bob’s axis $\beta _{j}$ (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s (...) and Bob’s spins are identified as $A_{ij}$ and $B_{ij}$ , even though their distributions are determined by, respectively, $\alpha _{i}$ alone and $\beta _{j}$ alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins $A_{ij},B_{ij}$ across all values of $\alpha _{i},\beta _{j}$ are constrained by fixing distributions of some subsets thereof. Of special interest among these subsets is the set of probabilistic connections, defined as the pairs $\left( A_{ij},A_{ij'}\right) $ and $\left( B_{ij},B_{i'j}\right) $ with $\alpha _{i}\not =\alpha _{i'}$ and $\beta _{j}\not =\beta _{j'}$ (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of $A_{ij}\not =A_{ij'}$ and $B_{ij}\not =B_{i'j}$ ). Thus, one can achieve a complete KPT characterization of the Bell-type inequalities, or Tsirelson’s inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs $\left( A_{ij},B_{ij}\right) $ that are subject to these inequalities. We show, however, that quantum-mechanical (QM) constraints are special. No-forcing theorem says that if a set of probabilistic connections is not compatible with correlations violating QM, then it is compatible only with the classical–mechanical correlations. No-matching theorem says that there are no subsets of the spin variables $A_{ij},B_{ij}$ whose distributions can be fixed to be compatible with and only with QM-compliant correlations. (shrink)

Within the framework of general relativity, in some cases at least, it is a delicate and interesting question just what it means to say that an extended body is or is not "rotating". It is so for two reasons. First, one can easily think of different criteria of rotation. Though they agree if the background spacetime structure is sufficiently simple, they do not do so in general. Second, none of the criteria fully answers to our classical intuitions. Each one exhibits (...) some feature or other that violates those intuitions in a significant and interesting way. The principal goal of the paper is to make the second claim precise in the form of a modest no-go theorem. (shrink)

Modal interpretations take quantum mechanics as a theory which assigns at all times definite values to magnitudes of quantum systems. In the case of single systems, modal interpretations manage to do so without falling prey to the Kochen and Specker no-go theorem, because they assign values only to a limited set of magnitudes. In this paper I present two further no-go theorems which prove that two modal interpretations become nevertheless problematic when applied to more than one system. The first (...) theorem proves that the modal interpretation proposed by Kochen and by Dieks cannot correlate the values simultaneously assigned to three systems. The second and new theorem proves that the atomic modal interpretation proposed by Bacciagaluppi and Dickson and by Dieks cannot correlate the values simultaneously and sequentially assigned to two systems if one assumes that these correlations are uniquely related to the dynamics of the state of the systems. (shrink)

Modal interpretations take quantum mechanics as a theory which assigns at all times definite values to magnitudes of quantum systems. In the case of single systems, modal interpretations manage to do so without falling prey to the Kochen and Specker no-go theorem, because they assign values only to a limited set of magnitudes. In this paper I present two further no-go theorems which prove that two modal interpretations become nevertheless problematic when applied to more than one system. The first (...) theorem proves that the modal interpretation proposed by Kochen and by Dieks cannot correlate the values simultaneously assigned to three systems. The second and new theorem proves that the atomic modal interpretation proposed by Bacciagaluppi and Dickson and by Dieks cannot correlate the values simultaneously and sequentially assigned to two systems if one assumes that these correlations are uniquely related to the dynamics of the state of the systems. (shrink)

This paper derives a no-trade theorem under Knightian uncertainty, which generalizes the theorem of Milgrom and Stokey by allowing general preference relations. It is shown that the no-trade theorem holds true as long as agents' preferences are dynamically consistent in the sense of Machina and Schmeidler, and satisfies the so-called piece-wise monotonicity axiom. A preference satisfying the piece-wise monotonicity axiom does not necessarily imply the additive utility representation, nor is necessarily based on beliefs.

The present paper investigates the philosophical relationship between John von Neumann’s Nohidden-variable theorem and Bell’s inequalities. Bell erroneously takes the axiomatic method as implying a finality claim and thus ignores von Neumann’s strongly pragmatist stance towards mathematical physics. If one considers, however, Hilbert’s axiomatic method as a critical enterprise, Bell’s theorem improves von Neumann’s by defining a more appropriate notion of ‘ hidden variable’ that permits one to include Bohm’s interpretation which recovers the predictive content of quantum mechanics. (...) Contrary to Bell’s belief, accepting this model does not require adopting the metaphysically realist Böhm picture. If one takes the latter as a physical research programme one sees that it only partly disputes a common domain of facts with the mathematically oriented research programme of von Neumann. (shrink)

Modal interpretations take quantum mechanics as a theory which assigns at all times definite values to magnitudes of quantum systems. In the case of single systems, modal interpretations manage to do so without falling prey to the Kochen and Specker no-go theorem, because they assign values only to a limited set of magnitudes. In this paper I present two further no-go theorems which prove that two modal interpretations become nevertheless problematic when applied to more than one system. The first (...) theorem proves that the modal interpretation proposed by Kochen and by Dieks cannot correlate the values simultaneously assigned to three systems. The second and new theorem proves that the atomic modal interpretation proposed by Bacciagaluppi and Dickson and by Dieks cannot correlate the values simultaneously and sequentially assigned to two systems if one assumes that these correlations are uniquely related to the dynamics of the state of the systems. (shrink)

Uncertainty about the actual orientation of the measurement device has been claimed to open a loophole for hidden variable models of quantum mechanics. In this paper I describe the statistics of inaccurate spin measurements by unsharp spin observables. A no‐go theorem for hidden variable models of the inaccurate measurement statistics follows: There is a finite set of directions for which not all results of inaccurate spin measurements can be predetermined in a non‐contextual way. In contrast to an earlier (...) class='Hi'>theorem (Breuer 2002) this result does not rely on the assigment of approximate truth values, and it holds under weaker assumptions on the measurement inaccuracy. (shrink)

Uncertainty about the actual orientation of the measurement device has been claimed to open a loophole for hidden variable models of quantum mechanics. In this paper I describe the statistics of inaccurate spin measurements by unsharp spin observables. A no-go theorem for hidden variable models of the inaccurate measurement statistics follows: There is a finite set of directions for which not all results of inaccurate spin measurements can be predetermined in a non-contextual way. In contrast to an earlier (...) class='Hi'>theorem [Breuer, Phys. Rev. Lett. 88(2002), 240402] this result does not rely on the assigment of approximate truth values, and it holds under weaker assumptions on the measurement inaccuracy. (shrink)

We present a toy theory that is based on a simple principle: the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge. Many quantum phenomena are found to have analogues within this toy theory. These include the noncommutativity of measurements, interference, the multiplicity of convex decompositions of a mixed state, the impossibility of discriminating nonorthogonal states, the impossibility of a universal (...) state inverter, the distinction between bipartite and tripartite entanglement, the monogamy of pure entanglement, no cloning, no broadcasting, remote steering, teleportation, entanglement swapping, dense coding, mutually unbiased bases, and many others. The diversity and quality of these analogies is taken as evidence for the view that quantum states are states of incomplete knowledge rather than states of reality. A consideration of the phenomena that the toy theory fails to reproduce, notably, violations of Bell inequalities and the existence of a Kochen-Specker theorem, provides clues for how to proceed with this research program. (shrink)

In the same way as the quantum no-cloning theorem and quantum key distribution in two preceding papers, entanglement-assisted quantum teleportation and Grover’s search algorithm are generalized by transferring them to an abstract setting, including usual quantum mechanics as a special case. This again shows that a much more general and abstract access to these quantum mechanical features is possible than commonly thought. A non-classical extension of conditional probability and, particularly, a very special type of state-independent conditional probability are (...) used instead of Hilbert spaces and wavefunctions. (shrink)

In the area of the foundations of quantum mechanics a true industry appears to have developed in the last decades, with the aim of proving as many results as possible concerning what there cannot be in the quantum realm. In principle, the significance of proving ‘no-go’ results should consist in clarifying the fundamental structure of the theory, by pointing out a class of basic constraints that the theory itself is supposed to satisfy. In the present paper I will discuss some (...) more recent no-go claims and I will argue against the deep significance of these results, with a two-fold strategy. First, I will consider three results concerning respectively local realism, quantum covariance and predictive power in quantum mechanics, and I will try to show how controversial the main conditions of the negative theorem turn out to be—something that strongly undermines the general relevance of these theorems. Second, I will try to discuss what I take to be a common feature of these theoretical enterprises, namely that of aiming at establishing negative results for quantum mechanics in absence of a deeper understanding of the overall ontological content and structure of the theory. I will argue that the only way toward such an understanding may be to cast in advance the problems in a clear and well-defined interpretational framework—which in my view means primarily to specify the ontology that quantum theory is supposed to be about—and after to wonder whether problems that seemed worth pursuing still are so in the framework. (shrink)

Conway and Kochen have presented a “free will theorem” [4, 6] which they claim shows that “if indeed we humans have free will, then [so do] elementary particles.” In a more precise fashion, they claim it shows that for certain quantum experiments in which the experimenters can choose between several options, no deterministic or stochastic model can account for the observed outcomes without violating a condition “MIN” motivated by relativistic symmetry. We point out that for stochastic models this conclusion (...) is not correct, while for deterministic models it is not new. In the way the free will theorem is formulated and proved, it only concerns deterministic models. But Conway and Kochen have argued [4, 5, 6, 7] that “randomness can’t help,” meaning that stochastic models are excluded as well if we insist on the conditions “SPIN”, “TWIN”, and “MIN”. We point out a mistake in their argument. Namely, the theorem is of the form deterministic model with SPIN & TWIN & MIN ⇒ contradiction , (1) and in order to derive the further claim, which is of the form stochastic model with SPIN & TWIN & MIN ⇒ contradiction , (2) Conway and Kochen propose a method for converting any stochastic model into a deterministic one [4]. (shrink)

The no free lunch theorem is a radicalized version of Hume’s induction skepticism. It asserts that relative to a uniform probability distribution over all possible worlds, all computable prediction algorithms—whether ‘clever’ inductive or ‘stupid’ guessing methods —have the same expected predictive success. This theorem seems to be in conflict with results about meta-induction. According to these results, certain meta-inductive prediction strategies may dominate other methods in their predictive success. In this article this conflict is analyzed and dissolved, by (...) means of probabilistic analysis and computer simulation. (shrink)

As duplicating techniques in art become more sophisticated and less easily detectable, Concern mounts as to what the status of these aesthetic clones should be. Can they be considered works of art; and to what extent do they share in the value of the original? this paper examines these questions, Using examples from audio and visual arts, And concludes that talk of an original and its copies has to do with the art market and not with the creative process. The (...) implication for aesthetic clones is that they are no different from their originals as far as aesthetic value is concerned. (shrink)

In [15], [16] G. Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals Φ A of order type 0 which realize the Herbrand normal form A H of A. Subsequently more perspicuous proofs of this fact via functional interpretation (combined with normalization) and cut-elimination were found. These proofs however (...) do not carry out the no-counterexample interpretation as a local proof interpretation and don't respect the modus ponens on the level of the no-counterexample interpretation of formulas A and A → B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (δ) and--at least not constructively-- (γ) which are part of the definition of an 'interpretation of a formal system' as formulated in [15]. In this paper we determine the complexity of the no-counterexample interpretation of the modus ponens rule for (i) PA-provable sentences, (ii) for arbitrary sentences A, B ∈ L(PA) uniformly in functionals satisfying the no-counterexample interpretation of (prenex normal forms of) A and A → B, and (iii) for arbitrary A, B ∈ L(PA) pointwise in given α( 0 ) -recursive functionals satisfying the no-counterexample interpretation of A and A → B. This yields in particular perspicuous proofs of new uniform versions of the conditions (γ), (δ). Finally we discuss a variant of the concept of an interpretation presented in [17] and show that it is incomparable with the concept studied in [15], [16]. In particular we show that the no-counterexample interpretation of PA n by α( n (ω))-recursive functionals (n ≥ 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (δ). (shrink)

This work-in-progress paper consists of four points which relate to the foundations and physical realization of quantum computing. The first point is that the qubit cannot be taken as the basic unit for quantum computing, because not every superposition of bit-strings of length n can be factored into a string of n-qubits. The second point is that the “No-cloning” theorem does not apply to the copying of one quantum register into another register, because the mathematical representation of this (...) copying is the identity operator, which is manifestly linear. The third point is that quantum parallelism is not destroyed only by environmental decoherence. There are two other forms of decoherence, which we call measurement decoherence and internal decoherence, that can also destroy quantum parallelism. The fourth point is that processing the contents of a quantum register “one qubit at a time” destroys entanglement. (shrink)

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey (...) class='Hi'>Theorem is not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$. (shrink)

If the concept of “free will” is reduced to that of “choice” all physical world share the latter quality. Anyway the “free will” can be distinguished from the “choice”: The “free will” involves implicitly certain preliminary goal, and the choice is only the mean, by which it can be achieved or not by the one who determines the goal. Thus, for example, an electron has always a choice but not free will unlike a human possessing both. Consequently, and paradoxically, the (...) determinism of classical physics is more subjective and more anthropomorphic than the indeterminism of quantum mechanics for the former presupposes certain deterministic goal implicitly following the model of human freewill behavior. The choice is usually linked to very complicated systems such as human brain or society and even often associated with consciousness. In its background, the material world is deterministic and absolutely devoid of choice. However, quantum mechanics introduces the choice in the fundament of physical world, in the only way, in which it can exist: All exists in the “phase transition” of the present between the uncertain future and the well-ordered past. Thus the present is forced to choose in order to be able to transform the coherent state of future into the well-ordering of past. The concept of choice as if suggests that there is one who chooses. However quantum mechanics involves a generalized case of choice, which can be called “subjectless”: There is certain choice, which originates from the transition of the future into the past. Thus that kind of choice is shared of all existing and does not need any subject: It can be considered as a low of nature. There are a few theorems in quantum mechanics directly relevant to the topic: two of them are called “free will theorems” by their authors, Conway and Kochen, and according to them: “Do we really have free will, or, as a few determined folk maintain, is it all an illusion? We don’t know, but will prove in this paper that if indeed there exist any experimenters with a modicum of free will, then elementary particles must have their own share of this valuable commodity” “The import of the free will theorem is that it is not only current quantum theory, but the world itself that is non-deterministic, so that no future theory can return us to a clockwork universe”. Those theorems can be considered as a continuation of the so-called theorems about the absence of “hidden variables” in quantum mechanics. (shrink)

Entangled states whose Wigner functions are non-negative may be viewed as being accounted for by local hidden variables (LHV). Recently, there were studies of Bell’s inequality violation (BIQV) for such states in conjunction with the well known theorem of Bell that precludes BIQV for theories that have LHV underpinning. We extend these studies to teleportation which is also based on entanglement. We investigate if, to what extent, and under what conditions may teleportation be accounted for via LHV theory. Our (...) study allows us to expose the role of various quantum requirements. These are, e.g., the uncertainty relation among non-commuting operators, and the no-cloning theorem which forces the complete elimination of the teleported state at its initial port. (shrink)

This chapter explains the interpersonal addition theorem. The theorem leads to two remarkable points. Firstly, it links the aggregation of good across the dimension of people with its aggregation across the dimension of states of nature. The result is that, in favourable circumstances, it links the value of equality in the distribution of good with the value of avoiding risk to good. The chapter also explains this link. The second point is even more remarkable. The theorem shows (...) that general utility is the sum of individual utilities. Since these utilities represent the general and individual betterness relations, this may seem, at first glance, to say that general good is the total of individual good. Joined with the addition theorem Bernoulli's hypothesis implies the utilitarian principle of distribution, which implies in turn that there is no value in avoiding inequality in the distribution of good between people. The chapter discusses the assessment of Bernoulli's hypothesis. (shrink)

We begin by considering two principles, each having the form causal completeness ergo screening-off. The first concerns a common cause of two or more effects; the second describes an intermediate link in a causal chain. They are logically independent of each other, each is independent of Reichenbach's principle of the common cause, and each is a consequence of the causal Markov condition. Simple examples show that causal incompleteness means that screening-off may fail to obtain. We derive a stronger result: in (...) a rather general setting, if the composite cause C1 & C2 & … & Cn screens-off one event from another, then each of the n component causes C1, C2, …, Cn must fail to screen-off. The idea that a cause may be ordinally invariant in its impact on different effects is defined; it plays an important role in establishing this no-go theorem. Along the way, we describe how composite and component causes can all screen-off when ordinal invariance fails. We argue that this theorem is relevant to assessing the plausibility of the two screening-off principles. The discovery of incomplete causes that screen-off is not evidence that causal completeness must engender screening-off. Formal and epistemic analogies between screening-off and determinism are discussed. (shrink)