Do we really understand quantum mechanics? Franck Laloë To cite this version: Franck Laloë. Do we really understand quantum mechanics?. American Journal of Physics, American Association of Physics Teachers, 2001, 69, pp.655 - 701. hal-00000001v2 HAL Id: hal-00000001 https://hal.archives-ouvertes.fr/hal-00000001v2 Submitted on 14 Nov 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Do we really understand quantum mechanics? Strange correlations, paradoxes and theorems. F. Lalo¨e Laboratoire de Physique de l’ENS, LKB, 24 rue Lhomond, F-75005 Paris, France February 4, 2011 Abstract This article presents a general discussion of several aspects of our present understanding of quantum mechanics. The emphasis is put on the very special correlations that this theory makes possible: they are forbidden by very general arguments based on realism and local causality. In fact, these correlations are completely impossible in any circumstance, except the very special situations designed by physicists especially to observe these purely quantum effects. Another general point that is emphasized is the necessity for the theory to predict the emergence of a single result in a single realization of an experiment. For this purpose, orthodox quantum mechanics introduces a special postu- late: the reduction of the state vector, which comes in addition to the Schr¨odinger evolution postulate. Nevertheless, the presence in parallel of two evolution processes of the same object (the state vector) may be a potential source for conflicts; various attitudes that are possible to avoid this problem are discussed in this text. After a brief historical introduction, recalling how the very special status of the state vector has emerged in quantum mechanics, various conceptual difficulties are introduced and discussed. The Einstein Podolsky Rosen (EPR) theo- rem is presented with the help of a botanical parable, in a way that emphasizes how deeply the EPR reasoning is rooted into what is often called “scientific method”. In another section the GHZ argument, the Hardy impossibilities, as well as the BKS theorem are introduced in simple terms. The final two sections attempt to give a summary of the present situation: one section discusses non-locality and entangle- ment as we see it presently, with brief mention of recent experiments; the last section contains a (non-exhaustive) list of various attitudes that are found among physicists, and that are helpful to alleviate the conceptual difficulties of quantum mechanics. 1 Contents 1 Historical perspective 4 1.1 Threeperiods........................... 5 1.1.1 Prehistory......................... 6 1.1.2 Theundulatoryperiod. 7 1.1.3 Emergence of the Copenhagen interpretation . 8 1.2 The status of the state vector . 9 1.2.1 Two extremes and the orthodox solution . 10 1.2.2 An illustration . 11 2 Difficulties, paradoxes 13 2.1 Von Neumann’s infinite regress . 14 2.2 Wigner’sfriend .......................... 16 2.3 Schr¨odinger’scat . 17 2.4 Unconvincing arguments . 19 3 Einstein, Podolsky and Rosen 20 3.1 Atheorem............................. 20 3.2 Ofpeas,podsandgenes . 22 3.2.1 Simple experiments; no conclusion yet. 22 3.2.2 Correlations; causes unveiled. 23 3.2.3 Transposition to physics . 25 4 Quantitative theorems: Bell, GHZ, Hardy, BKS 28 4.1 Bellinequalities.. .. .. .. .. .. .. .. 29 4.1.1 Two spins in a quantum singlet state . 29 4.1.2 Proof ........................... 30 4.1.3 Contradiction with quantum mechanics and with ex- periments ......................... 31 4.1.4 Generality of the theorem . 32 4.2 Hardy’simpossibilities . 35 4.3 GHZequality ........................... 37 4.4 Bell-Kochen-Specker; contextuality. 41 5 Non-locality and entanglement: where are we now? 44 5.1 Loopholes,conspiracies. 45 5.2 Locality, contrafactuality . 50 5.3 “All-or-nothing coherent states”; decoherence . 51 5.3.1 Definition and properties of the states . 51 5.3.2 Decoherence........................ 54 5.4 Quantum cryptography, teleportation . 57 5.4.1 Sharing cryptographic keys by quantum measurements 58 5.4.2 Teleporting a quantum state . 59 2 5.5 Quantum computing and information . 62 6 Various interpretations 64 6.1 Common ground; “correlation interpretation” . 65 6.2 Additional variables . 69 6.2.1 Generalframework . 69 6.2.2 Bohmian trajectories . 72 6.3 Modified (non-linear) Schr¨odinger dynamics . 74 6.3.1 Various forms of the theory . 75 6.3.2 Physical predictions . 77 6.4 History interpretation . 79 6.4.1 Histories, families of histories . 79 6.4.2 Consistent families . 81 6.4.3 Quantum evolution of an isolated system . 82 6.4.4 Comparison with other interpretations . 85 6.4.5 A profusion of points of view; discussion . 87 6.5 Everett interpretation . 89 Quantum mechanics describes physical systems through a mathematical object, the state vector Ψ >, which replaces positions and velocities of | classical mechanics. This is an enormous change, not only mathematically, but also conceptually. The relations between Ψ > and physical properties | are much less direct than in classical mechanics; the distance between the formalism and the experimental predictions leaves much more room for dis- cussions about the interpretation of the theory. Actually, many difficulties encountered by those who tried (or are still trying) to “really understand” quantum mechanics are related to questions pertaining to the exact status of Ψ >: for instance, does it describe the physical reality itself, or only some | partial knowledge that we might have of this reality? Does it fully describe ensemble of systems only (statistical description), or one single system as well (single events)? Assume that, indeed, Ψ > is affected by an imperfect | knowledge of the system; is it then not natural to expect that a better de- scription should exist, at least in principle? If so, what would be this deeper and more precise description of the reality? Another confusing feature of Ψ > is that, for systems extended in | space (for instance, a system made of two particles at very different loca- tions), it gives an overall description of all its physical properties in a single block from which the notion of space seems to have disappeared; in some cases, the physical properties of the two remote particles seem to be com- pletely “entangled” (the word was introduced by Schr¨odinger in the early 3 days of quantum mechanics) in a way where the usual notions of space- time and local events seem to become dimmed. Of course, one could think that this entanglement is just an innocent feature of the formalism with no special consequence: for instance, in classical electromagnetism, it is often convenient to introduce a choice of gauge for describing the fields in an in- termediate step, but we know very well that gauge invariance is actually fully preserved at the end. But, and as we will see below, it turns out that the situation is different in quantum mechanics: in fact, a mathematical entanglement in Ψ > can indeed have important physical consequences on | the result of experiments, and even lead to predictions that are, in a sense, contradictory with locality (we will see below in what sense). Without any doubt, the state vector is a rather curious object to describe reality; one purpose of this article is to describe some situations in which its use in quantum mechanics leads to predictions that are particularly unex- pected. As an introduction, and in order to set the stage for this discussion, we will start with a brief historical introduction, which will remind us of the successive steps from which the present status of Ψ > emerged. Paying | attention to history is not inappropriate in a field where the same recurrent ideas are so often rediscovered; they appear again and again, sometimes al- most identical over the years, sometimes remodelled or rephrased with new words, but in fact more or less unchanged. Therefore, a look at the past is not necessarily a waste of time! 1 Historical perspective The founding fathers of quantum mechanics had already perceived the essence of many aspects of the discussions on quantum mechanics; today, after al- most a century, the discussions are still lively and, if some very interesting new aspects have emerged, at a deeper level the questions have not changed so much. What is more recent, nevertheless, is a general change of atti- tude among physicists: until about 20 years ago, probably as a result of the famous discussions between Bohr, Einstein, Schr¨odinger, Heisenberg, Pauli, de Broglie and others (in particular at the famous Solvay meetings [1]), most physicists seemed to consider that “Bohr was right and proved his opponents to be wrong”, even if this was expressed with more nuance. In other words, the majority of physicists thought that the so called “Copenhagen interpre- tation” had clearly emerged from the infancy of quantum mechanics as the only sensible attitude for good scientists. As we all know, this interpretation introduced the idea that modern physics must contain indeterminacy as an essential ingredient: it is fundamentally impossible to predict the outcome of single microscopical events; it is impossible to go beyond the formalism of 4 the wave function (or its equivalent, the state vector Ψ >1) and complete | it; for some physicists, the Copenhagen interpretation also includes the diffi- cult notion of “complementarity”.... even if it is true that, depending on the context, complementarity comes in many varieties and has been interpreted in many different ways! By and large, the impression of the vast majority was that Bohr had eventually won the debate with Einstein, so that dis- cussing again the foundations of quantum mechanics after these giants was pretentious, useless, and maybe even bad taste.