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.
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.
Modern quantum physics now enables control of quantum systems at the level of individual trajectories, opening a new frontier that links quantum information theory, quantum many-body physics, and quantum thermodynamics, and uncovers novel non-equilibrium phenomena such as deep thermalization and measurement-induced entanglement. However, a central challenge remains: their characterization relies on measuring nonlinear properties of individual quantum states, a task tantamount to fine-grained cloning of a quantum ensemble. Here, the fundamental laws governing the cloning of quantum ensembles are investigated. First, a general no-cloning theorem for arbitrary ensembles is established from an information-theoretic perspective, even assuming multiple copies of the ensemble's purification. It is then shown that this barrier can be unexpectedly circumvented for physical ensembles generated by finite-time evolutions. Nevertheless, these tasks are proven to remain computationally intractable, even when the full circuit description of state preparation is known. This stands in sharp contrast to the conventional no-cloning theorem, which relies on the state being unknown. Together, these results establish new fundamental principles of quantum mechanics, reveal intrinsic trade-offs among sample complexity, computational complexity, and quantum measurements, and highlight the necessity of problem-specific strategies for probing measurement-induced quantum phenomena.
Quantum mechanics notoriously faces the measurement problem, the problem that if read thoroughly, it implies the nonexistence of definite outcomes in measurement procedures. A plausible reaction to this and to related problems is to regard a system’s quantum state |ψ> merely as an indication of our lack of knowledge about the system, i.e., to interpret it epistemically. However, there are radically different ways to spell out such an epistemic view of the quantum state. We here investigate new developments in the branch that introduces hidden variables λ in addition to the quantum state |ψ> and has its roots in Einstein’s views. In particular, we confront purported achievements of the view in the light of recent no-go results. It will be demonstrated that, despite the debatability of the premises of such results, they force the hidden variable theorist to make ad hoc moves which undermine the very conceptual basis of his own approach.
In most textbooks, measurement does not receive the full attention it deserves and sometimes is even not treated at all, apart a brief and cryptic mention of the “reduction of the wave packet.” However, in the last decades, the situation has profoundly changed and it is time to consider measurement a fundamental part of quantum mechanics, even, to a certain extent, an important generalization of the traditional theory (see also Chs. 14–15). This chapter consists of three major parts. In the first block (Secs. 9.1–9.4) we develop the main physical features of the measurement process: the heart of the argument is here represented by the aspects related to the interpretation. In the second part (Secs. 9.5–9.8) we discuss several special (and partly interdependent) topics of measurement: the heart here is represented by experimental aspects. In the third part (Secs. 9.9–9.12) we deal with the measurement process on a more formal plane, making use, in particular, of the generalization represented by the concepts of effect and positive operator valued measure (POVM). As we have said, the measurement problem is one of the most fundamental issues in the conceptual structure of quantum mechanics (as described in Sec. 9.1) and has a long history that will be examined in Sec. 9.2. In this context, the existence of apparently paradoxical quantum states comes about: the so-called Schrödinger cat states (see Sec. 9.3).
Hegerfeldt's theorem shows that the localization of a particle (or field) is not possible under very general assumptions of quantum mechanics and special relativity. This theorem, and further developments and associated theorems force penetrating questions on the proper understanding of the theory and its use. In particular, the use of the collapse postulate is called into question. The existence of particles and even localized fields are called into question. And, this in turn challenges local measurements. It generally forces one to consider the profound problems and even wrong results that arise from the collapse postulate. We give a qualitatively new example of such a wrong result. Researchers and students alike are bothered by these fundamental issues. Students' ability to learn the material, and even their interest in physics, can be compromised by such disturbing conundra that seem to make physics, or at least QFT and its QM limit, seem incomprehensible and even inconsistent. This article resolves these issues in a way accessible to undergraduates, by applying the natural (minimal) interpretation of quantum mechanics (QM) (the ensemble interpretation) and explicating and clarifying the basic principles and premises used in QM and the Hegerfeldt theorem and its application to measurements.
The newly revised edition of this well-known book categorizes prevalent views on measurement as described by quantum mechanics, discusses basics agreed to by many who on other issues disagree, and describes some of the problems. Although many paragraphs rest unchanged from the first (1991) edition, results found since 1991 are used to reorganize and sharpen the focus of the book, so that even the old paragraphs are now understood differently and more clearly. One advance in the second edition is a new and more general no-go theorem; another is separation of formal derivations from interpretation. The dozen or so definitions essential to following the book even casually are stated clearly. By this means the authors succeed pretty well in achieving their stated desire to make the book comprehensible to those who do not share their philosophic predispositions. While I differ in philosophy from the authors, attending to their discussion uncovered weaknesses in my own thinking and improved it. The mathematical language of quantum mechanics expresses a system under measurement by states (as rays in a Hilbert space) and a hamiltonian operator, distinct from the apparatus that produces outcomes, which is expressed by a positive operator-valued (POV) measure. In this way quantum language separates state preparation from state measurement. Often the Hilbert space is viewed as a tensor product of subspaces, allowing further conceptual separations. The authors take the stance that quantum mechanics calls for splitting the empirical world into parts, the most essential being (1) objective systems S (to be observed) and (2) apparatus A (preparation and registration devices that produces outcomes (or, as the authors write, definite pointer values) of the observed system). In some interpretations, one or both of an observer and an environment appear as additional parts. What is to be included in the system S versus the apparatus A is not specified by the language, but is open to the theoretician; a probe, for example, can be analysed as either part of the apparatus or as part of the system, as was discussed by von Neumann in work cited by the authors. With what should a theory of measurement be concerned? A broad division of approaches is defined by how one answers the question: should one try to do away with the cut between measuring apparatus and system? The authors phrase this as a choice of REFERENT, a term which in my mind puts them on shaky semantic ground; anyway The Quantum Theory of Measurement is concerned with implications of the `yes'. The book unfolds with the power of a tragedy, in which the logical consequences lead, step-by-step, to the statement and proof of theorem 6.2.1, which, roughly speaking, asserts the non-existence of a solution to the problem of getting definite pointer values from measuring apparatus that exhibits the superposition demanded by its description as a quantum system. Possible responses to this theorem are discussed, among them the many-worlds interpretation and a modification of quantum mechanics that elevates decoherence to a principle. Citations of the literature are thorough, and many points of view are crisply summarized. The book is indispensable to both those who work in the broad direction chosen by the authors and to those who wish to set any other direction in context.