Introduction

In many situations, learning from the results of measurements can be regarded as updating one's probability distributions over certain variables. According to Bayesians, this updating should be carried out according to the rule of conditionalization. In the theory of quantum mechanics, there is a rule that tells us how to update the state of a system, given observation of a measurement result. The state of a quantum system is closely related to probability distributions over potential measurements. Therefore we might expect there to be some relation between Bayesian conditionalization and the quantum state-update rule. There have been several suggestions that the state change just is Bayesian conditionalization, appropriately understood, or that it is closely analogous.

Bub was the first to make the connection between quantum measurement and Bayesian conditionalization in a 1977 paper, using an approach based on quantum logic. The connection is renewed in discussions by Fuchs and also Jacobs in 2002, where again the analogy between the quantum state update and Bayesian conditionalization is pointed out. At the same time, Fuchs draws attention to a disanalogy – namely that there is an “extra unitary” transformation as part of the measurement in the quantum case. In this chapter, I will first review the proposals of Bub, Jacobs, and Fuchs. I will then show that the presence of the extra unitaries in quantum measurement leads to a difference between classical and quantum measurement in terms of information gain, drawing on results by Nielsen and Fuchs and Jacobs.

Introduction

Quantum mechanics is, without any doubt, a tremendously successful theory: it started by explaining black-body radiation and the photoelectric effect, it explained the spectra of atoms, and then went on to explain chemical bonds, the structure of atoms and of the atomic nucleus, the properties of crystals and the elementary particles, and a myriad of other phenomena. Yet it is safe to say that we still lack a deep understanding of quantum mechanics – surprising and even puzzling new effects continue to be discovered with regularity. That we are surprised and puzzled is the best sign that we still don't understand; however, the veil over the mysteries of quantum mechanics is starting to lift a little.

One of the strangest things microscopic particles do is to follow non-local dynamics and to yield non-local correlations. That particles follow non-local equations of motion was discovered by Aharonov and Bohm, while non-local correlations – which are the subject of this chapter – were discovered by John Bell and first cast in a form that has physical meaning, i.e., that can be experimentally tested, by Clauser, Horne, Shimony, and Holt. When they were discovered, both phenomena seemed to be quite exotic and at the fringe of quantum mechanics. By now we understand that they are some of the most important aspects of quantummechanical behavior.

Introduction

Quantum information and quantum foundations

Quantum information science is about the processing of information by the exploitation of some distinguishing features of quantum systems, such as electrons, photons, ions. In recent years a lot has been promised in the domain of quantum information. In quantum computing it was promised that NP-problems would be solved in polynomial time. In quantum cryptography there were claims that protocols would have practically 100% security. At the moment it is too early to say anything definitive regarding the final results of this great project.

In quantum computing a few quantum algorithms and developed devices, “quantum pre-computers” with a few quantum registers, were created. However, difficulties could no longer be ignored. For some reason it was impossible to create numerous quantum algorithms that could be applied to various problems. Up to now the whole project is based on two or three types of algorithm, and among them one, namely, the algorithms for prime factorization, might be interesting for real-world application. There is a general tendency to consider this situation with quantum algorithms as an occasional difficulty. But, as the years pass, one might start to think that there is something fundamentally wrong. The same feelings are induced by developments in quantum hardware. It seems that the complexity of the problem of creation of a device with a large number N of quantum registers increases extremely non-linearly with increasing N. In quantum cryptography the situation is opposite to that of quantum computing. There were tremendous successes in the development of technologies for production and transmission of quantum information, especially pairs of entangled photons.

Introduction

Quantum information theory is the study of how the peculiar features of quantum mechanics can be exploited for the purposes of information processing and transmission. A central theme of such a study is the ways in which quantum mechanics opens up possibilities that go beyond what can be achieved classically. This has in turn led to a renewed interest in, and a new perspective on, the differences between the classical and the quantum. Although much of the work along these lines has been motivated by quantum information theory – and some of it has been motivated by the conviction that quantum theory is essentially about possibilities of information processing and transmission – the results obtained, and the frameworks developed, have interest even for those of us who are not of that conviction. Indeed, much of the recent work echoes, and builds upon, work that predates the inception of quantum information theory. The significance of such work extends beyond the setting of quantum information theory; the work done on distinguishing the quantum from the classical in the context of frameworks that embrace both is something worthy of the attention of anyone interested in the foundational issues surrounding quantum theory.

One of the striking features of quantum mechanics lies in its probabilistic character. A quantum state yields, not a definite prediction of the outcome of an experiment, but a probability measure on the space of possible outcomes. Of course, probabilities occur also in a classical context.

We live, we are told, in an information age. We are told this, perhaps, less often than once we were; but no doubt only because the phrase has become worn from use. If ours is an age of information, then quantum information theory is a field propitiously in tune with the spirit of the times: a rich and sophisticated physical theory that seeks to tame quantum mysteries (no less!) and turn them to ingenious computational and communication ends. It is a theory that hints, moreover, at the possibility of finally rendering the quantum unmysterious; or at least this is a conclusion that many have been tempted to draw.

Yet, for all its timeliness, some of the most intriguing of the prospects that quantum information science presents are to be found intertwining with some surprisingly old and familiar philosophical themes. These themes are immaterialism and instrumentalism; and in this chapter we shall be exploring how these old ideas feature in the context of two of the most tantalizing new questions that have arisen with the advent of this field. Does quantum information theory finally help us to resolve the conceptual conundrums of quantum mechanics? And does the theory indicate a new way of thinking about the world – one in which the material as the fundamental subject matter of physical theory is seen to be replaced by the immaterial: information?

Introduction

Many philosophers and physicists have expressed great hope that quantum information theory will help us understand the nature of the quantum world. The general problem is that there is no widespread agreement on what quantum information is. Hence, such pronouncements regarding quantum information theory as the savior of the philosophy of physics are hard to evaluate. Much work has been done producing and evaluating concepts of information.

In I have defended and articulated the Schumacher concept of quantum information. Roughly speaking, quantum information is construed as the statistical behavior associated with the measurement of a quantum system. Hence it is a coarse-grained operational description of quantum systems, with no recourse to the fundamental ontological features of quantum systems responsible for such behavior. From this perspective, construing quantum mechanics as a theory of quantum information departs from the traditional interpretive endeavors of philosophers and physicists. The question is whether there is any motivation for taking such a view.

The theorem of Clifton, Bub, and Halvorson (CBH) provides just such a motivation. The theorem guarantees that, if a theory T satisfies certain conditions, there will exist an empirically equivalent C*-algebraic theory that has a concrete representation in Hilbert space, which it is notoriously difficult to interpret as a constructive or mechanical theory. In such a case, any underlying ontologies philosophers develop that are compatible with T will be undermined by the C*-algebraic equivalent. Bub suggests in light of this in-principle uncertainty regarding ontology that we re-conceive of quantum mechanics as a theory about quantum information.

Recently there has emerged an exciting and rapidly growing field of research known as quantum information theory. This interdisciplinary field is unified by the following two goals: first, the possibility of harnessing the principles and laws of quantum mechanics to aid in the acquisition, transmission, and processing of information; and second, the potential that these new technologies have for deepening our understanding of the foundations of quantum mechanics and computation. Many of the new technologies and discoveries emerging from quantum information theory are challenging the adequacy of our old concepts of entanglement, non-locality, and information. This research suggests that the time is ripe for a reconsideration of the foundations – and philosophical implications – of quantum information theory.

Historically, apart from a small group of physicists working on foundational issues, it was philosophers of physics who recognized the importance of the concepts of entanglement and non-locality long before the mainstream physics community. Prior to the 1980s, discussions of the infamous “EPR” paper and John Bell's seminal papers on quantum non-locality were carried out more often by such philosophers than by ordinary physicists. In the 1990s that situation rapidly changed, once the larger community of physicists had begun to realize that entanglement and non-locality were not just quirky features of quantum mechanics, but physical resources that could be harnessed for the performance of various practical tasks. Since then, a large body of literature has emerged in physics, revealing many new dimensions to our concepts of entanglement and non-locality, particularly in relation to information. Regrettably, however, only a few philosophers have followed these more recent developments, and many philosophical discussions still end with Bell's work.

Entanglement can be understood as an extraordinary degree of correlation between states of quantum systems – a correlation that cannot be given an explanation in terms of something like a common cause. Entanglement can occur between two or more quantum systems, and the most interesting case is when these correlations occur between systems that are space-like separated, meaning that changes made to one system are immediately correlated with changes in a distant system even though there is no time for a signal to travel between them. In this case one says that quantum entanglement leads to non-local correlations, or non-locality.

More precisely, entanglement can be defined in the following way. Consider two particles, A and B, whose (pure) states can be represented by the state vectors ψA and ψB. Instead of representing the state of each particle individually, one can represent the composite two-particle system by another wavefunction, ΨAB. If the two particles are unentangled, then the composite state is just the tensor product of the states of the components: ΨAB = ψA ⊗ ψB; the state is then said to be factorable (or separable). If the particles are entangled, however, then the state of the composite system cannot be written as such a product of a definite state for A and a definite state for B. This is how an entangled state is defined for pure states: a state is entangled if and only if it cannot be factored: ΨAB ≠ ψA ⊗ ψB.

Introduction

More than a century after its birth, quantum mechanics (QM) remains mysterious. We still don't have general principles from which to derive its remarkable mathematical framework, as happened for the amazing Lorentz transformations, which were rederived by Einstein from the invariance of physical laws in inertial frames and from the constancy of the speed of light.

Despite the utmost relevance of the problem of deriving QM from operational principles, research efforts in this direction have been sporadic. The deepest of the early attacks on the problem were the works of Birkhoff, von Neumann, Jordan, and Wigner, attempting to derive QM from a set of axioms with as much physical significance as possible. The general idea in Ref. is to regard QM as a new kind of prepositional calculus, a proposal that spawned the research line of quantum logic, which is based on von Neumann's observation that the two-valued observables – represented in his formulation of QM by orthogonal projection operators – constitute a kind of “logic” of experimental propositions. After a hiatus of two decades of neglect, interest in quantum logic was revived by Varadarajan, and most notably by Mackey, who axiomatized QM within an operational framework, with the single exception of an admittedly ad hoc postulate, which represents the propositional calculus mathematically in the form of an orthomodular lattice. The most significant extension of Mackey's work is the general representation theorem of Piron.

Qubits are not the only information carriers that can be used for quantum information processing. In this chapter, we will focus on quantum communication with ‘continuous quantum variables’, or continuous variables for short. In the context of quantum information processing we will also call continuous variables ‘qunats’. We have seen in Chapter 2 that a natural representation of a continuous variable is given by the position of a particle. The conjugate continuous variable is then the momentum of the particle. Unfortunately, the eigenstates of the position and momentum operators are not physical, and we have to construct suitable approximations to these states that can be created in the laboratory. Any practical information processing device must then take into account the deviation of the actual states from the ideal position and momentum eigenstates. Rather than the position and momentum of a particle, we will consider here the two position and momentum quadratures of an electromagnetic field mode. These operators obey the same commutation relations as the canonical position and momentum operators, but they are not the physical position and momentum of field excitations. Approximate eigenstates of the quadrature operators can be constructed in the form of squeezed coherent states. We define a quantum mechanical phase space for quadrature operators, similar to a classical phase space for position and momentum. Probability distributions in the classical phase space then become quasi-probability distributions over the quadrature phase space.We will develop one of these distributions, namely the Wigner function, and identify certain phase-space transformations of the Wigner function with linearoptical and squeezing operations.