Unknown Quantum States and Operations, a Bayesian View

The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. In this paper, we motivate and review two results that generalize de Finetti's theorem to the quantum mechanical setting: Namely a de Finetti theorem for quantum states and a de Finetti theorem for quantum operations. The quantum-state theorem, in a closely analogous fashion to the original de Finetti theorem, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an "unknown quantum state" in quantum-state tomography. Similarly, the quantum-operation theorem gives an operational definition of an "unknown quantum operation" in quantum-process tomography. These results are especially important for a Bayesian interpretation of quantum mechanics, where quantum states and (at least some) quantum operations are taken to be states of belief rather than states of nature.Comment: 37 pages, 3 figures, to appear in "Quantum Estimation Theory," edited by M.G.A. Paris and J. Rehacek (Springer-Verlag, Berlin, 2004

On Participatory Realism

In the Philosophical Investigations, Ludwig Wittgenstein wrote, " 'I' is not the name of a person, nor 'here' of a place, .... But they are connected with names. ... [And] it is characteristic of physics not to use these words." This statement expresses the dominant way of thinking in physics: Physics is about the impersonal laws of nature; the "I" never makes an appearance in it. Since the advent of quantum theory, however, there has always been a nagging pressure to insert a first-person perspective into the heart of physics. In incarnations of lesser or greater strength, one may consider the "Copenhagen" views of Bohr, Heisenberg, and Pauli, the observer-participator view of John Wheeler, the informational interpretation of Anton Zeilinger and Caslav Brukner, the relational interpretation of Carlo Rovelli, and, most radically, the QBism of N. David Mermin, Ruediger Schack, and the present author, as acceding to the pressure. These views have lately been termed "participatory realism" to emphasize that rather than relinquishing the idea of reality (as they are often accused of), they are saying that reality is more than any third-person perspective can capture. Thus, far from instances of instrumentalism or antirealism, these views of quantum theory should be regarded as attempts to make a deep statement about the nature of reality. This paper explicates the idea for the case of QBism. As well, it highlights the influence of John Wheeler's "law without law" on QBism's formulation.Comment: 23 pages, to appear in "Information & Interaction: Eddington, Wheeler, and the Limits of Knowledge", edited by Ian T. Durham and Dean Rickles; v3 corrects word omissions from the Wheeler notebook page

Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations

The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the elusive symmetric informationally complete quantum measurements (SICs). We define a family of negativity measures which includes Zhu's as a special case and consider another member of the family which we call "sum negativity." We prove a sufficient condition for local maxima in sum negativity and find exact global maxima in dimensions $3$ and $4$. Notably, we find that Zhu's result on the SICs does not generally extend to sum negativity, although the analogous result does hold in dimension $4$. Finally, the Hoggar lines in dimension $8$ make an appearance in a conjecture on sum negativity.Comment: 21 pages. v2: journal version, added reference