PIRSA:09080011

# Operational structures and Natural Postulates for Quantum Theory

### APA

Hardy, L. (2009). Operational structures and Natural Postulates for Quantum Theory. Perimeter Institute. https://pirsa.org/09080011

### MLA

Hardy, Lucien. Operational structures and Natural Postulates for Quantum Theory. Perimeter Institute, Aug. 14, 2009, https://pirsa.org/09080011

### BibTex

@misc{ pirsa_09080011, doi = {10.48660/09080011}, url = {https://pirsa.org/09080011}, author = {Hardy, Lucien}, keywords = {Quantum Foundations}, language = {en}, title = {Operational structures and Natural Postulates for Quantum Theory}, publisher = {Perimeter Institute}, year = {2009}, month = {aug}, note = {PIRSA:09080011 see, \url{https://pirsa.org}} }

Lucien Hardy Perimeter Institute for Theoretical Physics

## Abstract

In this talk we provide four postulates that are more natural than the usual postulates of QT. The postulates require the predefinition of two integers, K and N, for a system. K is the number of probabilities that must be listed to specify the state. N is the maximum number of states that can be distinguished in a single shot measurement and consequently log N is the information carrying capacity. The postulates areP1 Information: Systems having, or constrained to have, a given information carrying capacity have the same properties.P2 Composites: For a composite system, AB, we have N_AB=N_A N_B and K_AB=K_A K_B.

P3 Continuity: There exists a continuous reversible transformation between any two pure states.

P4 Simplicity: For each N, K takes the smallest value consistent with the other postulates.

Note that P2 is equivalent to requiring that information carrying capacity be additive and that the state of a composite system can be determined by measurements on the components alone (local tomography is possible). We can prove a reconstruction theorem: the standard formalism of QT (for finite N) follows from these postulates. This includes the properties that quantum states can be represented by density operators on a complex Hilbert space, evolution is given by completely positive maps (of which unitary evolution is a special case), and that composite systems are formed using the tensor product. We derive the Born rule (or, equivalently, the trace rule) for calculating probabilities. If the single word “continuous” is dropped from P3 the postulates are consistent with both Classical Probability Theory and Quantum Theory. In this talk we will place particular emphasis on laying the operational foundations for such postulates. Then we will provide some highlights of the proof. Finally we will speculate on what needs to be changed for a theory of quantum gravity.