Abstract

Standard textbook Operational Quantum Theory is not formulated in a time symmetric way.  For example, the space of states is a cone with a flat top, whilst the space of effects is a double cone (with the identity effect at the apex).  Does this time asymmetry point to a fundamental time asymmetry in nature?  In fact, we will see that it is possible to formulate Operational Quantum Theory in a time symmetric manner and, further, this time symmetric theory is fully equivalent to the usual time forward textbook formulation.  We can simply think of changing perspective between a time forward and a time symmetric point of view (and, indeed, a time backward point of view).  Arguably the time symmetric perspective is more fundamental.  To set up the time symmetric theory, we have to go back to the basic building block of Operational Theories – the operation – and make it time symmetric.  We do this by adding “incomes” which are the time reverse of “outcomes” (whereby the results of measurements are recorded).  Operations are subject to two types of constraints (i) positivity and (ii) causality.   These constraints dictate the space of allowed operations (such as the space of states and effects mentioned above).  Positivity is the constraint that closed circuits have non-negative probability.  In the time symmetric theory, causality consists of two parts: forward causality (that it is impossible to send information back in time without conditioning on the future), and backward causality (that it is impossible to send information forward in time without conditioning on the past).    If I have time I will also speak about causally complex operations (as opposed to causally simple operations for which inputs are strictly to the past of outputs).   Thes can be a model for the physics in arbitrary shaped regions of spacetime.