Course

Non-local quantum computation (NLQC) is a subject within quantum information theory. NLQC considers, in a certain setting, with how local interactions can be simulated with distributed entanglement plus communication. NLQC has recently become well connected to several other areas, including communication complexity, cryptography, AdS/CFT, and computational complexity theory. This course will focus on learning the basics of NLQC, and then on understanding its applications in these other areas.

This course will cover the basics from my book, https://arxiv.org/abs/2603.12076. It is about operational probabilistic theories. The standard approach in such theories is, implicitly, from a time forward perspective. On the other hand, we will mostly take a time symmetric perspective. The course will consists of two parts: (1) a "simple part" about simple operations having simple causal structure (where all the inputs are before all the outputs); and (2) a "complex part" about complex operations that can have complicated causal structure (a complex operation comes equipped with a causal diagram). For the simple case we are able to show that the time symmetric perspective is equivalent to the time forward perspective. In each of these two parts we set up (A) operational probabilistic theories (OPTs) in terms of operations, (B) Operational Quantum Theory (OQT) in terms of operator tensors which correspond to operations, and (C) the theory of Hilbert objects which can be doubled up to give operator tensors. Operations are required to be physical. Physicality guarantees that circuits built out of operations have probabilities between 0 and 1 and that certain causality conditions are met. We prove composition theorems for both simple and complex operations -- that when we wire together operations the resulting networks are also physical (these theorems are especially interesting in the case of complex operations).The theory of complex operations can be used to model physics happening in (discrete) spacetime. We use this to address Sorkin's impossible measurements. It turns out that if the operations are physical then there is no anomalous signaling. We develop new diagrammatic notation to deal with Hilbert objects, particularly in the complex case. We discuss the conjuposition group of transformations on Hilbert objects. This includes mirrors to notate doubling up and some mirror theorems. We use this framework to prove time symmetric causal dilation theorems for a variety of causal diagrams.

Virtual Participation Link: https://pitp.zoom.us/j/93634737051?pwd=bJkB6HrVbOsrpCFInt76DNVlx7lwiS.1.

Location & Building Access: Tue, 11.00-12.30, Sky Room Wed, 11.00-12.30, Alice Room

Participants who do not have an access card for Perimeter Institute must sign in at the security desk before each session. For information on parking or accessibility please contact [email protected].

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