Abstract

In a recent paper, Chang et al. have proposed studying “quantum F u n ”: the q ↦ 1 limit of modal quantum theories over finite fields F q , motivated by the fact that such limit theories can be naturally interpreted in classical quantum theory. In this letter, we first make a number of rectifications of statements made in that paper. For instance, we show that quantum theory over F 1 does have a natural analogon of an inner product, and so orthogonality is a well-defined notion, contrary to what was claimed in Chang et al. Starting from that formalism, we introduce time evolution operators and observables in quantum F u n , and we determine the corresponding unitary group. Next, we obtain a typical no-cloning result in the general realm of quantum F u n . Finally, we obtain a no-deletion result as well. Remarkably, we show that we can perform quantum deletion by almost unitary operators, with a probability tending to 1. Although we develop the construction in quantum F u n , it is also valid in any other quantum theory (and thus also in classical quantum theory in complex Hilbert spaces).

Highlights

  • In a recent paper, Chang et al have proposed studying “quantum Fun ”: the q 7→ 1 limit of modal quantum theories over finite fields Fq, motivated by the fact that such limit theories can be naturally interpreted in classical quantum theory

  • In the last ten years, there has been an increasing interest in the field with one element; this nonexisting object is contained in every field, and its geometric theory is an “absolute theory” which is present in any geometric theory over a field

  • If we want the combinatorics of projective wave space available, we argued in [9] that the approach of general quantum theories (GQTs) is the most general possible, since combinatorial projective spaces in dimension at least 3 are always coordinatized over division rings

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Summary

A Virtual Deletion Machine in all Quantum Theories

The principle of superposition is a fundamental property in quantum mechanics; if two evolving states |si and |si solve the Schrödinger equation, an arbitrary linear combination a|si1 + b|si is a solution. The author has shown that both no-theorems still hold, and superposition remains to be a key in the proofs [9]. As F1 -theory lacks addition on the algebraic level (see Section 3), a major basic question is whether similar no-cloning and no-deletion results will still hold in quantum Fun. And whether the diagram (1) remains to have a meaning in the context of such more advanced questions. We will obtain the no-cloning and no-deletion theorems in quantum Fun. On the other hand, after introducing almost unitary operators (which are allowed to be singular), we obtain a quantum deletion theory which deletes one copy of any two given state rays with a probability tending to 1. The diagram (1) does apply to this result, so that we virtually obtain deletion in classical quantum theory

Overview
A Quick Review of General Quantum Theory
Hermitian Forms
F1 and F12
The Standard Form
Orthogonality
Time Evolution and Hermitian Operators
The Frobenius Maps
Standard Examples of Absolute Quantum Theories
Dictionary
One Cannot Clone an Unknown State in Absolute Quantum Theory
Quantum Deletion in the Absolute and Actual Context
Deleting Probability
Finite Case
Interpretation
Conclusions
Full Text
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