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arXiv:2310.04397

Cloning, deleting, and hiding in modal quantum theory

Diamond, Phillip, Benjamin W. Schumacher

Published Oct 6, 2023• 0 citations
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Abstract

We examine the toy model of modal quantum theory (MQT), an analogue of actual quantum theory based on finite fields. In particular, we investigate how several essential ``no-go'' results (for cloning, deleting and hiding processes) work in MQT. Cloning and deleting are still forbidden in MQT, though the details of these results are somewhat different in the new context. However, the information of a modal qubit can be completely hidden in the correlations between two entangled modal qubits.

DOI
10.48550/arxiv.2310.04397

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We examine the toy model of modal quantum theory (MQT), an analogue of actual quantum theory based on finite fields. In particular, we investigate how several essential ``no-go'' results (for cloning, deleting and hiding processes) work in MQT. Cloning and deleting are still forbidden in MQT, though the details of these results are somewhat different in the new context. However, the information of a modal qubit can be completely hidden in the correlations between two entangled modal qubits.
Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The proofs of these analogue theorems are similar to, but much simpler than, the originals. We also show that the structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.
Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.
Quantum mechanics put restriction on performing some task which we can do classically. One such restriction is that we cannot copy an arbitrary quantum state. This is known as No-cloning theorem. Although quantum mechanics forbid us to construct a perfect cloner, there is no restriction to construct an imperfect cloner. Therefore, we have constructed and studied a special kind of quantum cloning machine named as Hybrid quantum cloning machine which is nothing but a different combination of pre-existing quantum cloning machines. We showed that this hybrid quantum cloning machine sometime gives better quality copy than the pre-existing quantum cloners. Quantum entanglement is a very astonishing property in a multi-partite system and it is a very useful ingredient of quantum information theory. Therefore broadcasting (cloning) of entanglement is a very important subject to study. In this regard we have studied the broadcasting of entanglement using state dependent quantum cloning machine. Further, we have investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners. Linearity of quantum mechanics put another restriction on deletion of an arbitrary quantum state i.e. an arbitrary quantum state against a copy cannot be perfectly deleted. Like cloning machine one can construct an approximate deletion machine also. Therefore, we have constructed an universal deletion machine in the sense that the fidelity of deletion does not depend on the input state and showed that the fidelity of deletion is 3/4 . Further we showed that the fidelity of deletion depends on the blank state at the output and it can be improved from the value 3/4 for certain blank state. Moreover, we have studied the case where two cloned copies are given at the input of the deletion machine.
In recent work, Benjamin Schumacher and Michael D. Westmoreland investigate a version of quantum mechanics which they call modal quantum theory. This theory is obtained by instantiating the mathematical framework of Hilbert spaces with a finite field instead of the field of complex numbers. This instantiation collapses much the structure of actual quantum mechanics but retains several of its distinguishing characteristics including the notions of superposition, interference, and entanglement. Furthermore, modal quantum theory excludes local hidden variable models, has a no-cloning theorem, and can express natural counterparts of quantum information protocols such as superdense coding and teleportation. We show that the problem of UNIQUE-SAT --- which decides whether a given Boolean formula is unsatisfiable or has exactly one satisfying assignment --- is deterministically solvable in any modal quantum theory in constant time. The solution exploits the lack of orthogonality in modal quantum theories and is not directly applicable to actual quantum theory.