Quantum theory from questions

Abstract

We reconstruct the explicit formalism of qubit quantum theory from elementary rules on an observer's information acquisition. Our approach is purely operational: we consider an observer O interrogating a system S with binary questions and define S's state as O's 'catalogue of knowledge' about S. From the rules we derive the state spaces for N elementary systems and show that (a) they coincide with the set of density matrices over an N -qubit Hilbert space C 2 N ; (b) states evolve unitarily under the group PSU(2 N ) according to the von Neumann evolution equation; and (c) O's binary questions correspond to projective Pauli operator measurements with outcome probabilities given by the Born rule. As a by-product, this results in a propositional formulation of quantum theory. Aside from offering an informational explanation for the theory's architecture, the reconstruction also unravels new structural insights. We show that, in a derived quadratic information measure, (d) qubits satisfy inequalities which bound the information content in any set of mutually complementary questions to 1 bit; and (e) maximal sets of mutually complementary questions for one and two qubits must carry precisely 1 bit of information in pure states. The latter relations constitute conserved informational charges which define the unitary groups and, together with their conservation conditions, the sets of pure quantum states. These results highlight information as a 'charge of quantum theory' and the benefits of this informational approach. This work emphasizes the sufficiency of restricting to an observer's information to reconstruct the theory and completes the quantum reconstruction initiated in [1].

Quantum Theory from Rules on Information Acquisition

“…Remarkably, they cannot distinguish between complex and real numbers. Namely, the two solutions are qubit and rebit quantum theory, i.e., two-level systems over real Hilbert spaces [1,2]. Since the latter is both mathematically and physically a subcase of the former, these five rules can be regarded as sufficient.…”

“…(Without Principle 5, we would still obtain the structure of an informationally complete set Q M N , finding that it encodes a basis of projective Pauli operator measurements [2]; Principle 5 legalizes all such measurements. )…”

“…Effectively, these rules constitute a set of 'coordinates' for quantum theory on this landscape. The rules are spelled out first colloquially, then mathematically and are motivated in more detail in [1,2].…”

“…In this manuscript, we shall review and summarize how the quantum formalism for arbitrarily many qubits can be reconstructed from operational rules restricting an observer's acquisition of information about a set of observed systems [1,2]. The goal of this summary is to provide a didactical and easily-accessible overview of this reconstruction.…”

“…In particular, the goal should be to find an operational justification for the textbook axioms, i.e., ultimately for complex Hilbert spaces, unitary dynamics, tensor product structure for composite systems, Born rule, and so on. The result would be a reconstruction of quantum theory from operational axioms [1][2][3][4][5][6][7][8][9][10] and should ideally yield a better understanding of what quantum theory tells us about Nature; and why it is the way it is.…”