We study general physical systems with a notion of sequential measurement that satisfies a few simple and intuitive conditions. We show that, apart from two exceptional cases, the systems satisfying these conditions are exactly the real, complex or quaternionic quantum systems. Moreover, only the complex quantum systems remain if systems must be allowed to be composed together in a locally tomographic manner. In other words, quantum theory is the unique locally tomographic theory allowing sequential measurement.
We study general physical systems with a notion of sequential measurement that satisfies a few simple and intuitive conditions. We show that, apart from two exceptional cases, the systems satisfying these conditions are exactly the real, complex or quaternionic quantum systems. Moreover, only the complex quantum systems remain if systems must be allowed to be composed together in a locally tomographic manner. In other words, quantum theory is the unique locally tomographic theory allowing sequential measurement.
We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the composition of many systems; and second, it is closer to experiment by having absence of higher-order interference as a postulate, which is currently the subject of experimental investigation.We give three postulates-no higher-order interference, classical decomposability of states, and strong symmetry-and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with threelevel octonionic quantum theory and ball state spaces of arbitrary dimension.Then we show that adding observability of energy as a fourth postulate yields complex quantum theory as the unique solution, relating the emergence of the complex numbers to the possibility of Hamiltonian dynamics.We also show that there may be interesting non-quantum theories satisfying only the first two of our postulates, which would allow for higher-order interference in experiments while still respecting the contextuality analogue of the local orthogonality principle.
From the general difficulty of simulating quantum systems using classical systems, and in particular the existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is intrinsically more powerful than classical computation. At present, the best upper bound known for the power of quantum computation is that , where is a classical complexity class (known to be included in , hence ). This work investigates limits on computational power that are imposed by simple physical, or information theoretic, principles. To this end, we define a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and ask: what is the minimal set of physical assumptions under which the above inclusions still hold? We show that given only an assumption of tomographic locality (roughly, that multipartite states and transformations can be characterized by local measurements), efficient computations are contained in . This inclusion still holds even without assuming a basic notion of causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future measurement choices). Following Aaronson, we extend the computational model by allowing post-selection on measurement outcomes. Aaronson showed that the corresponding quantum complexity class, , is equal to . Given only the assumption of tomographic locality, the inclusion in still holds for post-selected computation in general theories. Hence in a world with post-selection, quantum theory is optimal for computation in the space of all operational theories. We then consider whether one can obtain relativized complexity results for general theories. It is not obvious how to define a sensible notion of a computational oracle in the general framework that reduces to the standard notion in the quantum case. Nevertheless, it is possible to define computation relative to a 'classical oracle'. Then, we show there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption does not include .
We show that any jointly measurable pair of quantum observables can be obtained in a sequential measurement scheme, even if the second observable will be decided after the first measurement. This means that it is possible to perform a measurement of any quantum observable in a way that does not disturb the subsequent measurements more than is dictated by joint measurability. Only measurements with a specific structure have this universality feature. As a supplementing result, we provide a characterization of all possible joint measurements obtained from a sequential measurement lacking universality.
Quantum physics provides us with a formalism with which we can try to understand the microscopic world. However, the underpinning concepts of the basic principles of quantum mechanics itself are surprisingly ill understood. Even the ubiquitous concept of the wavefunction collapsing post measurement, lives outside the formal theory of quantum mechanics and must be taken as a postulate. The Schr\"odinger equation, which governs the evolution of quantum systems, does not describe the system during the measurement process. This is referred to as the measurement problem, and has been debated since the inception of quantum theory. Many different interpretations of quantum mechanics try to elucidate what happens during wavefunction collapse, such as the famous many-worlds interpretation. However, these interpretations either give rise to the same physical outcome or simply are not testable, and so cannot be physically differentiated from one another. We attempt to gain insight into the measurement problem through experiments involving sequential measurements, both in the weak and strong regime. We find surprising outcomes such as the persistence of time-ordering asymmetry in the weak measurement limit, and the survival of coherence in the strong measurement limit.
From the general difficulty of simulating quantum systems using classical systems, and in particular \nthe existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is \nintrinsically more powerful than classical computation. At present, the best upper bound known for \nthe power of quantum computation is that BQP AWPP ⊆ , where AWPP is a classical complexity \nclass (known to be included in PP, hence PSPACE). This work investigates limits on computational \npower that are imposed by simple physical, or information theoretic, principles. To this end, we define \na circuit-based model of computation in a class of operationally-defined theories more general than \nquantum theory, and ask: what is the minimal set of physical assumptions under which the above \ninclusions still hold? We show that given only an assumption of tomographic locality (roughly, that \nmultipartite states and transformations can be characterized by local measurements), efficient \ncomputations are contained in AWPP. This inclusion still holds even without assuming a basic notion \nof causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future \nmeasurement choices). Following Aaronson, we extend the computational model by allowing postselection \non measurement outcomes. Aaronson showed that the corresponding quantum complexity \nclass, PostBQP, is equal to PP. Given only the assumption of tomographic locality, the inclusion in \nPP still holds for post-selected computation in general theories. Hence in a world with post-selection, \nquantum theory is optimal for computation in the space of all operational theories. We then consider \nwhether one can obtain relativized complexity results for general theories. It is not obvious how to \ndefine a sensible notion of a computational oracle in the general framework that reduces to the \nstandard notion in the quantum case. Nevertheless, it is possible to define computation relative to a \n‘classical oracle’. Then, we show there exists a classical oracle relative to which efficient computation in \nany theory satisfying the causality assumption does not include NP.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.