Textbook quantum superposition refers to the feature that certain linear combinations of Hilbert space rays, each representing a valid quantum state, are themselves valid states. This notion is not operational, and it relies on the underlying Hilbert space formalism. Recent proposals for experimental tests of indefinite causal order, as well as tests probing the non-classicality of gravity, pivot on superposition, thereby calling for a theory-independent, operational formalisation of the concept. Here, we define superposition within the framework of Generalised Probabilistic Theories, based on observed statistics in prepare-and-measure experiments. Using this, we formulate three superposition principles to investigate which structural features of quantum theory carry over to other theories. We study conditions under which these principles carry over from subsystems to their compositions; to this end, we show that the quantum tensor product emerges as the largest composition rule for quantum systems respecting all three principles. Furthermore, we show how non-classical features such as entanglement and preparational uncertainty can be viewed as special forms of superposition.
Textbook quantum superposition refers to the feature that certain linear combinations of Hilbert space rays, each representing a valid quantum state, are themselves valid states. This notion is not operational, and it relies on the underlying Hilbert space formalism. Recent proposals for experimental tests of indefinite causal order, as well as tests probing the non-classicality of gravity, pivot on superposition, thereby calling for a theory-independent, operational formalisation of the concept. Here, we define superposition within the framework of Generalised Probabilistic Theories, based on observed statistics in prepare-and-measure experiments. Using this, we formulate three superposition principles to investigate which structural features of quantum theory carry over to other theories. We study conditions under which these principles carry over from subsystems to their compositions; to this end, we show that the quantum tensor product emerges as the largest composition rule for quantum systems respecting all three principles. Furthermore, we show how non-classical features such as entanglement and preparational uncertainty can be viewed as special forms of superposition.
Quantum theory allows for the superposition of causal orders between operations, i.e., for an indefinite causal order; an implication of the principle of quantum superposition. Since a higher theory might also admit this feature, an understanding of superposition and indefinite causal order in a generalised probabilistic framework is needed. We present a possible notion of superposition for such a framework and show that in maximal theories, respecting non-signalling relations, single system state-spaces do not admit superposition; however, composite systems do. Additionally, we show that superposition does not imply entanglement. Next, we provide a concrete example of a maximally Bell-nonlocal theory, which not only admits the presented notion of superposition, but also allows for post-quantum violations of theory-independent inequalities that certify indefinite causal order; even up to an algebraic bound. These findings might point towards potential connections between a theory's ability to admit indefinite causal order, Bell-nonlocal correlations and the structure of its state spaces.
In this work, we give rigorous operational meaning to superposition of causal orders. This fits within a recent effort to understand how the standard operational perspective on quantum theory could be extended to include indefinite causality. The mainstream view, that of "process matrices", takes a top-down approach to the problem, considering all causal correlations that are compatible with local quantum experiments. Conversely, we pursue a bottom-up approach, investigating how the concept of indefiniteness emerges from specific characteristics of generic operational theories. Specifically, we pin down the operational phenomenology of the notion of non-classical (e.g. "coherent") control, which we then use to formalise a theory-independent notion of control (e.g. "superposition") of causal orders. To validate our framework, we show how salient examples from the literature can be captured in our framework.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Quantum theory provides a significant example of two intermingling hallmarks of science: the ability to consistently combine physical systems and study them compositely, and the power to extract predictions in the form of correlations. A striking consequence of this facet is the violation of Bell inequalities, which has been experimentally demonstrated via Bell tests. The prediction of this phenomenon originates as quantum systems are prescribed to combine according to the composition postulate, i.e. the tensor-product rule. This rule has also an operationally salient formulation given in terms of discriminability of composite states via local measurements. However, both the theoretical and the empirical status of such a postulate have been repeatedly challenged, questioning its independence from other physical principles -- most notably from quantum postulates pertaining solely to single systems. Is the composition postulate the only viable way to combine quantum systems into a consistent physical theory? Here, this long-standing problem is resolved by answering in the negative. This is achieved by adopting an operational approach to physical theories and exhibiting a family of theories that differ from standard quantum theory in their system-composition rule. These theories have the same predictions as standard quantum theory as far as Bell-like correlation scenarios are concerned. Quantum theory is thus established to embody genuinely more than quantum correlations. As a result, foundational programmes based on single-system principles only, or on mere Bell-like correlations, are operationally incomplete. On the experimental side, ascertaining the independence of postulates is a fundamental step to adjudicate between quantum theory and alternative physical theories: hence, the composition postulate calls for experimental scrutiny independently of the other features of quantum theory.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.