We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamio\l{}kowski isomorphism in quantum theory. Such an isomorphism allows one to prove most of the basic features of quantum theory, like, e.g., existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.
We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamio\l{}kowski isomorphism in quantum theory. Such an isomorphism allows one to prove most of the basic features of quantum theory, like, e.g., existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.
We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, namely that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamiolkowski isomorphism in quantum mechanics. Such an isomorphism allows one to prove most of the basic features of quantum mechanics, like e.g. existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.
Entanglement is one of the most striking features of quantum mechanics, and yet it is not specifically quantum.More specific to quantum mechanics is the connection between entanglement and thermodynamics, which leads to an identification between entropies and measures of pure state entanglement.Here we search for the roots of this connection, investigating the relation between entanglement and thermodynamics in the framework of general probabilistic theories.We first address the question whether an entangled state can be transformed into another by means of local operations and classical communication.Under two operational requirements, we prove a general version of the Lo-Popescu theorem, which lies at the foundations of the theory of pure-state entanglement.We then consider a resource theory of purity where free operations are random reversible transformations, modelling the scenario where an agent has limited control over the dynamics of a closed system.Our key result is a duality between the resource theory of entanglement and the resource theory of purity, valid for every physical theory where all processes arise from pure states and reversible interactions at the fundamental level.As an application of the main result, we establish a one-to-one correspondence between entropies and measures of pure bipartite entanglement.The correspondence is then used to define entanglement measures in the general probabilistic framework.Finally, we show a duality between the task of information erasure and the task of entanglement generation, whereby the existence of entropy sinks (systems that can absorb arbitrary amounts of information) becomes equivalent to the existence of entanglement sources (correlated systems from which arbitrary amounts of entanglement can be extracted).
The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states). The key assumption is that each system A can be paired with an isomorphic conjugate system, A¯ , by means of a non-signaling bipartite state ηA perfectly and uniformly correlating each basic measurement on A with its counterpart on A¯ . In the case of a quantum-mechanical system associated with a complex Hilbert space H , the conjugate system is that associated with the conjugate Hilbert space H , and ηA corresponds to the standard maximally entangled EPR state on H⊗H¯ . A second ingredient is the notion of a reversible filter , that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement. In addition to offering more flexibility than most existing reconstructions of finite-dimensional quantum theory, the approach taken here has the advantage of not relying on any form of the ``no restriction" hypothesis. That is, it is not assumed that arbitrary effects are physically measurable, nor that arbitrary families of physically measurable effects summing to the unit effect, represent physically accessible observables. (An appendix shows how a version of Hardy's ``subpace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.)
We study the task of lifting arbitrary quantum states and channels to purifications and Stinespring dilations, respectively, in both the probabilistic exact and deterministic approximate settings. We formalize this task through a general framework of quantum purification machines that, given a finite number of copies or uses of a black-box input, aim to output a corresponding purification or Stinespring dilation. In the probabilistic exact setting, we show that universality is not necessary to rule out such transformations: the simple requirement that a machine purifies two inputs of different rank with non-zero probability already implies that it cannot be described by a linear positive map. This simple argument captures a fundamental obstruction of quantum theory and recovers the impossibility of universal probabilistic purification from finitely many copies. In the approximate setting, we allow for general machines that are not required, in general, to produce a pure output. Using the minimum average error as our figure of merit, we derive analytical expressions for the performance of several physically motivated strategies as well as a general upper bound on the achievable error, which is tight in a specific regime. Our analysis reveals a trade-off: strategies that produce a pure output - among which we prove the optimal to be a strategy that produces as a fixed output a maximally entangled purification of the fully depolarizing channel - perform optimally between those considered for large environment dimension, while append-environment strategies that generally produce non-pure outputs perform better at small environment dimension.
It is not surprising that our language should be incapable of describing processes occurring within atoms, for, as has been remarked, it was invented to describe the experiences of daily life, … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme–the quantum theory–which seems entirely adequate for the treatment of atomic processes. Werner Heisenberg, The Physical Principles of the Quantum Theory (1930) In this chapter we introduce the mathematical structure of quantum theory, for a finite number of systems and finite dimension, starting from three simple Hilbert-space postulates, and based on the intuitive notions of state, transformation, and effect, within a causal context. Such notions will be formalized in a general non-causal context in Chapter 3, which is devoted to the informational framework. We will assume the reader has a basic knowledge of matrix analysis and of operators over Hilbert spaces. Comments about the extension to infinite dimensions, continuous outcome-spaces, and infinitely many systems can be found in the notes at the end of the chapter. From the three simple Hilbert-space postulates, the present chapter extracts six principles, which will become the postulates for quantum theory in the axiomatic derivation presented in Part III of the book. In addition, we will take the opportunity to present and explore many relevant features of the theory – including its convex and causal structure, the Choi–Jamiołkowski isomorphism, and relevant notions, such as those of POVM, complete positivity of maps, quantum operation and channels, entangled states, state discrimination, tomography of states, and transformations. The purification principle will open the stage to the dilation theorems of the theory of quantum open systems, along with other consequences, such as the existence of faithful and steering states, the theorem of no information without disturbance, quantum teleportation, and more. We will review the quantum no-cloning and no-programming theorems, and finally provide a thorough derivation of the theory of quantum channel inversion, which is the backbone of the general theory of quantum error correction in quantum information. Additional interesting topics will be classical information theory as a restriction of quantum information theory, and a critical reconsideration of the von Neumann postulate and the notion of observable.
In quantum theory every state can be diagonalized, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This elementary structure plays an ubiquitous role in quantum mechanics, quantum information theory, and quantum statistical mechanics, where it provides the foundation for the notions of majorization and entropy. A natural question then arises: can we reconstruct these notions from purely operational axioms? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that every state can be diagonalized. The first axiom is Causality, which ensures that the marginal of a bipartite state is well defined. Then, Purity Preservation states that the set of pure transformations is closed under composition. The third axiom is Purification, which allows to assign a pure state to the composition of a system with its environment. Finally, we introduce the axiom of Pure Sharpness, stating that for every system there exists at least one pure effect occurring with unit probability on some state. For theories satisfying our four axioms, we show a constructive algorithm for diagonalizing every given state. The diagonalization result allows us to formulate a majorization criterion that captures the convertibility of states in the operational resource theory of purity, where random reversible transformations are regarded as free operations.