http://www.sciencenews.org/view/generic/id/332557/title/Quantum_theory_gets_physical
Quantum theory gets physical New work finds physical basis for quantum mechanics By Devin Powell Web edition : Tuesday, July 19th, 2011 Physicists in Canada and Italy have derived quantum mechanics from physical principles related to the storage, manipulation and retrieval of information. The new work is a step in a long, ongoing effort to find fundamental physical motivation for the math of quantum physics, which describes processes in the atomic and subatomic realms with unerring accuracy but defies commonsense understanding. “We’d like to have a set of axioms that give us a little better physical understanding of quantum mechanics,” says Michael Westmoreland, a mathematician at Denison University in Granville, Ohio. Quantum theory’s foundations currently rest on abstract mathematical formulations known as Hilbert spaces and C* algebras. These abstractions work well for calculating the probability of a particular outcome in an experiment. But they lack the intuitive physical meaning that physicists crave -- the elegance of Einstein’s theory of special relativity, for instance, which says that the speed of light is constant and that laws of physics don’t change from one reference frame to the next. Giulio Chiribella, a theoretical physicist at the Perimeter Institute for Theoretical Physics in Ontario, Canada, and colleagues based their approach on a postulate called “purification.” A system with uncertain properties (a “mixed state”) is always part of a larger “pure state” that can, in principle, be completely known, the team proposes in the July Physical Review A. Consider the pion. This particle, which has a spin of zero, can decay into two spinning photons. Each single photon is in a mixed state – it has an equal chance of spinning up or down. The pair of photons together, though, comprise a pure state in which they must always spin in opposite directions. “We can be ignorant of the part, but we can have maximal knowledge of the whole,” says Chiribella. This purification principle requires the quantum phenomenon known as entanglement, which connects the parts to the whole. It also explains why quantum information can’t be copied without destroying it but can be “teleported” -- replicated at a distant location after being destroyed at its point of origin. Building on this principle, Chiribella and colleagues reproduced the mathematical structure of quantum mechanics with the aid of five additional axioms related to information processing. Their axioms include causality, the idea that a measurement now can’t be influenced by future measurements, and “ideal compression,” meaning that information can be encoded in a physical system and then decoded without error. Other axioms involve the ability to distinguish states from each other and the ability of measurements to create pure states. “They nail it,” says Christopher Fuchs, a theoretical physicist at the Perimeter Institute. “This now approaches something that I think is along the lines of trying to find a crisp physical principle.” Whether this new derivation of quantum theory will prove to the simplest and most physically meaningful remains to be seen. “What is simple or physically plausible is a matter of taste,” says Časlav Brukner, a physicist at the University of Vienna in Austria who has developed an alternative set of axioms. Some speculate that recasting quantum theory in terms of information could help to solve outstanding problems in physics, such as how to unify quantum mechanics and gravity. “If you have lots of formulations of the same theory, you’re more likely to have one that leads to whatever the next physics is,” says Ben Schumacher, a theoretical physicist at Kenyon College in Gambier, Ohio. SUGGESTED READING : L. Hardy. Quantum theory from five reasonable axioms. arXiv:quant-ph/0101012v4 Posted Sep. 25, 2001. [Go to] B. Dakić and C. Brukner. Quantum theory and beyond: Is entanglement special? arXiv:0911.0695v1 Posted Nov. 3, 2009. [Go to] C. Brukner. Questioning the rules of the game. Physics. Published online July 11, 2011. doi:10.1103/Physics.4.55. [Go to] CITATIONS & REFERENCES : G. Chiribella, G.M. D’Ariano and P. Perinotti. Informational derivation of quantum theory. Physical Review A. Vol. 84, July 2011, p. 012311-1. doi:10.1103/PhysrevA.84.012311. [Go to] http://arxiv.org/abs/1011.6451 http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.6451v3.pdf 47 pages Informational derivation of Quantum Theory Giulio Chiribella ∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Ontario, Canada N2L 2Y5. † Giacomo Mauro D’Ariano ‡ and Paolo Perinotti § QUIT Group, Dipartimento di Fisica “A. Volta” and INFN Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy ¶ (Dated: July 18, 2011) We derive Quantum Theory from purely informational principles. Five elementary axioms -- causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning -- define a broad class of theories of information-processing that can be regarded as standard. One postulate -- purification -- singles out quantum theory within this class. PACS numbers: 03.67.-a, 03.67.Ac, 03.65.Ta ∗ gchiribe...@perimeterinstitute.ca † http://www.perimeterinstitute.ca ‡ dari...@unipv.it § paolo.perino...@unipv.it ¶ http://www.qubit.it CONTENTS I. Introduction 1 II. The framework 4 A. Circuits with outcomes 4 B. Probabilistic structure: states, effects and transformations 5 C. Basic definitions in the operational-probabilistic framework 6 1. Coarse-graining, refinement, atomic transformations, pure, mixed and completely mixed states 6 2. Examples in in quantum theory 7 D. Operational principles 7 III. The principles 8 A. Axioms 8 1. Causality 8 2. Perfect distinguishability 9 3. Ideal compression 10 4. Local distinguishability 10 5. Pure conditioning 10 B. The purification postulate 11 IV. First consequences of the principles 11 A. Results about ideal compression 11 B. Results about purification 12 C. Results about the combination of compression and purification 14 D. Teleportation and the link product 14 E. No information without disturbance 15 V. Perfectly distinguishable states 16 VI. Duality between pure states and atomic effects 18 VII. Dimension 19 VIII. Decomposition into perfectly distinguishable pure states 20 IX. Teleportation revisited 23 A. Probability of teleportation 23 B. Isotropic states and effects 23 C. Dimension of the state space 24 X. Derivation of the qubit 26 XI. Projections 27 A. Orthogonal faces and orthogonal complements 28 B. Projections 30 C. Projection of a pure state on two orthogonal faces 33 XII. The superposition principle 34 A. Completeness for purification 35 B. Equivalence of systems with equal dimension 35 C. Reversible operations of perfectly distinguishable pure states 35 XIII. Derivation of the density matrix formalism 36 A. The basis 36 B. The matrices 37 C. Choice of axes for a two-qubit system 38 D. Positivity of the matrices 43 E. Quantum theory in finite dimensions 44 XIV. Conclusion 46 Acknowledgments 46 References 47 "In this paper we provide a complete derivation of finite dimensional quantum theory based of purely operational principles. Our principles do not refer to abstract properties of the mathematical structures that we use to represent states, transformations or measurements, but only to the way in which states, transformations and measurements combine with each other. More specifically, our principles are of informational nature: they assert basic properties of information-processing, such as the possibility or impossibility to carry out certain tasks by manipulating physical systems. In this approach the rules by which information can be processed determine the physical theory, in accordance with Wheeler’s program “it from bit”, for which he argued that “all things physical are information-theoretic in origin” [22]. Note that, however, our axiomatization of quantum theory is relevant, as a rigorous result, also for those who do not share Wheeler’s ideas on the informational origin of physics. In particular, in the process of deriving quantum theory we provide alternative proofs for many key features of the Hilbert space formalism, such as the spectral decomposition of self-adjoint operators or the existence of projections. The interesting feature of these proofs is that they are obtained by manipulation of the principles, without assuming Hilbert spaces form the start. The main message of our work is simple: within a standard class of theories of information processing, quantum theory is uniquely identified by a single postulate: purification. The purification postulate, introduced in Ref. [21], expresses a distinctive feature of quantum theory, namely that the ignorance about a part is always compatible with the maximal knowledge of the whole. The key role of this feature was noticed already in 1935 by Schr¨odinger in his discussion about entanglement [23], of which he famously wrote “I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought”. In a sense, our work can be viewed as the concrete realization of Schr¨odinger’s claim: the fact that every physical state can be viewed as the marginal of some pure state of a compound system is indeed the key to single out quantum theory within a standard set of possible theories. It is worth stressing, however, that the purification principle assumed in this paper includes a requirement that was not explicitly mentioned in Schr¨odinger’s discussion: if two pure states of a composite system AB have the same marginal on system A, then they are connected by some reversible transformation on system B. In other words, we assume that all purifications of a given mixed state are equivalent under local reversible operations [24]. The purification principle expresses a law of conservation of information, stating that at least in principle, irreversibility can always be reduced to the lack of control over an environment. More precisely, the purification principle is equivalent to the statement that every irreversible process can be simulated in an essentially unique way by a reversible interaction of the system with an environment, which is initially in a pure state [21]. This statement can also be extended to include the case of measurement processes, and in that case it implies the possibility of arbitrarily shifting the cut between the observer and the observed system [21]. The possibility of such a shift was considered by von Neumann as a “fundamental requirement of the scientific viewpoint” (see p. 418 of [2]) and his discussion of the measurement process was exactly aimed to show that quantum theory fulfils this requirement. Besides Schr¨odinger’s discussion on entanglement and von Neumann’s discussion of the measurement process, the purification principle is deeply rooted in the structure of quantum theory. At the purely mathematical level, it plays a crucial role in the theory of C*-algebras of operators on separable Hilbert spaces, where the purification principle is equivalent to the Gelfand-Naimark-Segal (GNS) construction [25] and implies the celebrated Stinespring’s theorem [26]. On the other hand, purification is a cornerstone of quantum information, lying at the origin of most quantum protocols. As it was shown in Ref. [21], the purification principle directly implies crucial features like no-cloning, teleportation, no-information without disturbance, error correction, the impossibility of bit commitment, and the “no-programming” theorem of Ref. [27]. In addition to the purification postulate, our derivation of quantum theory is based on five informational axioms. The reason why we call them “axioms”, as opposed to the the purification “postulate”, is that they are not at all specific of quantum theory. These axioms represent standard features of information-processing that everyone would, more or less implicitly, assume. They define a class of theories of information-processing that includes, for example, classical information theory, quantum information theory, and quantum theory with superselection rules. The question whether there are other theories satisfying our five axioms and, in case of a positive answer, the full classification of these theories is currently an open problem. Here we informally illustrate the five axioms, leaving the more detailed description to the remaining part of the paper: 1. Causality: the probability of a measurement outcome at a certain time does not depend on the choice of measurements that will be performed later. 2. Perfect distinguishability: if a state is not completely mixed (i.e. if it cannot be obtained as a mixture from any other state), then there exists at least one state that can be perfectly distinguished from it. 3. Ideal compression: every source of information can be encoded in a suitable physical system in a lossless and maximally efficient fashion. Here lossless means that the information can be decoded without errors and maximally efficient means that every state of the encoding system represents a state in the information source. 4. Local distinguishability: if two states of a composite system are different, then we can distinguish between them from the statistics of local measurements on the component systems. 5. Pure conditioning: if a pure state of system AB undergoes an atomic measurement on system A, then each outcome of the measurement induces a pure state on system B. (Here atomic measurement means a measurement that cannot be obtained as a coarse-graining of another measurement). All these axioms are satisfied by classical information theory. Axiom 5 is even trivial for classical theory, because the only pure states of a composite system AB are the product of pure states of the component systems A and B, and hence the state of system B will be pure irrespectively of what we do on system A. A stronger version of axiom 5, introduced in Ref. [20], is the following: 5’ Atomicity of composition: the sequential composition of two atomic operations is atomic. (Here atomic transformation means a transformation that cannot be obtained from coarse-graining). However, it turns out that Axiom 5 is enough for our derivation: thanks to the purification postulate we will be able to show the non-trivial implication: Axiom 5 ⇒ Axiom 5’ (see lemma 16). The paper is organized as follows. In Sec. II we review the framework of operational-probabilistic theories introduced in Ref. [21]. This framework will provide the basic notions needed for the formulation of our principles. In Sec. III we introduce the principles from which we will derive Quantum Theory. In Sec. IV we prove some direct consequences of the principles that will be used later in the paper. In Sec. V we discuss the properties of perfectly distinguishable states, while in Sec. VI we prove the existence of a duality between pure states and atomic effects. The results about distinguishability and duality of pure states and atomic effects allow us to show in Sec. VII that every system has a well defined informational dimension -- the operational counterpart of the Hilbert space dimension. Sec. VIII contains the proof that every state can be decomposed as a convex combination of perfectly distinguishable pure states. Similarly, any element of the vector space spanned by the states can be written as a linear combination of perfectly distinguishable states. This result corresponds to the spectral theorem for self-adjoint operators on complex Hilbert spaces. In Sec. IX we prove some results about the maximum teleportation probability, which allow us to derive a functional relation between the dimension of the state space and the number of perfectly distinguishable states of the system. The mathematical representation of systems with two perfectly distinguishable states is derived in Sec. X, where we prove that such systems are indeed two-dimensional quantum systems -- a.k.a. qubits. In Sec. XI we construct projections on the faces of the state space of any system and prove their main properties. These results lead to the derivation of the operational analogue of the superposition principle in Sec. XII which allows to prove that systems with the same number of perfectly distinguishable states are operationally equivalent (Subsec. XII B). The properties of the projections and the superposition principle are then exploited in Sec. XIII -- where we extend the density matrix representation from qubits to higherdimensional systems, thus proving that a system with d perfectly distinguishable states is indeed a quantum system with d-dimensional Hilbert space. We conclude the paper with Sec. XIV, where we review our results, discussing future directions for this research. II. THE FRAMEWORK This Section provides a brief summary of the framework of operational-probabilistic theories, which was formulated in Ref. [21]. We refer to Ref. [21] for an exhaustive presentation of the details of the framework and of the ideas behind it. The operational-probabilistic framework combines the operational language of circuits with the toolbox of probability theory: on the one hand, experiments are described by circuits resulting from the connection of physical devices, on the other hand each device in the circuit can have classical outcomes and the theory provides the probability distribution of outcomes when the devices are connected to form closed circuits (that is, circuits that start with a preparation and end with a measurement). The notions discussed in this section will allow us to draw a precise distinction between principles with an operational content and exclusively mathematical principles: with the expression ”operational principle” we will mean a principle that can be expressed using only the basic notions of the the operational-probabilistic framework." "Proof. We proved that our theory has the same normalized states of quantum theory. On the other hand, quantum theory is a theory with purification and in quantum theory the possible physical transformations are quantum operations, i.e. completely positive trace-preserving maps. The thesis then follows from the fact that two theories with purification that have the same set of normalized states are necessarily the same (theorem 3). XIV. CONCLUSION Quantum theory can be derived from purely informational principles. In particular, it belongs to a broad class of theories of information-processing that includes classical and quantum information theory as special cases. Within this class, quantum theory is identified uniquely by the purification postulate, stating that the ignorance about a part is always compatible with the maximal knowledge of the whole in an essentially unique way. This postulate appears as the origin of the key features of quantum information processing, such as no-cloning, teleportation, and error correction (see also Ref. [21]). The general vision underlying the present work is that the main primitives of quantum information processing should be derived directly from the principles, without the abstract mathematics of Hilbert spaces, in order to make the revolutionary aspects of quantum information immediately accessible and to place them in the broader context of the fundamental laws of physics. Finally, we would like to comment on possible generalizations of our work. As in any axiomatic construction, one can ask how the results change when the principles are modified. For example, one may be interested in relaxing the local distinguishability axiom and in considering theories, like quantum theory on real Hilbert spaces, where global measurements are essential to characterize the state of a composite system. In this direction, the results of Ref. [21] suggest that also quantum theory on real Hilbert spaces can be derived from the purification principle, after that the local distinguishability requirement has been suitably relaxed. A possible way to weaken the local distinguishability requirement is to assume only the property of local distinguishability from pure states proposed in Ref. [21]: this property states that the probability of distinguishing two states by local measurements is larger than 1/2 whenever one of the two states is pure. A different way to relax local distinguishability would be to assume the property of 2-local tomography proposed in Ref. [39], which requires that the state of a multipartite system can be completely characterized using only measurements on bipartite subsystems. This property is equivalent to 2-local distinguishability, defined as the requirement that two different states of a multipartite system can be distinguished with probability of success larger than 1/2 using only local measurements or measurements on bipartite subsystems. A more radical generalization of our work would be to relax the assumption of causality. This would be particularly important for the discussion of quantum gravity scenarios, where the causal structure is not given a priori but is part of the dynamical variables of the theory. In this respect, the contribution of our work is twofold. First, it makes evident how fundamental is the assumption of causality in the ordinary formulation of quantum theory: the whole formalism of quantum states as density matrices with unit trace, quantum measurements as resolutions of the identity, and quantum channels as trace-preserving maps is crucially based on it. Technically speaking, the fact that the normalization of a state is given by a single linear functional (the trace, in quantum theory) is the signature of causality. This partly explains the troubles and paradoxes encountered when trying to combine the formalism of density matrices with non-causal evolutions, as in Deutsch’s model for close timelike curves [40, 41]. Moreover, given that the usual notion of normalization has to be abandoned in the noncausal scenario, and that the ordinary quantum formalism becomes inadequate, one may ask in what sense a theory of quantum gravity would be “quantum”. The suggestion coming from our work is that a “quantum” theory is a theory satisfying the purification principle, which can be suitably formulated even in the absence of causality [42]. The discussion of theories with purification in the non-causal scenario is an exciting avenue of future research." [21] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. 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