6 research outputs found
The Information Content of Systems in General Physical Theories
What kind of object is a quantum state? Is it an object that encodes an
exponentially growing amount of information (in the size of the system) or more
akin to a probability distribution? It turns out that these questions are
sensitive to what we do with the information. For example, Holevo's bound tells
us that n qubits only encode n bits of classical information but for certain
communication complexity tasks there is an exponential separation between
quantum and classical resources. Instead of just contrasting quantum and
classical physics, we can place both within a broad landscape of physical
theories and ask how non-quantum (and non-classical) theories are different
from, or more powerful than quantum theory. For example, in communication
complexity, certain (non-quantum) theories can trivialise all communication
complexity tasks. In recent work [C. M. Lee and M. J. Hoban, Proc. Royal Soc. A
472 (2190), 2016], we showed that the immense power of the information content
of states in general (non-quantum) physical theories is not limited to
communication complexity. We showed that, in general physical theories, states
can be taken as "advice" for computers in these theories and this advice allows
the computers to easily solve any decision problem. Aaronson has highlighted
the close connection between quantum communication complexity and quantum
computations that take quantum advice, and our work gives further indications
that this is a very general connection. In this work, we review the results in
our previous work and discuss the intricate relationship between communication
complexity and computers taking advice for general theories.Comment: In Proceedings PC 2016, arXiv:1606.0651
Device-independent certification of non-classical joint measurements via causal models
Quantum measurements are crucial for quantum technologies and give rise to
some of the most classically counter-intuitive quantum phenomena. As such, the
ability to certify the presence of genuinely non-classical joint measurements
in a device-independent fashion is vital. However, previous work has either
been non-device-independent, or has relied on post-selection---the ability to
discard all runs of an experiment in which a specific event did not occur. In
the case of entanglement, the post-selection approach applies an entangled
measurement to independent states and post-selects the outcome, inducing
non-classical correlations between the states that can be device-independently
certified using a Bell inequality. That is, it certifies measurement
non-classicality not by what it is, but by what it does. This paper remedies
this discrepancy by providing a novel notion of what measurement
non-classicality is, which, in analogy with Bell's theorem, corresponds to
measurement statistics being incompatible with an underlying classical causal
model. It is shown that this provides a more fine-grained notion of
non-classicality than post-selection, as it certifies the presence of
non-classicality that cannot be revealed by examining post-selected outcomes
alone.Comment: v3: updated in response to referee feedback. Close to published
version. 6 pages, 3 figures. v2: fixed typo in statement of Result 1 (missing
minus sign
Oracles and query lower bounds in generalised probabilistic theories
We investigate the connection between interference and computational power
within the operationally defined framework of generalised probabilistic
theories. To compare the computational abilities of different theories within
this framework we show that any theory satisfying three natural physical
principles possess a well-defined oracle model. Indeed, we prove a subroutine
theorem for oracles in such theories which is a necessary condition for the
oracle to be well-defined. The three principles are: causality (roughly, no
signalling from the future), purification (each mixed state arises as the
marginal of a pure state of a larger system), and strong symmetry existence of
non-trivial reversible transformations). Sorkin has defined a hierarchy of
conceivable interference behaviours, where the order in the hierarchy
corresponds to the number of paths that have an irreducible interaction in a
multi-slit experiment. Given our oracle model, we show that if a classical
computer requires at least n queries to solve a learning problem, then the
corresponding lower bound in theories lying at the kth level of Sorkin's
hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum
oracle needed to solve certain problems are not optimal in the space of all
generalised probabilistic theories, although it is not yet known whether the
optimal bounds are achievable in general. Hence searches for higher-order
interference are not only foundationally motivated, but constitute a search for
a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue
"Foundational Aspects of Quantum Information" in Foundations of Physic
Ruling out Higher-Order Interference from Purity Principles
As first noted by Rafael Sorkin, there is a limit to quantum interference. The interference pattern formed in a multi-slit experiment is a function of the interference patterns formed between pairs of slits; there are no genuinely new features resulting from considering three slits instead of two. Sorkin has introduced a hierarchy of mathematically conceivable higher-order interference behaviours, where classical theory lies at the first level of this hierarchy and quantum theory theory at the second. Informally, the order in this hierarchy corresponds to the number of slits on which the interference pattern has an irreducible dependence. Many authors have wondered why quantum interference is limited to the second level of this hierarchy. Does the existence of higher-order interference violate some natural physical principle that we believe should be fundamental? In the current work we show that such principles can be found which limit interference behaviour to second-order, or “quantum-like”, interference, but that do not restrict us to the entire quantum formalism. We work within the operational framework of generalised probabilistic theories, and prove that any theory satisfying Causality, Purity Preservation, Pure Sharpness, and Purification—four principles that formalise the fundamental character of purity in nature—exhibits at most second-order interference. Hence these theories are, at least conceptually, very “close” to quantum theory. Along the way we show that systems in such theories correspond to Euclidean Jordan algebras. Hence, they are self-dual and, moreover, multi-slit experiments in such theories are described by pure projectors
Device-independent certification of non-classical joint measurements via causal models
Quantum measurements are crucial for quantum technologies and give rise to some of the most classically counter-intuitive quantum phenomena. As such, the ability to certify the presence of genuinely non-classical joint measurements in a device-independent fashion is vital. However, previous work has either been non-device-independent, or has relied on post-selection—the ability to discard all runs of an experiment in which a specific event did not occur. In the case of entanglement, the post-selection approach applies an entangled measurement to independent states and post-selects the outcome, inducing non-classical correlations between the states that can be device-independently certified using a Bell inequality. That is, it certifies measurement non-classicality not by what it is, but by what it does. This paper remedies this discrepancy by providing a novel notion of what measurement non-classicality is, which, in analogy with Bell’s theorem, corresponds to measurement statistics being incompatible with an underlying classical causal model. It is shown that this provides a more fine-grained notion of non-classicality than post-selection, as it certifies the presence of non-classicality that cannot be revealed by examining post-selected outcomes alone
A no-go theorem for theories that decohere to quantum mechanics
To date, there has been no experimental evidence that invalidates quantum theory. Yet it may only be an effective description of the world, in the same way that classical physics is an effective description of the quantum world. We ask whether there exists an operationally defined theory superseding quantum theory, but which reduces to it via a decoherence-like mechanism. We prove that no such post-quantum theory exists if it is demanded that it satisfy two natural physical principles: causality and purification. Causality formalizes the statement that information propagates from present to future, and purification that each state of incomplete information arises in an essentially unique way due to lack of information about an environment. Hence, our result can be viewed either as evidence that the fundamental theory of Nature is quantum or as showing in a rigorous manner that any post-quantum theory must abandon causality, purification or both