Quantum Foundations

A geometric approach to investigation of quantum entanglement is advocated.
We discuss first the geometry of the (N^2-1)--dimensional convex body
of  mixed quantum states acting  on an N--dimensional Hilbert space
and study projections of this set into 2- and 3-dimensional spaces.
For composed dimensions, N=K^2, one consideres the subset
of separable states and shows that it has a positive measure.
Analyzing its properties contributes to our understanding of
quantum entanglement and its time evolution.

To describe observed phenomena in the lab and to apply superposition principle to gravity, quantum theory needs to be generalized to incorporate indefinite causal structure. Practically, indefinite causal structure offers advantage in communication and computation. Fundamentally, superposing causal structure is one approach to quantize gravity (spacetime metric is equivalent to causal structure plus conformal factor, so quantizing causal structure effectively quantizes gravity). 

We develop a framework to do Operational Probabilistic Theories (OPT) with indefinite causal structure. For the interest of quantum gravity, this framework gives a general prescription to quantize causal structure, assuming linearity is intact. For the interest of quantum foundations, this framework can support new experimental tests about the validity of quantum theory in complex Hilbert space. It also offers opportunities for constructing new OPT models to substitute ordinary quantum theory. Along this direction, we identify principles that single out the complex Hilbert space theory within the general framework.

The on-demand generation of bright entangled photon pairs is highly needed in quantum optics and emerging quantum information applications. However, a quantum light source combining both high fidelity and on-demand bright emission has proven elusive with current leading photon technologies. In this work we present a new bright nanoscale source of strongly entangled photon pairs generated with a position controlled nanowire quantum dot. The major breakthrough in the nanowire growth to achieve both bright photon emission and highly entangled photon pairs will be discussed [2, 3]. Recent experiments show the entanglement fidelity approaching unity, while enhancing the photon pair efficiency beyond state-of-the-art. We further demonstrate violation of the famous Clauser-Horne-Shimony-Holt inequality in the traditional linear basis [4]. This is the first bright nanoscale source of entangled photon pairs capable of violating Bell`s inequalities, opening up future experiments in quantum optics and developments in quantum information applications.

For long-distance quantum communication we convert polarization entangled photons generated by a single quantum dot into time-bin entangled photons by sending them through a polarization-time-bin interface [5]. Importantly, this conversion is performed without loss of entanglement strength. Time-bin entanglement is more robust for long-distance quantum communication than polarization entanglement, since time-bin entangled photons are insensitive to thermal and mechanical disturbances in optical fibers.

References

[1]      M. A. M. Versteegh, M. E Reimer, K. D Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, Nature Commun. 5, 5298 (2014).

[2]      D. Dalacu, K. Mnaymneh, J. Lapointe, X. Wu, P. J. Poole, G. Bulgarini, V. Zwiller, and M. E. Reimer, Nano Lett. 12 (11), 5919-5923 (2012).

[3]      M. E. Reimer et al., Phys. Rev. B 93, 195316 (2016).

[4]      K. D. Jöns et al., arXiv:1510.03897 (2015).

[5]      M. A. M. Versteegh, M. E. Reimer, A. A. van den Berg, G. Juska, V. Dimastrodonato, A. Gocalinska, E. Pelucchi, and V. Zwiller, Phys. Rev. A 92, 033802 (2015).

In this talk I will review the construction of space starting purely from quantum mechanics and without assuming that the notion of space is attached to a preconceived notion of classical reality. I will show that if one start with the simplest notion of a quantum system encoded into the Heisenberg group algebra one naturally obtain a  notion of space that generalizes the usual notion of Euclidean space. Along the way I will try to illustrate how this notion is in a way going back to the roots of the discovery of QM by Heisenberg-Born-Jordan which never made it past the original papers and giving a critical reading of the subsequent interpretation of space and QM that were put forward by Schrodinger and von Neumann. The notion of space that emerge from quantum mechanics is naturally modular in the sense of Aharonov, it also naturally possess a built-in length scale and renders possible to assign a new notion of locality to non-local superpositions. I will illustrate how  such space can allow reconciliation of relativity with the presence of a fundamental scale.

I will show how to construct such spaces following the original analysis by Mackay and also show that such modular spaces possess a beautiful geometrical structure that generalizes Riemanian geometry to phase space. A geometry we have named Born geometry. I hope this will open wild speculations on the nature of locality in the presence of quantum mechanics and more broadly the nature of classical reality viewed from a quantum perspective.

Can we decompose the information of a composite system into terms arising from its parts and their interactions?
For a bipartite system (X,Y), the joint entropy can be written as an algebraic sum of three terms: the entropy of X alone, the entropy of Y alone, and the mutual information of X and Y, which comes with an opposite sign. This suggests a set-theoretical analogy: mutual information is a sort of "intersection", and joint entropy is a sort of "union".
The same picture cannot be generalized to three or more parts in a straightforward way, and the problem is still considered open. Is there a deep reason for why the set-theoretical analogy fails?
Category theory can give an alternative, conceptual point of view on the problem. As Shannon already noted, information appears to be related to symmetry. This suggests a natural lattice structure for information, which is compatible with a set-theoretical picture only for bipartite systems.
The categorical approach favors objects with a structure in place of just numbers to describe information quantities. We hope that this can clarify the mathematical structure underlying information theory, and leave it open to wider generalizations.

In order to solve the problem of quantum gravity, we first need to pose the problem. In this talk I will argue that the problem of quantum gravity arises already in the domain of quantum mechanics and the relativity principle. Specifically, the relativity principle implies that the concept of inertial motion should extend also to those systems that are in quantum superpositions of inertial motions. By contrast, relativistic quantum field theory only considers the point of view of classical observers in states of definite relative motion (i.e. the observers of a quantum field do not include inertial observers in quantum superpositions). The problem is that, if we extend the class of inertial observers to include quantum observers, the manifold of local events becomes ill-defined, as `locality' itself becomes an observer-relative property of an event. Thus, the Relativity Principle and the Superposition Principle are jointly opposed to the concept of a space-time manifold of local events, and our understanding of relativistic quantum theory needs to be revised before gravity even enters the picture.

I will explain and prove the statement of the title. The proof relies on a recent result of Slofstra in combinatorial group theory and the hypergraph approach to contextuality.

Based on http://arxiv.org/abs/1607.05870.

In quantum information, we frequently consider (for instance, whenever we talk about entanglement) a composite system consisting of two separated subsystems. A standard axiom of quantum mechanics states that a composite system can be modeled as the tensor product of the two subsystems. However, there is another less restrictive way to model a composite system, which is used in quantum field theory: we can require only that the algebras of observables for each subsystem commute within some larger subalgebra. For finite-dimensional systems, these two axioms are equivalent, but this is not necessarily true for infinite-dimensional systems. Tsirelson's question (which comes in several variants) asks whether the correlations arising from commuting-operator models can always be represented by tensor-product models. I will give examples of linear system non-local games which cannot be played perfectly with tensor-product strategies, but can be played perfectly with commuting-operator strategies, resolving (one version of) Tsirelson's question in the negative. From these examples, we can also derive other consequences for the theory of non-local games, such as the undecidability of determining whether a non-local game has a perfect quantum strategy. 

Bell's theorem shows that our intuitive understanding of causation must be overturned in light of quantum correlations. Nevertheless, quantum mechanics does not permit signalling and hence a notion of cause remains. Understanding this notion is not only important at a fundamental level, but also for technological applications such as key distribution and randomness expansion. It has recently been shown that a useful way to determine which classical causal structures give rise to a given set of correlations is to use entropy vectors. We consider the question of whether such vectors can lead to useful certificates of non-classicality. We find that for a family of causal structures that include the usual bipartite Bell structure they do not, in spite of the existence of non-classical correlations. Furthermore, we find that for many causal structures non-Shannon entropic inequalities give additional constraints on the sets of possible entropy vectors in the classical case. They hence lead to tighter approximations of the set of realisable entropy vectors, which enables a sharper distinction of different causal structures. Whether these improved characterisations are also valid for the quantum case remains an open problem whose resolution would have implications for the discrimination of classical and quantum causes.

In this talk, I will outline the current state of the art in the study of the reality of the quantum state.  The main theme will be that, although you cannot derive the reality of the quantum state in an ontological model without additional assumptions, you can place constraints on the amount of overlap between probability measures that begin to make psi-epistemic theories look implausible.  These overlap bounds come from noncontextuality inequalities, and there are two types in the literature: those based on Cabello-Severini-Winter type inequalities and those based on Yu-Oh type inequalities.  The latter type of overlap bound was not originally derived from noncontextuality, but thinking of them this way yields a new proof of the Yu-Oh inequality and gives rise to family of related inequalities.  I will also explain why I think that most papers on ovelap bounds (including my own) have adopted sub-optimal measures, introduce better ones, and explain how this affects the choice of the best experimental protocol for demonstrating overlap bounds.  

The ideas of no-signalling, nonlocality, Bell inequalities, and quantum correlations can all be understood as implications of a presumed causal structure. In particular, the causal structure of the Bell scenario implies the Bell inequalities whenever the shared resource is presumed to act like a classical hidden random variable. If the shared resource in the scenario is a quantum system, however, then the quantum causal structure can give rise to a larger set of correlations, including probability distributions which violate Bell inequalities up to Tsirelson's bound. It is hard to generically distinguish between the classical and quantum correlations, though, because the standard method for computing Bell inequalities cannot be generalized to general causal scenarios. We therefore introduce a method (the "Inflation DAG" technique) to heuristically constrain the set of correlations compatible with a given classical causal structure, and we demonstrate how it may be used to derive explicit inequalities in terms of probabilities. We also discuss deriving physics-independent constraints, i.e. non-trivial inequalities which are nevertheless satisfied even by all quantum correlations, thereby quantifying some absolute limits as to what the universe allows.

[Unpublished results of E.W., Rob Spekkens, Tobias Fritz]