Research

Covariance

AdS/CFT

Chern-Simons

Error Correction

Universality

Entanglement

Polarization

Gambling

• Covariant Quantum Error Correction and an Approximate Eastin-Knill Theorem

  We studied the problem of error correction of various forms of "physical information", such as quantum reference frames. We showed that QEC of reference frame information is equivalent to the problem of group covariant quantum error correction, in which the encoding and decoding maps must both transform covariantly with respect to a symmetry group. The additional constraints imposed by this covariance condition could, in principle, be sufficiently restrictive so as to preclude the existence of covariant encoding and decoding maps. Indeed, we prove a no-go theorem showing that there are no finite-dimensional quantum codes that are covariant with respect to a continuous symmetry. However, we also provide explicit constructions of covariant codes in all other cases (finite groups, infinite dimensional Hilbert spaces, etc.) Our no-go theorem reproduces the main thrust of the Eastin-Knill theorem when we restrict our encoding operations to isometric embeddings. By circumventing our no-go theorem using infinite dimensional codes, we consequently showed that one can also circumvent the Eastin-Knill theorem using continuous variables. In an in depth follow up project, we generalized our theorem to approximate quantum error correcting codes, thereby proving a robust version of the Eastin-Knill theorem, and a general tension between continuous symmetries and quantum error correction. [arXiv:1709.04471], and [arXiv:1902.07714]

• Entanglement Wedge Reconstruction via Universal Recovery Channels

  The AdS/CFT correspondence is a duality between a bulk gravitational theory in anti-de Sitter space and a boundary conformal field theory in one lower spatial dimension. The correspondence constitutes one of our best toy models of a theory of quantum gravity. An important recent discovery in AdS/CFT is that it exhibits properties of quantum error correction, a discovery that sheds light on the longstanding question of what is dual (in the bulk gravity theory) to a subset of the conformal field theory. One facet of this discovery -- the entanglement wedge hypothesis -- posits that any local bulk operator living in the so-called entanglement wedge of a boundary subregion can be written as an operator with support only on that subregion. A key ingredient in proofs of the hypothesis is that relative entropies are exactly preserved between bulk and boundary, which is a strong assumption that is known not to be true in general. In joint work with Jordan Cotler, Patrick Hayden, Brian Swingle, and Michael Walter, we prove the entanglement wedge hypothesis by leveraging recent results in quantum information theory. In particular, we used the theory of universal recovery channels to show that AdS/CFT is robustly quantum error correcting, and we found an explicit formula for entanglement wedge reconstruction. We proved all our results at the level of algebras of observables by generalizing the universal recovery theorems to finite dimensional von Neumann algebras. [arXiv:1704.05839]

• Entanglement from topology in Chern-Simons theory

  In a general quantum field theory, the Euclidean path integral provides a way of preparing a state of the field. In a topological quantum field theory (TQFT), all quantities of interest depend only on the topology of the underlying manifold, and not on the geometry. As such, the Euclidean path integral in a TQFT prepares states that depend only on topological invariants of the manifold. We ask the question: by varying over all manifolds, which states can we prepare? In particular, we studied Chern-Simons theory (a type of TQFT) on 3-manifolds whose boundaries are a disjoint union of tori. Varying over all such 3-manifolds with any allowed link of Wilson lines generates a set of a states through the path integral. For the abelian theory U(1)_k, we showed that the set of states generated through the path integral is precisely the set of stabilizer states. In the non-abelian case, we showed for SO(3)_k at certain levels k that any state can be prepared, giving rise to a notion of "state universality". Our work was motivated by multi-boundary wormholes in AdS/CFT. [arXiv:1611.01516], or [Phys. Rev. D]

• Spacetime replication of continuous variable quantum information

  How does quantum information move through spacetime? Techniques like quantum teleportation and error correction show us that we can break quantum information into classical data (which can be cloned), and quantum correlations (which can persist non-locally), and later combine the two in order to recover the state at a different place and time. But what limits do the laws of physics place on the passage of quantum information through spacetime? The theory of relativity requires that no information travel faster than light, whereas the unitarity of quantum mechanics ensures that quantum information cannot be cloned. These conditions provide the basic constraints that appear in information replication tasks, which formalize aspects of the behavior of information in relativistic quantum mechanics. In our work, we provide continuous variable (CV) strategies for spacetime quantum information replication that are directly amenable to optical or mechanical implementation. We use a new class of homologically-constructed CV quantum error correcting codes to provide efficient solutions for the general case of information replication. We also provide an optimized five-mode strategy for replicating quantum information in a particular configuration of four spacetime regions designed not to be reducible to previously performed experiments, and we provide detailed encoding and decoding procedures using standard optical apparatus for this strategy. [arXiv:1601.02544], or [New J. Phys.]

• Universal Quantum Computation by Scattering in the Fermi-Hubbard Model

  The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of its dynamics remains beyond the reach of current numerical methods. We showed that general quantum computations can be encoded into the physics of wave packets propagating through a planar graph, with scattering interactions governed by the fermionic Hubbard model. Therefore, simulating the model on planar graphs is as hard as simulating quantum computation. We gave two different arguments, demonstrating that the simulation is difficult both for wave packets prepared as excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the t-J model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. In that sense, the construction provides a quantum mechanical analog of the Fredkin-Toffoli billiard ball computer. [arXiv:1409.3585], or [New J. Phys.]

• Acceleration-assisted entanglement harvesting and rangefinding

  By coupling locally to a quantum field, two unentangled quantum systems can become entangled through a process known as entanglement harvesting. If the systems accelerate uniformly they will experience Unruh radiation, which can interfere with the harvesting process. We studied the effects of acceleration on the entanglement that can be harvested by two Unruh-DeWitt detectors. We found that acceleration in the same direction (parallel acceleration) degrades the entanglement harvested process. Moreover, this degradation is identical to that caused by Gibbons-Hawking radiation experienced by two inertial detectors in a de Sitter universe. However, we also found that anti-parallel acceleration enhances the entanglement harvesting process, thereby allowing non-inertial detectors to become entangled while their inertial counterparts could not. [arXiv:1408.1395], or [New J. Phys.]

• Power spectrum of CMB polarization from cosmic string wakes

  We calculated the power spectrum of E and B-mode polarization of the CMB due to cosmic string wakes. Cosmic strings are topological defects formed during symmetry breaking transitions and are regions of trapped potential energy. When they move through space, they create a wake of overdensity to form along their path. These wakes affect the matter distribution and hence the CMB temperature. In particular, the CMB can be polarized by way of Thompson scattering, giving rise to the so-called E- and B-modes. We found that cosmic string wakes produce B-mode polarization at leading order, though with a power smaller than that of gravitational lensing. [arXiv:1308.5693] or [MSc. Thesis]

• Betting on quantum theory

  Decision making in the face of uncertainty is something we do each day. We gather and analyze data in order to make informed decisions when faced with uncertainty. An example of this process is gambling, in which the player cannot know the outcome of the game, but nevertheless wishes to maximize winnings. In fact, gambling provides a paradigm for studying decision making in the face of uncertainty. In this joint work with Robin Blume-Kohout (conducted during a summer undergraduate internship at the Perimeter Institute for Theoretical Physics), we attempted to generalize the paradigm of gambling to the quantum regime. We considered decision making in the face of quantum mechanical uncertainty by agents who can use quantum strategies. We first defined the notion of quantum gambling as a paradigm for studying this problem, and then succeeded in generalizing results from classical gambling (such as the Kelly Criterion) to the quantum case. I presented a seminar on this project at the Perimeter Institute; the recorded lecture is available online at https://pirsa.org/09090026.