Scientific Series

We will discuss work, joint with Victor Ginzburg, on the quantization (non-commutative deformation) of the Ngô morphism, a morphism of group schemes which plays a key role in Ngô’s proof of the fundamental lemma in the Langlands program. We will also discuss how the tools used to construct this morphism can be used to prove conjectures of Ben-Zvi—Gunningham, which predict that this morphism gives “spectral decomposition” of DG categories with an action of a reductive group over the coarse quotient of a maximal Cartan subalgebra by the affine Weyl group.

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The quantum difference equation (qde) is the $q$-difference equation which is proposed by Okounkov and Smirnov to encode the $K$-theoretic twisted quasimap counting for the Nakajima quiver varieties. In this talk, we will give a direct quantum toroidal algebra $U_{q,t}(\hat{\hat{\mf{sl}}}_{n})$ construction for the qde of the affine type $A$ quiver varieties. We will show that there is a really explicit and concise formula for the quantum difference operators. Moreover we will show that the degeneration limit of the quantum difference equation is equivalent to the Dubrovin connection for the quantum cohomology of the affine type A quiver varieties, which will give the description of the monodromy representation of the Dubrovin connection via the monodromy operators in the quantum difference equation.

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4 months after it opened, the Tacoma Narrows bridge, one of the world's
longest suspension bridges at the time, collapsed spectacularly in the first
storm that hit it. Though it was built to exceed all of the standrds at the
time, something  clearly went wrong. The failure was filmed from almost the
beginning to the end (about 1 hour), and that film has been shown to almost
all first year physics or engineering classes as an example of resonance, that
explanation is clearly nonsense. What happened? Why did it collapse. The
explanation is closely linked to, for example, the reason that clarinets or
flutes, or even violins, make their music. With Daniel Green (whom you may
remember from his time at Toronto) we were able to show in detail what
happened.

For continuous-variable systems, the negativities in the s-parametrized family of quasi-probability representations on a classical phase space establish a sort of hierarchy of non-classility measures. The coherent states, by design, display no negativity for any value of -1≤s≤1, meaning that sampling from the quantum probability distribution resulting from any measurement of a coherent state can be classically simulated, placing the coherent states as the most classical states according to this particular choice of phase space.

In this talk, I will describe how to construct s-ordered quasi-probability representations for finite-dimensional quantum systems when the phase space is equipped with more general group symmetries, focusing on the fermionic SO(2n) symmetry. Along the way, I will comment on an obstruction to an analogue of Hudson's theorem, namely that the only pure states that have positive s=0 Wigner functions are Gaussian states, and a possible remedy by giving up linearity in the phase-space correspondence.

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A Bell scenario can be conceptualized as a "communication" scenario with zero rounds of communication between parties, i.e., although each party can receive a system from its environment on which it can implement a measurement, it cannot send out any system to another party. Under this constraint, there is a strict hierarchy of correlation sets, namely, classical, quantum, and non-signalling. However, without any constraints on the number of communication rounds between the parties, they can realize arbitrary correlations by exchanging only classical systems. We consider a multipartite scenario where the parties can engage in at most a single round of communication, i.e., each party is allowed to receive a system once, implement any local intervention on it, and send out the resulting system once. Taking our cue from Bell nonlocality in the "zero rounds" scenario, we propose a notion of nonclassicality---termed antinomicity---for correlations in scenarios with a single round of communication. Similar to the zero rounds case, we establish a strict hierarchy of correlation sets classified by their antinomicity in single-round communication scenarios. Since we do not assume a global causal order between the parties, antinomicity serves as a notion of nonclassicality in the presence of indefinite causal order (as witnessed by causal inequality violations). A key contribution of this work is an explicit antinomicity witness that goes beyond causal inequalities, inspired by a modification of the Guess Your Neighbour's Input (GYNI) game that we term the Guess Your Neighbour's Input or NOT (GYNIN) game. Time permitting, I will speculate on why antinomicity is a strong notion of nonclassicality by interpreting it as an example of fine-tuning in classical models of indefinite causality.This is based on joint work with Ognyan Oreshkov, arXiv:2307.02565.

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Quantum measurement has conventionally been regarded as the final step in quantum information processing, which is essential for reading out the processed information but collapses the quantum state into a classical state. However, recent studies have shown that quantum measurement itself can induce novel quantum phenomena. One seminal example is a monitored random circuit, which can generate long-range entanglement faster than a random unitary circuit. Inspired by these results, in this talk, we address the following question: When quantum information is encoded in a quantum error-correcting code, how many physical qubits should be randomly measured to destroy the encoded information? We investigate this question for various quantum error-correcting codes and derive the necessary and sufficient conditions for destroying the information through measurements. In particular, we demonstrate that for a large class of quantum error-correcting codes, it is impossible to destroy the encoded information through random single-qubit Pauli measurements when a tiny portion of physical qubits is still unmeasured. Our results not only reveal the extraordinary robustness of quantum codes under measurement decoherence, but also suggest potential applications in quantum information processing tasks.

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The concept of symmetries is crucial in our comprehension of modern theoretical physics. The Corner Proposal introduces a novel framework where symmetries are reinstated as foundational principles in our understanding of gravity. This aims to describe gravity using a language that is more adapted to quantization. In this presentation, I will start by providing an overview of the essential results leading to the main ideas the proposal. This will then allow me to state the proposal in the general case to then specialize to 1+1 dimensional gravity.

Finally, I will present elements of our recent research applying the proposal to the case of 1+1 dimensional gravity. I will demonstrate the framework's promising potential by calculating the entanglement entropy between two spatial regions—a significant challenge in quantum gravity. The result is the 1+1 dimensional equivalent of the well-established Bekenstein-Hawking area law governing the entropy of gravitational systems with the expected behavior of the quantum corrections.

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This colloquium is presented in collaboration with the Foundations of Quantum Computational Advantage conference. 

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Abstract TBA

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Integrable field theories in two dimensions are known to originate as defect theories of 4d Chern-Simons theory and as symmetry reductions of the 4d anti-self-dual Yang-Mills equations. Based on ideas of Costello, it has been proposed in work of Bittleston and Skinner that these two approaches can be unified starting from holomorphic Chern-Simons theory in 6 dimensions. In this talk I will introduce the first complete description of this diamond of integrable theories for a family of deformed sigma models, going beyond the Dirichlet boundary conditions that have been considered thus far. The talk is based on the recent work https://arxiv.org/abs/2311.17551.

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Ground states as well as Gibbs states of many-body quantum Hamiltonians have been studied extensively since the inception of quantum mechanics. In contrast, the landscape of many-body quantum states that are not in thermal equilibrium is relatively less explored. In this talk I will discuss some of the recent progress in understanding decohered or dissipative quantum many-body states. One of the key ideas I will employ is that of "separability", i.e., whether a mixed state can be expressed as an ensemble of short-range entangled pure states. I will discuss several quantum phase transitions in topological phases of matter subjected to Markovian environmental noise from a separability viewpoint, and argue that such a framework also subsumes our understanding of pure quantum states as well as Gibbs states. Time permitting, I will also provide a brief overview of quantum spin-systems subjected to non-Markovian noise originating from an electronic bath, and discuss a new critical phase of matter where quantum coherence coexists with dissipation.

References: 2307.13889, 2309.11879, 2310.07286

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