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Long-range entangled mixed states are exotic many-body systems that exhibit intrinsically quantum phenomena despite extensive classical fluctuations. In the first part of the talk, I will show how they can be efficiently prepared with measurements and unitary feedback conditioned on the measurement outcome. For example, symmetry-protected topological phases can be universally converted into mixed states with long-range entanglement, and certain gapped topological states such as Chern insulators can be converted into mixed states with critical correlations in the bulk. In the second part of the talk, I will discuss how decoherence can drive interesting mixed-state entanglement transitions. By focusing on the toric codes in various space dimensions subject to certain types of decoherence, I will present the exact results of entanglement negativity, from which the universality class of entanglement transitions can be completely characterized.

For quantum phases of Hamiltonian ground states, the energy gap plays a central role in ensuring the stability of the phase as long as the gap remains finite. In this talk we introduce Markov length, the length scale at which the quantum conditional mutual information (CMI) decays exponentially, as an equally essential quantity characterizing mixed-state phases and transitions. For a state evolving under a local Lindbladian, we argue that if its Markov length remains finite along the evolution, then it remains in the same phase, meaning there exists another quasi-local Lindbladian evolution that can reverse the former one. We apply this diagnostic to toric code subject to decoherence and show that the Markov length is finite everywhere except at its decodability transition, at which it diverges. This implies that the mixed state phase transition coincides with the decodability transition and also suggests a quasi-local decoding channel.

Ground states as well as Gibbs states of many-body quantum Hamiltonians have been studied extensively for some time. In contrast, the landscape of mixed states that do not correspond to a system in thermal equilibrium is relatively less explored. In this talk I will motivate a rather coarse characterization of mixed quantum many-body states using the idea of "separability", i.e., whether a mixed state can be expressed as an ensemble of short-range entangled pure states. I will discuss several examples of decoherence-driven phase transitions from a separability viewpoint, and argue that such a framework also provides a potentially new view on Gibbs states. Based on work with Yu-Hsueh Chen. References: 2309.11879, 2310.07286, 2403.06553.

"Optimal fault tolerant error correction thresholds for CCS codes are traditionally obtained via mappings to classical statistical mechanics models, for example the 2d random bond Ising model for the 1d repetition code subject to bit-flip noise and faulty measurements. Here, we revisit the 1d repetition code, and develop an exact “stabilizer expansion” of the full time evolving density matrix under repeated rounds of (incoherent and coherent) noise and faulty stabilizer measurements. This expansion enables computation of the coherent information, indicating whether encoded information is retained under the noisy dynamics, and generates a dual representation of the (replicated) 2d random bond Ising model. However, in the fully generic case with both coherent noise and weak measurements, the stabilizer expansion breaks down (as does the canonical 2d random bond Ising model mapping). If the measurement results are thrown away all encoded information is lost at long times, but the evolution towards the trivial steady state reveals a signature of a quantum transition between an over and under damped regime. Implications for generic noisy dynamics in other CCS codes will be mentioned, including open issues."

We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary ‘clock register’ to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any ‘combinatorial state’ with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As an application, we give a new proof of the QMA-completeness of the local Hamiltonian problem (with logarithmic locality) and show that contracting injective tensor networks to additive error is BQP- hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth. Based on joint work with Anurag Anshu and Nikolas P. Breuckmann. (https://arxiv.org/abs/2309.16475)

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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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