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arXiv:1701.08630 · Physical Review A

Coarse graining the phase space of N qubits

Olivia Di Matteo, Luis Lorenzo Sánchez-Soto, Gerd Leuchs, Markus Grassl

Published Feb 28, 2017• 6 citations
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Abstract

We develop a systematic coarse-graining procedure for systems of $N$ qubits. We exploit the underlying geometrical structures of the associated discrete phase space to produce a coarse-grained version with reduced effective size. Our coarse-grained spaces inherit key properties of the original ones. In particular, our procedure naturally yields a subset of the original measurement operators, which can be used to construct a coarse discrete Wigner function. These operators also constitute a systematic choice of incomplete measurements for the tomographer wishing to probe an intractably large system.

DOI
10.1103/physreva.95.022340

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We develop a systematic coarse-graining procedure for systems of NNN qubits. We exploit the underlying geometrical structures of the associated discrete phase space to produce a coarse-grained version with reduced effective size. Our coarse-grained spaces inherit key properties of the original ones. In particular, our procedure naturally yields a subset of the original measurement operators, which can be used to construct a coarse discrete Wigner function. These operators also constitute a systematic choice of incomplete measurements for the tomographer wishing to probe an intractably large system.
We present a technique to coarse grain quantum states in a finite-dimensional Hilbert space. Our method is distinguished from other approaches by not relying on structures such as a preferred factorization of Hilbert space or a preferred set of operators (local or otherwise) in an associated algebra. Rather, we use the data corresponding to a given set of states, either specified independently or constructed from a single state evolving in time. Our technique is based on principle component analysis (PCA), and the resulting coarse-grained quantum states live in a lower-dimensional Hilbert space whose basis is defined using the underlying (isometric embedding) transformation of the set of fine-grained states we wish to coarse grain. Physically, the transformation can be interpreted to be an ``entanglement coarse-graining'' scheme that retains most of the global, useful entanglement structure of each state, while needing fewer degrees of freedom for its reconstruction. This scheme could be useful for efficiently describing collections of states whose number is much smaller than the dimension of Hilbert space, or a single state evolving over time.
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One can think of some physical evolutions as being the emergent-effective result of a microscopic discrete model. Inspired by classical coarse-graining procedures, we provide a simple procedure to coarse-grain color-blind quantum cellular automata that follow the so-called Goldilocks rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size NNN; (ii) projecting the states of a cell onto its borders, connecting them with the fine dynamics; (iii) describing the overall dynamics by the border states, that we call signals; and (iv) constructing the coarse-grained dynamics for different sizes NNN of the cells. A by-product of this simple toy model is a general discrete analogous of the Stokes law. Moreover we prove that in the space-time limit, the automaton converges to a Dirac free Hamiltonian. The QCA we introduce here can be implemented by present-day quantum platforms, such as Rydberg arrays, trapped ions, and superconducting qubits. We hope our study can pave the way to a richer understanding of those systems with limited resolution.
One can think of some physical evolutions as being the emergent-effective result of a microscopic discrete model. Inspired by classical coarse graining procedures, we provide a simple procedures to coarse-grain color-blind quantum cellular automata, following Goldilock rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size NNN, (ii) projecting the cells states onto its borders, connecting them with the fine dynamics, which is reminiscent of the edge-bulk correspondence in quantum field theory; (iii) describe the overall dynamics by the border states, that we call signals; (iv) construct the coarse-grained dynamics for different sizes NNN of the cells. A byproduct of this simple toy-model is a general discrete analogous of the Stokes law. Moreover, we proved that in the spacetime limit, the automaton converge to a Dirac free Hamiltonian. The QCA we introduced here, can be implemented by present-day quantum platforms, such as Rydberg arrays, trapped ions, and superconducting qbits. We hope our study can pave the way to a richer understanding of those systems with limited resolution.
We study a coarse-graining map arising from incomplete and imperfect addressing of particles in a multipartite quantum system. In its simplest form, corresponding to a two-qubit state, the resulting channel produces a convex mixture of the two partial traces. We derive the probability density of obtaining a given coarse-grained state, using geometric arguments for two qubits coarse-grained to one, and random-matrix methods for larger systems. As the number of qubits increases, the probability density sharply concentrates around the maximally mixed state, making nearly pure coarse-grained states increasingly unlikely. For two qubits, we also compute the inverse state needed to characterize the effective dynamics under coarse-graining and find that the average preimage of the maximally mixed state contains a finite singlet component. Finally, we validate the analytical predictions by inferring the underlying probabilities from Monte-Carlo-generated coarse-grained statistics.
Using the quantum map formalism, we provide a framework to construct fuzzy and coarse-grained quantum maps of many-body systems that account for limitations in the resolution of real measurement devices probing them. The first set of maps handles particle-indexing errors, while the second deals with the effects of detectors that can only resolve a fraction of the system constituents. We apply these maps to a spin-1/2XX1/2 \mathit{XX}1/2XX chain, obtaining a blurred picture of the entanglement generation and propagation in the system. By construction, both maps are simply related via a partial trace, which allows us to concentrate on the properties of the former. We fully characterize the fuzzy map, identifying its symmetries and invariant spaces. We show that the volume of the tomographically accessible states decreases at a double-exponential rate in the number of particles, imposing severe bounds on the ability to read and use information of a many-body quantum system.