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1. Error Correction for Correlated Quantum Systems

   https://doi.org/https://doi.org/10.1007/978-3-030-63591-6_34  

   Byrd, Mark; Gonzales, Alvin; Dilley, Dilley; Thakre, Purva (August 2021, International Conference on Applied Mathematics, Modeling and Computational Science, AMMCS 2019: Recent Developments in Mathematical, Statistical and Computational Sciences)

   Kilgour, D.M.; Kunze, H.; Makarov, R.; Melnik, R.; Wang, X. (Ed.)

   Modeling open quantum systems is a difficult task for many experiments. A standard method for modeling open system evolution uses an environment that is initially uncorrelated with the system in question, evolves the two unitarily, and then traces over the bath degrees of freedom to find an effective evolution of the system. This model can be insufficient for physical systems that have initial correlations. Specifically, there are evolutions ρS=trE(ρSE)→ρ′S=trE(UρSEU†) which cannot be modeled as ρS=trE(ρSE)→ρ′S=trE(UρS⊗ρEU†). An example of this is ρSE=|Φ+⟩⟨Φ+| and USE=CNOT with control on the environment. Unfortunately, there is no known method of modeling an open quantum system which is completely general. We first present some restrictions on the availability of completely positive (CP) maps via the standard prescription. We then discuss some implications a more general treatment would have for quantum control methods. In particular, we provide a theorem that restricts the reversibility of a map that is not completely positive (NCP). Let Φ be NCP and Φ~ be the corresponding CP map given by taking the absolute value of the coefficients in Φ. The theorem shows that the CP reversibility conditions for Φ~ do not provide reversibility conditions for Φ unless Φ is positive on the domain of the code space. 

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2. Entanglement-breaking superchannels

   https://doi.org/10.22331/q-2020-07-16-299  

   Chen, Senrui; Chitambar, Eric (July 2020, Quantum)

   In this paper we initiate the study of entanglement-breaking (EB) superchannels. These are processes that always yield separable maps when acting on one side of a bipartite completely positive (CP) map. EB superchannels are a generalization of the well-known EB channels. We give several equivalent characterizations of EB supermaps and superchannels. Unlike its channel counterpart, we find that not every EB superchannel can be implemented as a measure-and-prepare superchannel. We also demonstrate that many EB superchannels can be superactivated, in the sense that they can output non-separable channels when wired in series.We then introduce the notions of CPTP- and CP-complete images of a superchannel, which capture deterministic and probabilistic channel convertibility, respectively. This allows us to characterize the power of EB superchannels for generating CP maps in different scenarios, and it reveals some fundamental differences between channels and superchannels. Finally, we relax the definition of separable channels to include ( p , q ) -non-entangling channels, which are bipartite channels that cannot generate entanglement using p - and q -dimensional ancillary systems. By introducing and investigating k -EB maps, we construct examples of ( p , q ) -EB superchannels that are not fully entanglement breaking. Partial results on the characterization of ( p , q ) -EB superchannels are also provided. 

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3. Sufficient conditions and constraints for reversing general quantum errors

   https://doi.org/https://doi.org/10.1103/PhysRevA.102.062415  

   Gonzales, Alvin; Dilley, Daniel; Byrd, Mark (December 2020, Physical review and Physical review letters index)

   Abstract Reversing the effects of a quantum evolution, for example, as is done in error correction, is an important task for controlling quantum systems in order to produce reliable quantum devices. When the evolution is governed by a completely positive map, there exist reversibility conditions, known as the quantum error correcting code conditions, which are necessary and sufficient conditions for the reversibility of a quantum operation on a subspace, the code space. However, if we suppose that the evolution is not described by a completely positive map, necessary and sufficient conditions are not known. Here we consider evolutions that do not necessarily correspond to a completely positive map. We prove that the completely positive map error correcting code conditions can lead to a code space that is not in the domain of the map, meaning that the output of the map is not positive. A corollary to our theorem provides a class of relevant examples. Finally, we provide a set of sufficient conditions that will enable the use of quantum error correcting code conditions while ensuring positivity. 

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4. Determinantal Representations and the Image of the Principal Minor Map

   https://doi.org/10.1093/imrn/rnae038  

   Al_Ahmadieh, Abeer; Vinzant, Cynthia (March 2024, International Mathematics Research Notices)

   Abstract In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $$(\textrm{SL}_{2}(\mathbb{R}))^{n} \rtimes S_{n}$$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $$\mathbb{F}$$, there is no finite set of equations whose orbit under $$(\textrm{SL}_{2}(\mathbb{F}))^{n} \rtimes S_{n}$$ cuts out the image of $$n\times n$$ matrices over $${\mathbb{F}}$$ under the principal minor map for every $$n$$. 

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5. On reduction of differential inclusions and Lyapunov stability

   https://doi.org/10.1051/cocv/2019074  

   Kamalapurkar, Rushikesh; Dixon, Warren E.; Teel, Andrew R. (January 2020, ESAIM: Control, Optimisation and Calculus of Variations)

   In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is pointwise smaller (in the sense of set containment) than the original set-valued map. The corresponding reduced differential inclusion, defined by the reduced set-valued map, is utilized to develop a generalized notion of a derivative for locally Lipschitz candidate Lyapunov functions in the direction(s) of a set-valued map. The developed generalized derivative yields less conservative statements of Lyapunov stability theorems, invariance theorems, invariance-like results, and Matrosov theorems for differential inclusions. Included illustrative examples demonstrate the utility of the developed theory. 

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