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arXiv:2010.15783 · Journal of Physics A Mathematical and Theoretical

On construction of finite averaging sets for SL(2,C) via its Cartan decomposition

Marcin Markiewicz, Janusz Przewocki

Published Apr 22, 2021• 3 citations
arXivHTMLPDF

Abstract

Abstract Averaging physical quantities over Lie groups appears in many contexts across the rapidly developing branches of physics like quantum information science or quantum optics. Such an averaging process can be always represented as averaging with respect to a finite number of elements of the group, called a finite averaging set . In the previous research such sets, known as t -designs, were constructed only for the case of averaging over unitary groups (hence the name unitary t -designs ). In this work we investigate the problem of constructing finite averaging sets for averaging over general non-compact matrix Lie groups, which is much more subtle task due to the fact that the the uniform invariant measure on the group manifold (the Haar measure) is infinite. We provide a general construction of such sets based on the Cartan decomposition of the group, which splits the group into its compact and non-compact components. The averaging over the compact part can be done in a uniform way, whereas the averaging over the non-compact one has to be endowed with a suppressing weight function, and can be approached using generalized Gauss quadratures. This leads us to the general form of finite averaging sets for semisimple matrix Lie groups in the product form of finite averaging sets with respect to the compact and non-compact parts. We provide an explicit calculation of such sets for the group S L ( 2 , C ) , although our construction can be applied to other cases. Possible applications of our results cover finding finite ensembles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.

DOI
10.1088/1751-8121/abfa44

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Abstract Averaging physical quantities over Lie groups appears in many contexts across the rapidly developing branches of physics like quantum information science or quantum optics. Such an averaging process can be always represented as averaging with respect to a finite number of elements of the group, called a finite averaging set . In the previous research such sets, known as t -designs, were constructed only for the case of averaging over unitary groups (hence the name unitary t -designs ). In this work we investigate the problem of constructing finite averaging sets for averaging over general non-compact matrix Lie groups, which is much more subtle task due to the fact that the the uniform invariant measure on the group manifold (the Haar measure) is infinite. We provide a general construction of such sets based on the Cartan decomposition of the group, which splits the group into its compact and non-compact components. The averaging over the compact part can be done in a uniform way, whereas the averaging over the non-compact one has to be endowed with a suppressing weight function, and can be approached using generalized Gauss quadratures. This leads us to the general form of finite averaging sets for semisimple matrix Lie groups in the product form of finite averaging sets with respect to the compact and non-compact parts. We provide an explicit calculation of such sets for the group S L ( 2 , C ) , although our construction can be applied to other cases. Possible applications of our results cover finding finite ensembles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.
We construct finite averaging sets for twirling finite-dimensional quantum states over a broad class of symmetry transformations represented by reductive Lie groups. The averaging is defined through the Cartan decomposition of the group, with Haar-uniform integration over the compact components and an arbitrary normalized measure over the non-compact Abelian component. We show that the resulting twirling map can always be represented as a finite probabilistic mixture of unitary 1-design channels acting on irreducible sectors of the representation restricted to the maximal compact subgroup. Thus, the finite construction is determined by the compact-sector decomposition, while the non-compact Cartan component contributes only scalar sector weights. This distinction is important for implementation: no Schur transform for the full reductive group is required. Compared with standard t -design-based finite averaging, our construction generally uses fewer averaging operators, at the price of replacing local tensor-power sampling by sector-wise unitary averaging. This could be especially beneficial for simulations and quantum algorithms on higher-dimensional systems as reduction of the size of the averaging set relatively increases with dimension.
Twirling, i.e. averaging over symmetry actions, is a standard tool for reducing quantum states and channels to a symmetry-invariant form. We study channel twirling from the perspective of the channel-state duality and provide a constructive Choi-level description of the averaging map induced by arbitrary input/output representations. Our main technical result concerns the collective setting: for πin(U)=U⊗tinπ^{\textrm{in}}(U)=U^{\otimes t_{\textrm{in}}}πin(U)=U⊗tin​ and πout(U)=U⊗toutπ^{\textrm{out}}(U)=U^{\otimes t_{\textrm{out}}}πout(U)=U⊗tout​, we introduce a partial-transpose reduction that removes the contragredient action and converts the mixed (walled Brauer) channel twirl into an ordinary Schur-Weyl twirl of the partially transposed Choi operator under U⊗(tin+tout)U^{\otimes(t_{\textrm{in}}+t_{\textrm{out}})}U⊗(tin​+tout​), enabling explicit permutation-based formulas without constructing walled Brauer idempotents or mixed Schur transforms. Beyond compact symmetries, we extend channel twirling to reductive, generally non-unitary groups via Cartan decomposition and obtain an invariant-sector decomposition of the averaged Choi operator with weights determined solely by the Abelian Cartan component. Finally, we provide two finite realizations of channel averaging: a "dual" implementation as a convex mixture of unitary-111-design channels acting on invariant sectors, and a design-like reconstruction showing that weighted group ttt-designs induce channel ttt-designs for t=tin+toutt=t_{\textrm{in}}+t_{\textrm{out}}t=tin​+tout​.
We provide a generalization of the idea of unitary designs to cover finite averaging over much more general operations on quantum states. Namely, we construct finite averaging sets for averaging quantum states over arbitrary reductive Lie groups, on condition that the averaging is performed uniformly over the compact component of the group. Our construction comprises probabilistic mixtures of unitary 1-designs on specific operator subspaces. Provided construction is very general, competitive in the size of the averaging set when compared to other known constructions, and can be efficiently implemented in the quantum circuit model of computation.
Abstract It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging . The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs . In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.
Abstract Unitary t -designs are ‘good’ finite subsets of the unitary group that approximate the whole unitary group well. Unitary t -designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary t -design itself is a group then it is called a unitary t -group. Although it is known that unitary t -designs in exist for any t and d , the unitary t -groups do not exist for if , as it is shown by Guralnick–Tiep (Guralnick and Tiep 2005 Represent. Theory 9 138–208) and Bannai–Navarro–Rizo–Tiep (Bannai et al 2018 J. Math. Soc. Japan (accepted) ). Explicit constructions of exact unitary t -designs in are not easy in general. In particular, explicit constructions of unitary 4-designs in have been an open problem in quantum information theory. We prove that some exact unitary -designs in the unitary group are constructed from unitary t -groups in that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary 3-designs in from the unitary 2-group in and also unitary 4-designs in from the unitary 3-group in numerically. We also discuss some related problems and a few applications to physics problems.
A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.