Criterion for SLOCC equivalence of multipartite quantum states

Zhang, Tinggui; Zhao, Ming‐Jing; Huang, Xiaofen

doi:10.1088/1751-8113/49/40/405301

Cited by 12 publications

(9 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The open problem, which we state in this work, is the question, whether described duality of averaging holds if at least one component of the reductive pair is a non-compact symmetry group. An emblematic example of such a pair is the collective action of a special linear group SL(d, C), which physically represents collective SLOCC-type operations (stochastic local operations and classical communication) [21][22][23][24][25][26][27][28][29], and the symmetric group [1,30]. On the one hand there seem to appear two crucial obstacles.…”

Section: Duality Of Averagingmentioning

confidence: 99%

See 1 more Smart Citation

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

Markiewicz,

Przewocki

2023

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

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.

show abstract

Section: Duality Of Averagingmentioning

confidence: 99%

“…In this section we introduce the concept of averaging multipartite quantum states of a finite dimension d over the set of the most general quantum operations, namely SLOCC [21][22][23][24][25][26][27][28][29].…”

Section: Averaging Multipartite Quantum States Over Slocc Operationsmentioning

confidence: 99%

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

Markiewicz,

Przewocki

2023

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Finite averaging sets for unitary transformations found numerous applications in the theory of quantum information protocols, mainly as a finite source of local random operations [10], [11], [12], [13], [14], [21], [16], [18]. Now it turns out that the group SL(d, C) represents the most general class of local operations that do not increase global quantum correlations, namely Stochastic Local Operations and Classical Communication (SLOCC) on multipartite states with d degrees of freedom in local subsystems [22], [23], [24], [25], [26], [27]. Finite averaging sets with respect to such operations may find applications as finite source of the most general quantum operations not increasing correlations in the theory of general quantum circuits.…”

Section: Physical Motivation and Possible Applicationsmentioning

confidence: 99%

On construction of finite averaging sets for $SL(2, \mathbb{C})$ via its Cartan decomposition

Markiewicz,

Przewocki

2020

Preprint

View full text Add to dashboard Cite

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 suppresing weight function, and can be approached using generalised 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 noncompact parts. We provide an explicit calculation of such sets for the group SL(2, C), although our construction can be applied to other cases. Possible applications of our results cover finding finite ensambles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.

show abstract

“…The classification for multi-qubit pure states under SLOCC is attracting a great deal of attention. [5][6][7][8][9][10] However, there are an uncountable number of SLOCC inequivalent classes in n-qubit systems when n ≥ 4, so it is a formidable task to classify multipartite states under SLOCC.…”

Section: Introductionmentioning

confidence: 99%

The stabilizer for n -qubit symmetric states

Shi

2018

Chinese Phys. B

View full text Add to dashboard Cite

The stabilizer group for an n-qubit state |φ is the set of all invertible local operators (ILO) Gour et al. [Gour G, Kraus B and Wallach N R 2017 J. Math. Phys. 58 092204] presented that almost all n-qubit states |ψ own a trivial stabilizer group when n ≥ 5. In this article, we consider the case when the stabilizer group of an n-qubit symmetric pure state |ψ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state |φ is nontrivial when n ≤ 4. Then we present a class of n-qubit symmetric states |φ with a trivial stabilizer group when n ≥ 5. Finally, we propose a conjecture and prove that an n-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5 under the conjecture we make, which confirms the main result of Gour et al. partly.

show abstract

Criterion for SLOCC equivalence of multipartite quantum states

Cited by 12 publications

References 37 publications

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

On construction of finite averaging sets for $SL(2, \mathbb{C})$ via its Cartan decomposition

The stabilizer for n -qubit symmetric states

Contact Info

Product

Resources

About