A bstract In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form $\textrm{Rep}\left(\mathcal{H}\right)$ Rep H for $\mathcal{H}$ H a suitable Hopf algebra (which includes the special case Rep( G ) for G a finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on ${\mathcal{H}}^{\ast }$ H ∗ . We discuss how ordinary G orbifolds for finite groups G are a special case of the construction, corresponding to the fusion category Vec( G ) = Rep(ℂ[ G ] * ). For the cases Rep( S 3 ), Rep( D 4 ), and Rep( Q 8 ), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep( S 3 ), Rep( D 4 ), Rep( Q 8 ), and $\textrm{Rep}\left({\mathcal{H}}_8\right)$ Rep H 8 , and discuss applications in c = 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.A bstract In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form $$ \textrm{Rep}\left(\mathcal{H}\right) $$ Rep H for $$ \mathcal{H} $$ H a suitable Hopf algebra (which includes the special case Rep( G ) for G a finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on $$ {\mathcal{H}}^{\ast } $$ H ∗ . We discuss how ordinary G orbifolds for finite groups G are a special case of the construction, corresponding to the fusion category Vec( G ) = Rep(ℂ[ G ] * ). For the cases Rep( S 3 ), Rep( D 4 ), and Rep( Q 8 ), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep( S 3 ), Rep( D 4 ), Rep( Q 8 ), and $$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ Rep H 8 , and discuss applications in c = 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.

arXiv:2311.16230Journal of High Energy PhysicsFeb 21, 202439 citations

Probabilistic Remote Preparation of a Four-Particle Entangled W State for the General Case and for All Kinds of the Special Cases

Hong‐Yi Dai, Ming Zhang, Zu-Rong Zhang +1 more

We present a protocol for probabilistic remote preparation of a four-particle entangled W state. The quantum channel is composed of two partial entangled four-particle cluster states. We calculate the total successful probability and total classical communication cost required for the general case and for all kinds of the special cases, respectively. It is shown that for two maximally entangled four-particle cluster states, such a scheme for the general case has the total successful probability of 25% and only consumes the total classical communication of 1 bit, while this scheme for the special cases under certain conditions can possess successful probability of 50% or 100%, the required classical communication will only be 2 bits or 4 bits. Meantime, we give in detail all unitary transformations for the general case and for all kinds of the special cases, respectively.We present a protocol for probabilistic remote preparation of a four-particle entangled W state. The quantum channel is composed of two partial entangled four-particle cluster states. We calculate the total successful probability and total classical communication cost required for the general case and for all kinds of the special cases, respectively. It is shown that for two maximally entangled four-particle cluster states, such a scheme for the general case has the total successful probability of 25% and only consumes the total classical communication of 1 bit, while this scheme for the special cases under certain conditions can possess successful probability of 50% or 100%, the required classical communication will only be 2 bits or 4 bits. Meantime, we give in detail all unitary transformations for the general case and for all kinds of the special cases, respectively.

Communications in Theoretical PhysicsSep 1, 201326 citations

On a conjecture regarding quantum hypothesis testing

Zsombor Szilágyi

In this MSc thesis I consider the asymptotic behaviour of the symmetric error in composite hypothesis testing. In the classical case, when the null and alternative hypothesis are finite sets of states, the best achievable symmetric error exponent is simply the the best achievable symmetric error exponent corresponding to the "worst case pair". The conjecture -- raised several years ago -- is that this remains true in the quantum case, too. This is known to be true in some special case. However, all of the known special cases are in some sense "too nice", e.g., have certain symmetries. Hoping to find a counter-example, in my thesis I consider a new special case, which on one hand is as "asymmetrical" as possible, yet still analytically computable. However, as it turns out from some involved computation, the conjecture is also true in this case, and thus gives further evidence to this conjecture.In this MSc thesis I consider the asymptotic behaviour of the symmetric error in composite hypothesis testing. In the classical case, when the null and alternative hypothesis are finite sets of states, the best achievable symmetric error exponent is simply the the best achievable symmetric error exponent corresponding to the "worst case pair". The conjecture -- raised several years ago -- is that this remains true in the quantum case, too. This is known to be true in some special case. However, all of the known special cases are in some sense "too nice", e.g., have certain symmetries. Hoping to find a counter-example, in my thesis I consider a new special case, which on one hand is as "asymmetrical" as possible, yet still analytically computable. However, as it turns out from some involved computation, the conjecture is also true in this case, and thus gives further evidence to this conjecture.

arXiv:2011.03342Nov 5, 20201 citation

Elementary Cases

MASSIMO RUDAN

Springer eBooksDec 10, 20140 citations

A bstract In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form

\textrm{Rep}\left(\mathcal{H}\right)

Rep H for

\mathcal{H}

H a suitable Hopf algebra (which includes the special case Rep( G ) for G a finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on

{\mathcal{H}}^{\ast }

H ∗ . We discuss how ordinary G orbifolds for finite groups G are a special case of the construction, corresponding to the fusion category Vec( G ) = Rep(ℂ[ G ] * ). For the cases Rep( S 3 ), Rep( D 4 ), and Rep( Q 8 ), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep( S 3 ), Rep( D 4 ), Rep( Q 8 ), and

\textrm{Rep}\left({\mathcal{H}}_8\right)

Rep H 8 , and discuss applications in c = 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.