All Teleportation and Dense Coding Schemes

We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert–Schmidt scalar product and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed to be `tight' in the sense that all Hilbert spaces involved have the same finite dimension d , and the classical channel involved distinguishes d 2 signals. A general construction procedure for orthonormal bases of unitaries, involving Latin squares and complex Hadamard matrices is also presented.We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert–Schmidt scalar product and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed to be `tight' in the sense that all Hilbert spaces involved have the same finite dimension d , and the classical channel involved distinguishes d 2 signals. A general construction procedure for orthonormal bases of unitaries, involving Latin squares and complex Hadamard matrices is also presented.

arXiv:quant-ph/0003070Journal of Physics A Mathematical and GeneralAug 24, 2001270 citations

Entanglement-invertible channels

Dominic Verdon

In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].

arXiv:2204.04493Journal of Mathematical PhysicsJun 1, 20240 citations

Teleportation of entangled states and dense coding using a multiparticle quantum channel

В. Н. Горбачев, Andrew I. Trubilko, A. I. Zhiliba +1 more

A set of protocols for teleportation and dense coding tasks with the use of a N particle quantum channel, presented by entangled states of the GHZ class, is introduced, when N>2. Using a found representation for the multiparticle entangled states of the GHZ class, it has shown, that for dense coding schemes enhancement of the classical capacity of the channel due from entanglement is N/N-1. If N>2 there is no one - to -one correspondence between teleportation and dense coding schemes in comparison with the EPR channel is exploited. A set of schemes, for which two additional operations as entanglement and disentanglement are permitted, is considered.A set of protocols for teleportation and dense coding tasks with the use of a N particle quantum channel, presented by entangled states of the GHZ class, is introduced, when N>2. Using a found representation for the multiparticle entangled states of the GHZ class, it has shown, that for dense coding schemes enhancement of the classical capacity of the channel due from entanglement is N/N-1. If N>2 there is no one - to -one correspondence between teleportation and dense coding schemes in comparison with the EPR channel is exploited. A set of schemes, for which two additional operations as entanglement and disentanglement are permitted, is considered.

arXiv:quant-ph/0011124Nov 30, 200011 citations

Teleportation and dense coding via a multiparticle quantum channel of the GHZ-class

В. Н. Горбачев, A. I. Zhiliba, Andrew I. Trubilko +1 more

A set of protocols for teleportation and dense coding schemes based on a multiparticle quantum channel, represented by the $N$ -particle entangled states of the GHZ class, is introduced. Using a found representation for the GHZ states, it was shown that for dense coding schemes enhancement of the classical capacity of the channel due from entanglement is $N/N-1$ . Within the context of our schemes it becomes clear that there is no one-to one correspondence between teleportation and dense coding schemes in comparison when the EPR channel is exploited. A set of schemes, for which two additional operations as entanglement and disentanglement are permitted, is considered.A set of protocols for teleportation and dense coding schemes based on a multiparticle quantum channel, represented by the $N$-particle entangled states of the GHZ class, is introduced. Using a found representation for the GHZ states, it was shown that for dense coding schemes enhancement of the classical capacity of the channel due from entanglement is $N/N-1$. Within the context of our schemes it becomes clear that there is no one-to one correspondence between teleportation and dense coding schemes in comparison when the EPR channel is exploited. A set of schemes, for which two additional operations as entanglement and disentanglement are permitted, is considered.

Quantum Information and ComputationAug 1, 200212 citations

Exploring quantum teleportation through unitary error bases

Benjamin Musto

Unitary error bases have a great number of applications across quantum information and quantum computation, and are fundamentally linked to quantum teleportation, dense coding and quantum error correction. Werner's combinatorial construction builds a unitary error basis from a family of Hadamard matrices and a Latin square. In this dissertation, I give a new categorical axiomatisation of Latin squares, and use this to give a fully graphical presentation and proof of the correctness of Werner's construction. The categorical approach makes clear that some of the Latin square axioms are unnecessary for the construction to go through, and I propose a generalised construction scheme with the potential to create new classes of unitary error bases.Unitary error bases have a great number of applications across quantum information and quantum computation, and are fundamentally linked to quantum teleportation, dense coding and quantum error correction. Werner's combinatorial construction builds a unitary error basis from a family of Hadamard matrices and a Latin square. In this dissertation, I give a new categorical axiomatisation of Latin squares, and use this to give a fully graphical presentation and proof of the correctness of Werner's construction. The categorical approach makes clear that some of the Latin square axioms are unnecessary for the construction to go through, and I propose a generalised construction scheme with the potential to create new classes of unitary error bases.

arXiv:1608.04608Aug 16, 20162 citations

Entanglement-equivalences of covariant channels

Dominic Verdon

We define two channels N and N' to be entanglement-equivalent if there exists an entanglement-assisted channel coding scheme for N from N' and vice versa. For example, the noiseless channel on a d^2-level classical system and the noiseless channel on a d-dimensional quantum system are entanglement-equivalent by teleportation and dense coding. We provide a construction of entanglement-equivalences of channels based on the principle of covariance. Let G be a compact quantum group of Kac type, let F': Rep(G) -> Hilb be a fibre functor accessible from the canonical fibre functor F: Rep(G) -> Hilb by a unitary pseudonatural transformation, and let G' be the compact quantum group of Kac type obtained from the fibre functor F' by Tannaka duality. Then for any G-covariant channel N we construct a G'-covariant channel N' and entanglement-assisted channel coding schemes witnessing an entanglement-equivalence of N and N'. This construction induces an equivalence of categories E: Chan(G) -> Chan(G'), where Chan(G) is the category whose objects are G-C*-algebras and whose morphisms are covariant channels. We show that all tight teleportation and dense coding schemes are a special case of our construction, using covariance for the action of the quantum permutation group of the algebra M_d(\mathbb{C}). Where G is a finite abelian group, we explicitly compute the effect of the equivalence E on algebras and channels. We thereby produce concrete new examples of entanglement-equivalences of channels in the case where G=Z_2 \times Z_2 and the noncanonical fibre functor is specified by the Pauli matrices.We define two channels N and N' to be entanglement-equivalent if there exists an entanglement-assisted channel coding scheme for N from N' and vice versa. For example, the noiseless channel on a d^2-level classical system and the noiseless channel on a d-dimensional quantum system are entanglement-equivalent by teleportation and dense coding. We provide a construction of entanglement-equivalences of channels based on the principle of covariance. Let G be a compact quantum group of Kac type, let F': Rep(G) -> Hilb be a fibre functor accessible from the canonical fibre functor F: Rep(G) -> Hilb by a unitary pseudonatural transformation, and let G' be the compact quantum group of Kac type obtained from the fibre functor F' by Tannaka duality. Then for any G-covariant channel N we construct a G'-covariant channel N' and entanglement-assisted channel coding schemes witnessing an entanglement-equivalence of N and N'. This construction induces an equivalence of categories E: Chan(G) -> Chan(G'), where Chan(G) is the category whose objects are G-C*-algebras and whose morphisms are covariant channels. We show that all tight teleportation and dense coding schemes are a special case of our construction, using covariance for the action of the quantum permutation group of the algebra M_d(\mathbb{C}). Where G is a finite abelian group, we explicitly compute the effect of the equivalence E on algebras and channels. We thereby produce concrete new examples of entanglement-equivalences of channels in the case where G=Z_2 \times Z_2 and the noncanonical fibre functor is specified by the Pauli matrices.

Dec 10, 20201 citation

A set of protocols for teleportation and dense coding schemes based on a multiparticle quantum channel, represented by the

N

-particle entangled states of the GHZ class, is introduced. Using a found representation for the GHZ states, it was shown that for dense coding schemes enhancement of the classical capacity of the channel due from entanglement is

N/N-1

. Within the context of our schemes it becomes clear that there is no one-to one correspondence between teleportation and dense coding schemes in comparison when the EPR channel is exploited. A set of schemes, for which two additional operations as entanglement and disentanglement are permitted, is considered.