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arXiv:2204.04493 · Journal of Mathematical Physics

Entanglement-invertible channels

Dominic Verdon

Published Jun 1, 2024• 0 citations
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Abstract

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)].

DOI
10.1063/5.0159504

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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)].
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.
Quantum teleportation is a quantum communication primitive that allows a long-distance quantum channel to be built using pre-shared entanglement and one-way classical communication. However, the quality of the established channel crucially depends on the quality of the pre-shared entanglement. In this work, we revisit the problem of using noisy entanglement for the task of teleportation. We first show how this problem can be rephrased as a state discrimination problem. In this picture, a quantitative duality between teleportation and dense coding emerges in which every Alice-to-Bob teleportation protocol can be repurposed as a Bob-to-Alice dense coding protocol, and the quality of each protocol can be measured by the success probability in the same state discrimination problem. One of our main results provides a complete characterization of the states that offer no advantage in one-way teleportation protocols over classical states, thereby offering a new and intriguing perspective on the long-standing open problem of identifying such states. This also yields a new proof of the known fact that bound entangled states cannot exceed the classical teleportation threshold. Moreover, our established duality between teleportation and dense coding can be used to show that the exact same states are unable to provide a non-classical advantage for dense coding as well. We also discuss the duality from a communication capacity point of view, deriving upper and lower bounds on the accessible information of a dense coding protocol in terms of the fidelity of its associated teleportation protocol. A corollary of this discussion is a simple proof of the previously established fact that bound entangled states do not provide any advantage in dense coding.
Quantum teleportation is a quantum communication primitive that allows a long-distance quantum channel to be built using pre-shared entanglement and one-way classical communication. However, the quality of the established channel crucially depends on the quality of the pre-shared entanglement. In this work, we revisit the problem of using noisy entanglement for the task of teleportation. We first show how this problem can be rephrased as a state discrimination problem. In this picture, a quantitative duality between teleportation and dense coding emerges in which every Alice-to-Bob teleportation protocol can be repurposed as a Bob-to-Alice dense coding protocol, and the quality of each protocol can be measured by the success probability in the same state discrimination problem. One of our main results provides a complete characterization of the states that offer no advantage in one-way teleportation protocols over classical states, thereby offering a new and intriguing perspective on the long-standing open problem of identifying such states. This also yields a new proof of the known fact that bound entangled states cannot exceed the classical teleportation threshold. Moreover, our established duality between teleportation and dense coding can be used to show that the exact same states are unable to provide a non-classical advantage for dense coding as well. We also discuss the duality from a communication capacity point of view, deriving upper and lower bounds on the accessible information of a dense coding protocol in terms of the fidelity of its associated teleportation protocol. A corollary of this discussion is a simple proof of the previously established fact that bound entangled states do not provide any advantage in dense coding.
The beauty of the quantum circuit lies in the fact that it can perform certain tasks which are impossible in the classical world. For example, consider teleportation and superdense coding. Teleportation is a quantum task in which an unknown quantum state is transmitted from a sender (Alice) to a spatially separated receiver (Bob) via an entangled quantum channel and with the help of some classical communications. Dense coding is another communication scheme in which if Alice and Bob share a prior entanglement then one of them can send n bits of classical information to the other by communicating m-qubits (m < n). Before we go into the technical details of these two schemes let us try to lucidly visualize the uniqueness of teleportation. Suppose Alice and Bob are spatially separated and Alice wants to send an unknown quantum state |ψ〉 = α|0〉 + β|1〉 to Bob. For some reason she is not allowed to send the state directly through a quantum channel. She cannot measure it and send the information to Bob because the moment she will try to measure the state, it will collapse either to |0〉 or to |1〉. But the state can be sent by using a simple quantum circuit (pre-shared entanglement) and some classical communications. In the process the unknown quantum state gets destroyed at Alice’s end and reappears at Bob’s end. The state never exists in the communication channel. This process is called quantum teleportation. The uniqueness of quantum teleportation will be clearer if we note that the Fax is not an example of teleportation because the original copy remains with the sender. Again a letter posted in a letter box is not an example of teleportation because the letter exists (travels) in its original form in the communication channel. In brief, there is no classical analogue of quantum teleportation. The same is true for dense coding. In this chapter we will describe these two unique quantum communication schemes and discuss their significance and applications.
Abstract We show for any d,m≥2d,m\ge 2d,m≥2 with (d,m)≠(2,2)(d,m)\neq (2,2)(d,m)=(2,2), the matrix-valued generalization of the (tensor product) quantum correlation set of ddd inputs and mmm outputs is not closed. Our argument uses a reformulation of super-dense coding and teleportation in terms of C∗C^*C∗-algebra isomorphisms. Namely, we prove that for certain actions of cyclic group Zd{{\mathbb{Z}}}_dZd​, $$\begin{equation*}M_d(C^*({{\mathbb{F}}}_{d^2}))\cong{{\mathcal{B}}}_d\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d , M_d({{\mathcal{B}}}_d)\cong C^*({{\mathbb{F}}}_{d^2})\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d,\end{equation*}$$where Bd{{\mathcal{B}}}_dBd​ is the universal unital C∗C^*C∗-algebra generated by the elements ujk, 0≤i,j≤d−1u_{jk}, \, 0 \le i, j \le d-1ujk​,0≤i,j≤d−1, satisfying the relations that [uj,k][u_{j,k}][uj,k​] is a unitary operator, and C∗(Fd2)C^*({{\mathbb{F}}}_{d^2})C∗(Fd2​) is the universal C∗C^*C∗-algebra of d2d^2d2 unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.
We investigate dense coding by imposing various locality restrictions to our decoder by employing the resource theory of asymmetry framework. In this task, the sender Alice and the receiver Bob share an entangled state. She encodes the classical information into it using a symmetric preserving encoder and sends the encoded state to Bob through a noiseless quantum channel. The decoder is limited to a measurement to satisfy a certain locality condition on the bipartite system composed of the receiving system and the preshared entanglement half. Our contributions are summarized as follows: First, we derive an achievable transmission rate for this task dependently of conditions of encoder and decoder. Surprisingly, we show that the obtained rate cannot be improved even when the decoder is relaxed to local measurements, two-way LOCCs, separable measurements, or partial transpose positive (PPT) measurements for the bipartite system. Moreover, depending on the class of allowed measurements with a locality condition, we relax the class of encoding operations to super-quantum encoders in the framework of general probability theory (GPT). That is, when our decoder is restricted to a separable measurement, theoretically, a positive operation is allowed as an encoding operation. Surprisingly, even under this type of super-quantum relaxation, the transmission rate cannot be improved. This fact highlights the universal validity of our analysis beyond quantum theory.