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Physical Review A

General teleportation channel, singlet fraction, and quasidistillation

Michał Horodecki, Paweł Horodecki, Ryszard Horodecki

Published Sep 1, 1999• 964 citations
Publisher

Abstract

We prove a theorem on direct relation between the optimal fidelity ${f}_{\mathrm{max}}$ of teleportation and the maximal singlet fraction ${F}_{\mathrm{max}}$ attainable by means of trace-preserving local quantum and classical communication (LQCC) action. For a given bipartite state acting on ${C}^{d}\ensuremath{\bigotimes}{C}^{d}$ we have ${f}_{\mathrm{max}}{=(F}_{\mathrm{max}}d+1)/(d+1)$. We assume completely general teleportation scheme (trace preserving LQCC action over the pair and the third particle in unknown state). The proof involves the isomorphism between quantum channels and a class of bipartite states. We also exploit the technique of $U\ensuremath{\bigotimes}{U}^{*}$ twirling states (random application of unitary transformation of the above form) and the introduced analogous twirling of channels. We illustrate the power of the theorem by showing that any bound entangled state does not provide better fidelity of teleportation than for the purely classical channel. Subsequently, we apply our tools to the problem of the so-called conclusive teleportation, then reduced to the question of optimal conclusive increasing of singlet fraction. We provide an example of state for which Alice and Bob have no chance to obtain perfect singlet by LQCC action, but still singlet fraction arbitrarily close to unity can be obtained with nonzero probability. We show that a slight modification of the state has a threshold for singlet fraction, which cannot be exceeded anymore.

DOI
10.1103/physreva.60.1888

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We prove a theorem on direct relation between the optimal fidelity fmaxf_{max}fmax​ of teleportation and the maximal singlet fraction FmaxF_{max}Fmax​ attainable by means of trace-preserving LQCC action (local quantum and classical communication). For a given bipartite state acting on Cd⊗CdC^d\otimes C^dCd⊗Cd we have fmax=Fmaxd+1d+1f_{max}= {F_{max}d+1\over d+1}fmax​=d+1Fmax​d+1​. We assume completely general teleportation scheme (trace preserving LQCC action over the pair and the third particle in unknown state). The proof involves the isomorphism between quantum channels and a class of bipartite states. We also exploit the technique of U⊗U∗U\otimes U^*U⊗U∗ twirling states (random application of unitary transformation of the above form) and the introduced analogous twirling of channels. We illustrate the power of the theorem by showing that {\it any} bound entangled state does not provide better fidelity of teleportation than for the purely classical channel. Subsequently, we apply our tools to the problem of the so-called conclusive teleportation, then reduced to the question of optimal conclusive increasing of singlet fraction. We provide an example of state for which Alice and Bob have no chance to obtain perfect singlet by LQCC action, but still singlet fraction arbitrarily close to unity can be obtained with nonzero probability. We show that a slight modification of the state has a threshold for singlet fraction which cannot be exceeded anymore.
Quantum Teleportation is a very useful scheme for transferring quantum information. Given that the quantum information is encoded in a state of a system of distinguishable particles, and given that the shared bi-partite entangled state is also that of a system of distinguishable particles, the optimal teleportation fidelity\textit{optimal teleportation fidelity}optimal teleportation fidelity of the shared state is known to be (Fmaxd+1)/(d+1)(F_{max}d+1)/(d+1)(Fmax​d+1)/(d+1) with FmaxF_{max}Fmax​ being the `maximal singlet fraction' of the shared state. However, Parity Superselection Rule (PSSR) in Fermionic Quantum Theory (FQT) puts constraint on the allowed set of physical states and operations, and thereby, leads to a different notion of Quantum entanglement - locally accessible and locally inaccessible (topological correlation). In the present work, we derive an expression for the optimal teleportation fidelity\textit{optimal teleportation fidelity}optimal teleportation fidelity of locally accessible entanglement preservation, given that the quantum information to be teleported is encoded in fermionic modes of dimension 2N×2N2^N \times 2^N2N×2N using 2N×2N2^N \times 2^N2N×2N-dim shared fermionic resource between the sender and receiver. To get the optimal teleportation fidelity in FQT, we introduce PSSR restricted twirling operations and establish fermionic state-channel isomorphism. Remarkably, we notice that the structure of the canonical form of twirl invariant fermionic shared state differs from that of the isotropic state\textit{isotropic state}isotropic state -- the corresponding canonical invariant form for teleportation in Standard Quantum Theory (SQT). In this context, we also introduce restricted Clifford twirling operation that constitute the Unitary 2-design in case of FQT for experimentally validating such optimal average fidelity. Finally, we discuss the preservation of locally inaccessible entanglement for a class of fermionic teleportation channel.
Maximally entangled states---a resource for quantum information processing---can only be shared through noiseless quantum channels, whereas in practice channels are noisy. Here we ask: Given a noisy quantum channel, what is the maximum attainable purity (measured by singlet fraction) of shared entanglement for single channel use and local trace preserving operations? We find an exact formula of the maximum singlet fraction attainable for a qubit channel and give an explicit protocol to achieve the optimal value. The protocol distinguishes between unital and nonunital channels and requires no local postprocessing. In particular, the optimal singlet fraction is achieved by transmitting part of an appropriate pure entangled state, which is maximally entangled if and only if the channel is unital. A linear function of the optimal singlet fraction is also shown to be an upper bound on the distillable entanglement of the mixed state dual to the channel.
We consider the problem of establishing entangled states of optimal singlet fraction and negativity between two remote parties for every use of a noisy quantum channel and trace-preserving local operations and classical communication (LOCC) under the assumption that the parties do not share prior correlations. We show that for a family of quantum channels in every finite dimension $d\ensuremath{\ge}3$, one-shot optimal singlet fraction and entanglement negativity are attained only with appropriate nonmaximally entangled states. A consequence of our results is that the ordering of entangled states in all finite dimensions may not be preserved under trace-preserving LOCC.
Quantum teleportation efficiently transfers quantum information between distant locations by utilizing a pre-established composite system. Assessing the effectiveness of teleportation hinges on its fidelity, representing the similarity between input and output states. This fidelity, in turn, relies on a singlet fraction, quantifying the resemblance of the composite system to maximally entangled states. The relation between teleportation fidelity and singlet fraction given by Horodecki et al. [Phys. Rev. A 60, 1888 (1999)] does not hold for distinguishable particles with multiple degrees of freedom or indistinguishable particles with single or multiple degrees of freedom. In this paper, we propose generalized expressions for teleportation fidelity and singlet fraction and derive their relations, applicable to both distinguishable and indistinguishable particles with single or multiple degrees of freedom. We derive an upper bound for the generalized singlet fraction for distinguishable particles using the monogamy of singlet fraction by Kay et al. [Phys. Rev. Lett. 103, 050501 (2009)]. We also show how our relation helps characterize different types of composite states in terms of their distinguishability, separability, the presence of maximally entangled structure, and the number of degrees of freedom. We complement our theory with two practical illustrations. First, we demonstrate two counter-intuitive values of generalized singlet fraction using our optical circuit and the circuit of Li et al. [Phys. Rev. Lett. 120, 050404 (2018)]. Finally, we show that using an additional degree of freedom as an ancilla instead of a particle can be advantageous in quantum cryptographic protocols.
We study the Braunstein-Kimble protocol for the continuous variable teleportation of a coherent state. We determine lower and upper bounds for the optimal fidelity of teleportation, maximized over all local Gaussian operations for a given entanglement of the two-mode Gaussian state shared by the sender (Alice) and the receiver (Bob). We also determine the optimal local transformations at Alice and Bob sites and the corresponding maximum fidelity when one restricts to local trace-preserving Gaussian completely positive maps.