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arXiv:2412.16992 · APL Quantum

Generalized teleportation fidelity and singlet fraction and their relation for (in)distinguishable particles and its applications

Soumya Das, Goutam Paul, Anindya Banerji

Published May 16, 2025• 0 citations
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Abstract

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.

DOI
10.1063/5.0254301

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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.
Recently, an explicit protocol E0{\mathcal{E}}_{0}E0​ for faithfully teleporting arbitrary two-qubit states using genuine four-qubit entangled states was presented by us [Phys. Rev. Lett. 96, 060502 (2006)]. Here, we show that E0{\mathcal{E}}_{0}E0​ with an arbitrary four-qubit mixed-state resource $\ensuremath{\Xi}$ is equivalent to a generalized depolarizing bichannel with probabilities given by the maximally entangled components of the resource. These are defined in terms of our four-qubit entangled states. We define the generalized singlet fraction $\mathcal{G}[\ensuremath{\Xi}]$ and illustrate its physical significance with several examples. We argue that in order to teleport arbitrary two-qubit states with average fidelity better than is classically possible, we have to demand that $\mathcal{G}[\ensuremath{\Xi}]>1∕2$. In addition, we conjecture that when $\mathcal{G}[\ensuremath{\Xi}]<1∕4$, then no entanglement could be teleported. It is shown that to determine the usefulness of $\ensuremath{\Xi}$ for E0{\mathcal{E}}_{0}E0​, it is necessary to analyze $\mathcal{G}[\ensuremath{\Xi}]$.
Quantum teleportation provides a "disembodied" way to transfer quantum states from one object to another at a distant location, assisted by priorly shared entangled states and a classical communication channel. In addition to its fundamental interest, teleportation has been recognized as an important element in long-distance quantum communication, distributed quantum networks and measurement-based quantum computation. There have been numerous demonstrations of teleportation in different physical systems such as photons, atoms, ions, electrons, and superconducting circuits. Yet, all the previous experiments were limited to teleportation of one degree of freedom (DoF) only. However, a single quantum particle can naturally possess various DoFs -- internal and external -- and with coherent coupling among them. A fundamental open challenge is to simultaneously teleport multiple DoFs, which is necessary to fully describe a quantum particle, thereby truly teleporting it intactly. Here, we demonstrate the first teleportation of the composite quantum states of a single photon encoded in both the spin and orbital angular momentum. We develop a method to project and discriminate hyper-entangled Bell states exploiting probabilistic quantum non-demolition measurement, which can be extended to more DoFs. We verify the teleportation for both spin-orbit product states and hybrid entangled state, and achieve a teleportation fidelity ranging from 0.57 to 0.68, above the classical limit. Our work moves a step toward teleportation of more complex quantum systems, and demonstrates an enhanced capability for scalable quantum technologies.
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.
We analyze quantum teleportation for composite systems, specifically for concatenated teleporation (decomposing a large composite state into smaller states of dimension commensurate with the channel) and partial teleportation (teleporting one component of a larger quantum state). We obtain an exact expression for teleportation fidelity that depends solely on the dimension and singlet fraction for the entanglement channel and entanglement (measures by III concurrence) for the state; in fact quantum teleportation for composite systems provides an operational interpretation for III concurrence. In addition we obtain tight bounds on teleportation fidelity and prove that the average fidelity approaches the lower bound of teleportation fidelity in the high-dimension limit.
We prove a theorem on direct relation between the optimal fidelity fmax{f}_{\mathrm{max}}fmax​ of teleportation and the maximal singlet fraction Fmax{F}_{\mathrm{max}}Fmax​ 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 fmax=(Fmaxd+1)/(d+1){f}_{\mathrm{max}}{=(F}_{\mathrm{max}}d+1)/(d+1)fmax​=(Fmax​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.
The monogamy inequality in terms of the concurrence, called the Coffman-Kundu-Wootters inequality [V. Coffman et al., Phys. Rev. A 61, 052306 (2000)], and its generalization [T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006)] hold on general nnn-qubit states including mixed ones. In this Brief Report, we consider the monogamy inequalities in terms of the fully entangled fraction and the teleportation fidelity. We show that the monogamy inequalities do not hold on general mixed states, while the inequalities hold on nnn-qubit pure states.