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Annalen der Physik

Entanglement and Magic Correlations Generated by Discrete Beam‐Splitters

Lingxuan Feng, Shunlong Luo

Published Aug 15, 2025• 0 citations
Publisher

Abstract

Abstract Beam‐splitters are basic and instrumental tools not only in the study of foundational issues in quantum mechanics, but also in various applications of quantum information processing. In the continuous‐variable scenario involving quantum optical states and Gaussian quantum information, interference phenomena are widely investigated and characterized via beam‐splitters. In this work, within the framework of stabilizer quantum computation in finite quantum systems, entanglement and magic correlations generated by discrete beam‐splitters are explored. Two types of magic correlations are distinguished: mutual magic (analogous to quantum mutual information) and non‐local magic. Explicit formulas for these correlations are derived in the qubit scenario where the beam‐splitter is implemented by the controlled‐NOT gate, with the ‐type magic states playing a particularly prominent role as input states. Extending to prime‐dimensional qudit systems, it is shown that fiducial states of mutually unbiased bases also play a remarkable role. Basic properties of these magic correlations are revealed and illustrated through some typical examples.

DOI
10.1002/andp.202500288

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Abstract Beam‐splitters are basic and instrumental tools not only in the study of foundational issues in quantum mechanics, but also in various applications of quantum information processing. In the continuous‐variable scenario involving quantum optical states and Gaussian quantum information, interference phenomena are widely investigated and characterized via beam‐splitters. In this work, within the framework of stabilizer quantum computation in finite quantum systems, entanglement and magic correlations generated by discrete beam‐splitters are explored. Two types of magic correlations are distinguished: mutual magic (analogous to quantum mutual information) and non‐local magic. Explicit formulas for these correlations are derived in the qubit scenario where the beam‐splitter is implemented by the controlled‐NOT gate, with the ‐type magic states playing a particularly prominent role as input states. Extending to prime‐dimensional qudit systems, it is shown that fiducial states of mutually unbiased bases also play a remarkable role. Basic properties of these magic correlations are revealed and illustrated through some typical examples.
Magic (non-stabilizerness) is a key resource for achieving universal fault-tolerant quantum computation beyond classical computation. While previous studies have primarily focused on magic in single systems, its interactions and distribution in multipartite settings remain largely unexplored. In this work, we introduce mutual mana as a measure of magic correlations defined in close analogy with quantum mutual information. Our definition builds upon mana, which is the established quantifier of magic based on discrete Wigner function negativity. We characterize magic correlations generated by discrete beamsplitters, whose Gaussian counterparts are fundamental components in quantum optics and quantum technologies. We show that coupling a magic state with a stabilizer vacuum state via a discrete beamsplitter will induce a full conversion of local magic into mutual mana, thereby establishing a mechanism for redistributing magic resources as magic correlations. We reveal the fundamental properties of mutual mana and derive its explicit expressions for several prototypical qutrit states subject to a discrete beamsplitter. We make a comparative study of mutual mana with several established quantifiers of correlations generated by the qutrit beamsplitter, including quantum mutual information, mutual L1L^1L1-norm magic, and mutual stabilizer 2-Rényi entropy.
We investigate the private capacity of quantum channels using the recently proposed quantum convolution theory for discrete-variable quantum systems. We focus on the role of the magic resource played in this framework. Firstly, for a large class of convolutional channels, we find that the private capacity is zero if the fixed environmental state is a stabilizer state. Moreover, we show that the private capacity can be nonzero for some magic environmental states. Furthermore, we show that the private capacity of a discrete beam splitter unitary is upper-bounded by the amount of magic of the environmental state. In addition, if the environmental state exhibits a certain symmetric structure, even if it is magic, the corresponding private capacity will also vanish for a class of convolution. These results emphasize the role of magic resources in quantum communication
Abstract A beam splitter operator is a very important linear device in the field of quantum optics and quantum information. It can not only be used to prepare complete representations of quantum mechanics, entangled state representation, but it can also be used to simulate the dissipative environment of quantum systems. In this paper, by combining the transform relation of the beam splitter operator and the technique of integration within the product of the operator, we present the coherent state representation of the operator and the corresponding normal ordering form. Based on this, we consider the applications of the coherent state representation of the beam splitter operator, such as deriving some operator identities and entangled state representation preparation with continuous-discrete variables. Furthermore, we extend our investigation to two single and two-mode cascaded beam splitter operators, giving the corresponding coherent state representation and its normal ordering form. In addition, the application of a beam splitter to prepare entangled states in quantum teleportation is further investigated, and the fidelity is discussed. The above results provide good theoretical value in the fields of quantum optics and quantum information.
Entanglement is an important ingredient for quantum information processing. We discuss some sources of entanglement, namely a beam splitter and a thermal field. For the generation of entangled continuous-variable states, we consider a beam splitter and find some conditions for input fields to see entanglement in the output. While a beam splitter is a unitary device to generate an entangled state for a bipartite continuous-variable system, a thermal field is shown to mediate entanglement of two qubits.
A beam splitter is a basic linear optical element appearing in many optics experiments and is frequently used as a continuous-variable entangler transforming a pair of input modes from a separable Gaussian state into an entangled state. However, a beam splitter is a passive operation that can create entanglement from Gaussian states only under certain conditions. One such condition is that the input light is suitably squeezed. We demonstrate, experimentally, that a beam splitter can create entanglement even from modes which do not possess such a squeezing provided that they are correlated to, but not entangled with, a third mode. Specifically, we show that a beam splitter can create three-mode entanglement by acting on two modes of a three-mode fully separable Gaussian state without entangling the two modes themselves. This beam splitter property is a key mechanism behind the performance of the protocol for entanglement distribution by separable states. Moreover, the property also finds application in collaborative quantum dense coding in which decoding of transmitted information is assisted by interference with a mode of the collaborating party.
We investigate the role of magic resource in the quantum capacity of channels. We consider the quantum channel of the recently proposed discrete beam splitter with the fixed environmental state. We find that if the fixed environmental state is a stabilizer state, then the quantum capacity is zero. Moreover, we find that the quantum capacity is nonzero for some magic states, and the quantum capacity increases linearly with respect to the number of single-qudit magic states in the environment. We also bound the maximal quantum capacity of the discrete beam splitter in terms of the amount of magic resource in the environmental states. These results suggest that magic resource can increase the quantum capacity of channels; it sheds new insight into the role of stabilizer and magic states in quantum communication.