Quantum teleportation in higher dimension and entanglement distribution via quantum switches

Abstract

High‐dimensional quantum states, or ‘qudits’, provide significant advantages over traditional qubits in quantum communication, such as increased information capacity, enhanced noise resilience, and reduced information loss. Despite these benefits, their implementation has been constrained by challenges in generation, transmission, and detection. This paper presents a novel theoretical framework for transmitting quantum information using qudit entanglement distribution over a superposition of causal orders in two quantum channels. Using this model, a quantum switch operation for 2‐qudit systems is introduced, which facilitates enhanced fidelity of entanglement distribution and quantum teleportation. The results demonstrate that the use of qudits in entanglement distribution achieves a fidelity improvement from 0.5 (for qubit‐based systems) to 0.94 for 20‐dimensional qudits, even under noisy channel conditions. This enhancement is achieved by exploiting the increased Hilbert space of high‐dimensional states and the inherent noise‐resilience properties of quantum switches operating in superpositions of causal orders. The findings underscore the potential of qudit‐based quantum systems in achieving robust and high‐fidelity communication in environments where traditional qubit‐based systems face limitations.

INTRODUCTION

Communication networks are gradually emerging as critical infrastructure for all walks of life. Presently, the choicest mode of communication for any network is combination of radio frequency (RF) wireless links for fronthaul and optical fibre links for backhaul connection. However, accommodating increasing number of users and applications in the RF band will be impossible in terms of carrier frequency and bandwidth. On the other hand, optical fibre links impose high latency over long-distance connections. Quantum communication networks, in future, will be the most promising alternative that can solve said problems with wireless and fibre links [1].

Quantum communication between any two points is viable through a property called ‘shared entanglement’. Shared entanglement can be accomplished by distributing an Einstein-Podolsky-Rosen (EPR) pair between two classically or quantumly operated nodes communicating through a quantum channel [2]. Entangled qubits are distributed between two nodes with high fidelity. Since quantum states cannot be copied or amplified, success of quantum communication over long distances strongly depends on the use of one or more quantum repeaters that forward information by entanglement distribution [3].

Quantum particles experiencing two quantum processes in superposition of alternative orders (i.e. which channel is traversed before the other is not imperative) are capable of conveying more information than when travelling through each process of arbitrary order separately and consecutively [4]. For example, in normal condition, no quantum information can be transmitted over either of the pair of dephasing channels [5]. However, a qubit travelling through the superposition of two dephasing channels of alternative orders can reach the receiver error-free with 25% probability. The success of an error-free transmission can only be validated by the outcome of a measurement on the order of the qubit [6], where the order depends on the state of the quantum degree of freedom. Similarly, for partially dephasing channels, quantum capacity of two superposed channels is larger than that achievable over individual channels.

The travelling quantum particles can arrive intact at the receiver with a finite probability of success. This advantage can be capitalised on through the implementation of a quantum ‘switch’ in a quantum teleportation chain for entanglement distribution, a phenomenon by which a pair of maximally entangled qubits must be generated and shared between the transmitter and the receiver [7]. In a quantum switch, the transmitted quantum particle experiences two quantum channels $$\mathcal{A}$$ and $$\mathcal{B}$$ in a superposition of different orders, $$\mathcal{A}\to \mathcal{B}$$ and $$\mathcal{B}\to \mathcal{A}$$. The order depends on the state of a quantum degree of freedom, which can be two-dimensional for qubits.

This phenomenon can be leveraged to create a perfect noiseless communication channel through quantum superposition of two noisy channels of alternative orders. As a result, fidelity of a teleported qubit via an ordinary channel can be enhanced by using a maximally entangled state as a quantum resource in a superposition of alternative orders [8]. Moreover, quantum entanglement is easily affected by noise resulting in degeneration of the transmitted information, an effect that can be significantly ameliorated by the application of quantum switch enabling superposition of noisy quantum channels of different causal orders [9].

Qubits can only exploit two photonic degrees of freedom and are therefore limited in channel capacity, data throughput and noise resilience, thereby lowering the possibility of breaking the Shannon's limit on achievable capacity. This has spurred a recent interest in teleporting qudits as units for information exchange with arbitrary dimensions exploiting $$d$$-degrees of freedom [10]. Teleportation of photonic qudits increases the quantum information sent per carrier photon. Teleportation of qudits can be achieved either by preparing $$d$$ additional photons in a highly entangled state [11] or by transcribing qudit encoded on a single photon to $$d$$ qubits carried by light modes which propagate along different optical paths [12]. Moreover, qudits can carry information over a plethora of media ranging from free space optical links to multi-core and multi-mode optical fibres, and peer-to-peer underwater acoustic communications [13].

Though qudits have garnered significant interest in quantum communication due to their ability to encode larger amounts of information and their inherent robustness to noise. However, the practical realisation of such states presents several challenges across their generation, propagation, and measurement. The generation of qudits necessitates precise manipulation of photonic degrees of freedom, such as orbital angular momentum (OAM), time-bin, and frequency encoding [14]. These processes require advanced experimental setups and are constrained by the current technological limits on the number of dimensions that can be coherently controlled, which hampers the scalability of high-dimensional systems.

The transmission of qudits through communication channels, including free-space links, optical fibres, and underwater paths, introduces significant complexities. Free-space transmission is susceptible to beam divergence, while fibre-based systems face challenges such as cross-talk in multicore fibres and the need for low-loss single-mode fibres. Underwater channels are subject to severe attenuation, scattering, and absorption of photons, particularly in optical wavelengths. These effects result in significant signal loss over relatively short distances, making the reliable transmission of high-dimensional quantum states particularly difficult. Existing optical fibre infrastructures, optimised for single-mode operations, are not naturally suited for the multidimensional encoding of qudits. Moreover, the detection of high-dimensional quantum states requires specialised equipment capable of distinguishing among a larger set of states. This limitation often reduces measurement efficiency, particularly in experimental setups constrained to project onto a single state at a time, impeding the realisation of complete Bell-state measurements. However, qudits exhibit greater resilience to quantum cloning attacks, and technological and noise-related constraints as compared to qubits. Therefore, advantages offered by qudits outweigh the challenges offered by their practical implementation in terms of robust quantum links, and error correction schemes [15].

Recent advancements in quantum communication have significantly explored the advantages and applications of high-dimensional quantum systems and quantum switches. The study in ref. [16] highlights the role of high-dimensional quantum states in enhancing the robustness of quantum key distribution protocols, showcasing their resilience to environmental noise and increased channel capacity. Similarly, Sephton et al. [17] provide evidence of high-dimensional entanglement enabling more efficient and secure quantum network operations, emphasising its potential for next-generation quantum Internet frameworks. Furthermore, the detailed analysis in ref. [18] elaborates on the implementation of quantum switches to achieve noise-resilient communication over noisy quantum channels, underscoring their transformative impact on distributed quantum networks. These findings collectively underline the potential of integrating quantum switches and high-dimensional quantum states for scalable and noise-tolerant quantum communication systems.

The concept of a quantum switch, enabling the superposition of causal orders, has been shown to improve the transmission of quantum information over noisy channels. Works such as refs. [19, 20] have demonstrated that the use of quantum switches can extend quantum channel capacity beyond the limitations imposed by classical trajectories. Furthermore, studies like refs. [21, 22] provide a theoretical foundation for the noise-resilient properties of quantum switches. These works have established that quantum superposition of causal orders can transform non-communicative channels into communicative ones, thereby unlocking novel applications in quantum communication protocols. Moreover, practical implications of quantum switches in high-dimensional systems have been explored in ref. [23]. These studies provide experimental validation of theoretical models, showcasing fidelity improvements and enhanced error correction capabilities in real-world settings. These works have established that quantum superposition of causal orders can transform non-communicative channels into communicative ones, thereby unlocking novel applications in quantum communication protocols. Recent studies like ref. [24] continue to push the boundaries of quantum switch applications, offering insights into optimising multi-qudit teleportation systems for robust quantum network architectures. Complementary to the preliminary efforts of the quantum physics [25] and quantum communication engineering community [26] in exploiting quantum switch for entanglement distribution of 2-qubits system, we investigate the possibility of employing quantum switch for entanglement distribution of qudits teleportation system. To achieve this, we formulate the theoretical system model for quantum switch for entanglement distribution of 2-qudit systems. We also derive closed-form expressions for connecting the teleported qudit received by Bob to the degradation suffered by the entangled pair during the distribution process.

The results presented in this work have significant implications for distributed quantum computing and quantum Internet architectures. In distributed quantum computing settings, as described in ref. [27], the ability to distribute high-fidelity entanglement across multiple nodes is critical for implementing scalable gate-model quantum operations. The demonstrated fidelity improvements and noise-resilience of qudits in this study provide a pathway for robust inter-node communication, enabling more efficient quantum circuit executions across geographically distributed quantum processors. Similarly, in the context of the quantum Internet, the principles outlined in [28, 29] highlight the importance of enhancing channel capacities and maintaining entanglement over noisy channels. The noise-scaled stability bounds and entanglement rate maximisation strategies discussed in ref. [30] align closely with the findings of this study, further underscoring the potential of quantum switches and high-dimensional quantum states to enable efficient, scalable, and resilient quantum communication networks. The rest of the paper is organised as below. Section 2 introduces the concept and mathematical expressions for the functionality of a quantum switch for qudits. Section 3 develops the basic framework for quantum teleportation of qudits. Section 4 follows up with the introduction to different circuit configurations possible for realising a qudit switch and the theoretical framework for entanglement distribution of qudits. Finally, we develop the mathematical formulation for teleportation of qudits through the implementation of entanglement distribution of a 2-qudit system by means of a qudit-based quantum switch in Section 5. The concluding remarks are provided in Section 6.

QUANTUM SWITCH—CONCEPT FOR QUDITS

Let us consider an arbitrary qudit $$\vert \psi \rangle ={\left\{\vert k\rangle \right\}}_{k=0}^{d-1}$$ of dimension $$d$$ and unknown state travelling through two noisy quantum channels $$\mathcal{A}$$ and $$\mathcal{B}$$, and let us assume that $$\mathcal{A}$$ is the cyclic shift channel and $$\mathcal{B}$$ is the cyclic clock channel. The cyclic shift channel $$\mathcal{A}$$ performs a cyclic permutation of the basis states and maps the set, $$\left\{\vert 0\rangle ,\vert 1\rangle ,\text{\ldots },\vert d-1\rangle \right\}$$ to $$\left\{\vert 1\rangle ,\vert 2\rangle ,\text{\ldots },\vert 0\rangle \right\}$$ and vice-versa with a probability $$p$$, leaving the qudit unaltered with probability $$1-p$$, such that, 1 $$\mathcal{A}(\vert \psi \rangle )=(1-p)\vert \psi \rangle +p{X}_{d}\vert \psi \rangle $$ where $${X}_{d}$$ denotes the generalised Pauli-X gate given in Table 1. Here Equation (1) is the qudit generalisation of the bit-flip channel that introduces a flip of the state of a qubit from $$\vert 0\rangle $$ to $$\vert 1\rangle $$ with a certain probability.

TABLE 1 Higher dimensional quantum gates.

Gate	Symbol	Equation	Description
Generalised Pauli-$$X$$	$${X}_{d}$$	$${X}_{d}^{q}\vert k\rangle =\vert k\oplus q\rangle $$	$$q$$ is the shift change in standard basis
Generalised Pauli-$$Z$$	$${Z}_{d}$$	$${Z}_{d}^{l}\vert k\rangle ={\omega }^{lk}\vert k\rangle $$	$$l$$ is the phase change in standard basis
Generalised Hadamard	$${H}_{d}$$	$${H}_{d}\vert k\rangle =\frac{1}{\sqrt{d}}{\sum }_{m=0}^{d-1}{x}_{m}$$, $${x}_{m}={(-1)}^{k\cdot m}\vert m\rangle $$	$$k\cdot m={\oplus }_{\rho =0}^{n-1}{k}_{\rho }{m}_{\rho }$$ is the bitwise inner product
CNOT right-shift	$${R}_{{c}_{d}}$$	$${{R}_{c}}_{d}(\vert k\rangle \otimes \vert x\rangle )=\vert k\rangle \otimes \vert x\oplus k\rangle $$	$$\oplus $$ denotes modulo-$$d$$ addition and $$\vert k\rangle \otimes \vert x\rangle $$ is the standard basis state of $${\mathcal{H}}_{\mathcal{A}}\otimes {\mathcal{H}}_{\mathcal{B}}$$
CNOT left-shift	$${L}_{{c}_{d}}$$	$${{L}_{c}}_{d}(\vert k\rangle \otimes \vert x\rangle )=\vert k\rangle \otimes \vert x\ominus k\rangle $$	$$\ominus $$ denotes modulo-$$d$$ subtraction
Generalised XOR (GXOR)	$${\text{GXOR}}_{d}$$	$${\text{GXOR}}_{d}\vert k,l\rangle =\vert k,k\ominus l\rangle $$	GXOR is the generalised quantum XOR-gate

The cyclic clock channel $$\mathcal{B}$$ introduces relative phase shift of $$\omega ={e}^{2\pi i/d}$$ between complex set of amplitudes with probability $$q$$, leaving the qudit unaltered with probability $$1-q$$, such that, 2 $$\mathcal{B}(\vert \psi \rangle )=(1-q)\vert \psi \rangle +q{Z}_{d}\vert \psi \rangle $$ where $${Z}_{d}$$ denotes the generalised Pauli-Z gate given in Table 1. Here, Equation (2) is the qudit generalisation of the phase-flip channel that introduces a relative phase-shift between complex amplitudes. The complex set of amplitudes are given by $${\alpha }_{k}$$ such that, $$\vert \psi \rangle ={\sum }_{k=0}^{d-1}{\alpha }_{k}\vert k\rangle $$, where $${\alpha }_{k}\in {\mathbb{C}}^{d},{\sum }_{k=0}^{d-1}\vert {\alpha }_{k}{\vert }^{2}=1$$, $${\mathbb{C}}^{d}$$ is the computational basis of $$d$$-dimensional Hilbert space, $${\mathcal{H}}^{d}$$. A $$n$$-qudit state, in the tensor product Hilbert space is given by, $${\mathcal{H}}^{\otimes n}={({\mathbb{C}}^{d})}^{\otimes n}$$.

The standard basis of $${\mathcal{H}}^{\otimes n}$$ is the orthonormal basis given by the $${d}^{n}$$ classical $$n$$-qudits, such that 3 $$\vert {k}_{1}\text{\ldots }{k}_{n}\rangle =\vert {k}_{1}\rangle \otimes \text{\ldots }\vert {k}_{n}\rangle $$ where $$0\le {k}_{i}\le d-1$$ (for $$i=1,\text{\ldots },n$$). Since a qudit itself is an arbitrary superposition of orthonormal basis states $$\vert k\rangle $$ for $$d$$-dimensional quantum system, quantum switch for qudits will operate in a slightly different way with respect to quantum switch for qubits. For example, quantum circuit architectures completely described by instances of the CNOT gate cannot implement a transposition of a pair of qudits for dimension $$d > 2$$. Therefore, in order to design quantum switch for qudits, we need to consider the problem of generalising the design beyond the qubit settings.

Next, we introduce the concept of quantum switch for qudits where the traversing particles experience superposition of different orders, depending on the $$d$$ degrees of freedom. Whenever the control qudit is initialised to one of the basis states $$\vert \xi \rangle $$, such that $$\vert \xi \rangle $$ is a prime $$\vert \xi \rangle \ge 2$$, the quantum switch enables the message $$m$$ to experience the alternative classical trajectory $$\mathcal{B}\to \mathcal{A}$$ representing channel $$\mathcal{B}$$ being traversed before channel $$\mathcal{A}$$. Whenever the control qudit is initialised to a non-prime basis state, the quantum switch enables the message $$m$$ to experience the classical trajectory $$\mathcal{A}\to \mathcal{B}$$ representing channel $$\mathcal{A}$$ being traversed before channel $$\mathcal{B}$$. Whenever the control qudit is initialised to a superposition of the basis states, such as $$\vert \varphi \rangle =\frac{1}{\sqrt{d}}{\sum }_{j=0}^{d-1}{e}^{2\pi i\varphi j/d}\vert j\rangle $$, where $$\varphi $$ is an integer from the set $$\left\{0,\text{\ldots },d-1\right\}$$ and $$\vert \varphi \rangle $$ is an eigenstate of the cyclic shift operator $${X}_{d}$$. Now, since a quantum superposition of two alternative orders of noisy channels, specifically $$\mathcal{A}\to \mathcal{B}$$ and $$\mathcal{B}\to \mathcal{A}$$, can behave as a perfect quantum communication channel [4], we will use it to conceptualise quantum teleportation of qudits through entanglement generation and distribution via the application of quantum switch for qudits.

QUANTUM TELEPORTATION USING QUDITS

Our model of qudit teleportation protocol consists of two processes: Alice and Bob. The sender Alice possesses a qudit of unknown state 4 $$\vert {\psi \rangle }_{\alpha }=\sum\limits _{k=0}^{d-1}{\alpha }_{k}\vert {k\rangle }_{d}$$ that is to be teleported to Bob. This qudit is teleported using maximally entangled EPR pair of $$\vert {{\Phi }}^{+}\rangle $$ generated by entangling qudits $$\vert {{\Phi }\rangle }_{1}$$ and $$\vert {{\Phi }\rangle }_{2}$$ such that, $$\vert {{\Phi }\rangle }_{12}=\frac{1}{\sqrt{d}}{\sum }_{k=0}^{d-1}\vert {k\rangle }_{1}\vert {k\rangle }_{2}$$, through local quantum operations and classical communications [31] as shown in Figure 1.

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Firstly, CNOT Left shift (defined in Table 1) operation is applied to qudits $$\vert {\psi \rangle }_{\alpha }$$ and $$\vert {\Phi }+\rangle $$ followed by Hadamard operation (also defined in Table 1) applied to $$\vert {\psi \rangle }_{\alpha }$$. Finally, the qudits $$\vert {\psi \rangle }_{\alpha }$$ and $$\vert {{\Phi }}^{+}\rangle $$ are measured resulting in classical values ranging between 0 and $$d-1$$. Using the classical values Bob can perform necessary unitary operations to recover the original state $$\vert {\psi \rangle }_{\alpha }$$. To begin with, Alice will make a joint von Neumann measurement of the qudit $$\vert {\psi \rangle }_{\alpha }$$ and the qudit $$\vert {{\Phi }\rangle }_{1}$$. To make this measurement, Alice will use the Bell basis of the qudits, 5 $$\vert {{\Psi }}_{yz}\rangle =\frac{1}{\sqrt{d}}\sum\limits _{k=0}^{d-1}{e}^{2\pi iyk/d}\vert k\oplus z\rangle \vert k\rangle $$ where $$k\oplus z$$ means sum of $$k$$ and $$z$$ modulo $$d$$, $$y,z=0,1,\text{\ldots },d-1$$, $$y$$ and $$z$$ can take integer values between 0 and $$d-1$$ and form the Bell basis for the qudits such that $$\vert {{\Psi }}_{yz}\rangle $$ form a set of basis vectors for a two-qudit systems. $$\vert {{\Psi }}_{yz}\rangle $$ can be inverted to obtain, 6 $$\vert st\rangle =\frac{1}{\sqrt{d}}\sum\limits _{y,w=0}^{d-1}{e}^{-2\pi iyk/d}{\delta }_{t,s\oplus w}\vert {{\Phi }}^{yw}\rangle .$$ The combined state of the system $$\vert {\psi \rangle }_{\alpha }\vert {{\Phi }\rangle }_{12}$$ in terms of the above Bell basis vectors for the system $$\vert {\psi \rangle }_{\alpha }\vert {{\Phi }\rangle }_{1}$$ is as follows: 7 $$\vert {\psi \rangle }_{\alpha }\vert {{\Phi }\rangle }_{12}=\frac{1}{d}\sum\limits _{y,v=0}^{d-1}\vert {{{\Psi }}_{yv}\rangle }_{\alpha 1}{U}_{yv}^{{\dagger}}\vert {\chi \rangle }_{2}$$ where these unitary operators $${U}_{vw}$$ are given by the following equation: 8 $${U}_{vw}=\sum\limits _{k=0}^{d-1}{e}^{2\pi ivk/d}\vert k\rangle \langle k\oplus w\vert $$ and obey the following orthogonality conditions, $${U}^{{\dagger}}$$ is the adjoint of $$U$$ and $${U}^{{\dagger}}$$ is the conjugate transpose of $$U$$ in basis states, $$\text{Tr}\left({U}_{vw}^{{\dagger}}{U}_{yz}\right)=d\hspace*{.5em}{\delta }_{vy}{\delta }_{wz}$$, where $$\text{Tr}$$ denotes trace of a square matrix and $${\delta }_{vy}$$ and $${\delta }_{wz}$$ are Kronecker delta functions. We introduced the indices $$v$$ and $$w$$, such that they can take any integer between 0 and $$d-1$$. We introduced these indices to differentiate between the unitary operators used by Alice $$\left({U}_{yv}\right)$$ and Bob $$\left({U}_{vw}\right)$$. Alice uses the unitary operator to control the qudit state for teleportation and Bob uses a different set of operators to convert its qudit state back to the input state.

In summary, Alice makes the von Neumann measurement in the Bell basis to obtain one of the possible $${d}^{2}$$ results. She conveys the results of her measurement to Bob by sending $$2{\log }_{2}d$$ classical bits of information. The measurement outcomes must be communicated to Bob through classical communication channels, as this information is crucial for Bob to apply the appropriate unitary operations and recover the original state $$\vert {\psi \rangle }_{\alpha }$$. This dependence on classical communication underscores its indispensable role in completing the quantum teleportation process. Without the transfer of classical measurement results, Bob would be unable to reconstruct the teleported quantum state, thereby halting the teleportation. After receiving this information Bob uses appropriate unitary operator $${U}_{vw}$$ on his qudit to convert its state to that of the input state, thereby completing the standard teleportation of a qudit $$\vert {\psi \rangle }_{\alpha }$$ of arbitrary state over quantum channels. Though quantum teleportation with the aid of quantum entanglement looks promising, it is a very fragile resource and is easily degraded by noise [32], resulting in loss of teleported information. However, the situation can be ameliorated by exploiting quantum superposition of different causal orders realised through a quantum switch.

ENTANGLEMENT DISTRIBUTION OF QUDITS

We aim to engineer entanglement distribution of qudits and thereby develop the theoretical framework for entanglement distribution of qudits via quantum switch from a communications engineer's point of view, and different circuit realisations to achieve this are given in Figure 2.

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The operator $$U$$ routes the entanglement carrier $$\vert {\Phi }+\rangle 2$$ through either channel $$\mathcal{A}$$ or $$\mathcal{B}$$ depending on the state of $$\vert {\varphi }_{c}\rangle $$. The combination of $${X}_{d}$$ and GXOR gates performs the function of routing the entanglement carrier through the other portion of the circuit through the trajectories $$\mathcal{A}\to \mathcal{B}$$ and $$\mathcal{B}\to \mathcal{A}$$ respectively (refer to Figure 2-Configuration 1). The $${U}^{{\dagger}}$$-gate is responsible for recombining the $$\mathcal{A}\to \mathcal{B}$$ trajectories and generating the final set of EPR pairs to be distributed between the transmitter and the receiver. It is worth-mentioning here that the GXOR is the generalised quantum XOR-gate (refer to Table 1) working according to the following: 9 $${\text{GXOR}}_{d}\vert k,l\rangle =\vert k,k\ominus l\rangle .$$ A quantum switch for qudits can be implemented through the generalised SWAP gate operation. The generalised SWAP gate is not a single basic quantum gate for qudits; it has to be engineered using combinations of basic qudit gates. The main circuit idea for qudit SWAP, in this paper, is taken from ref. [33] where three GXOR and two $${X}_{d}$$ gates are implemented for interchanging control and target qudits between two traversing quantum channels.

Generalised SWAP gate can also be realised through several other combinations of basic higher dimensional quantum gates. A few common combinations involve CNOT Right-Shift $$\left({R}_{{c}_{d}}\right)$$, CNOT Left-Shift $$\left({L}_{{c}_{d}}\right)$$ and generalised Pauli-X $$\left({X}_{d}\right)$$ gates (refer to Table 1). Using these combinations, we extend our design for quantum switch in Figure 2-Configuration 1 to different implementational ways depicted in Figure 2-Configuration 2, 2-Configuration 3 and 2-Configuration 4. The SWAP Gate for qudits can be implemented in many different ways depending on the requirement. The design can involve from at least 4 gates to 6 gates or more. The qudit GXOR gate can be combined with qudit Pauli $$X$$-gates in a cyclic fashion to generate the SWAP operation [34–36]. On the other hand, the qudit CNOT gates can also be combined with Pauli-$$X$$ gates in a cyclic fashion without the involvement of any GXOR gates. In that case, they are extensions of SWAP operation for qubits to qudits. This is because a combination of a single CNOT gate and a single Pauli X-gate can only be used to implement the SWAP operation for qubits, not for qudits. It is worth-mentioning here, that three CNOT gates can be concatenated to implement two-qubit SWAP operation, where the control and the target qubits are exchanged in the second gate [37]. However, the choice of the design of the SWAP gate will have an impact on the overall operation of the quantum switch. By comparing the quantum fidelity and channel capacity achievable through qudit teleportation over each of the proposed design for quantum SWAP gates (refer to Figure 2), we can choose the best possible combination for the design of a quantum switch. We leave this interesting design detail for our future work.

In this paper, we employ a permutational definition of the SWAP gate for qudits to facilitate the controlled exchange of quantum states between two channels in a superposition of causal orders. This simplified construction combines generalised qudit gates, such as $${X}_{d}$$, $${Z}_{d}$$, and, and GXOR gates, to form a modular and experimentally realisable framework. While this definition aligns with the operational requirements of the proposed protocol, it does not fully capture the generality of the SWAP gate as established in the literature. The general definition of the SWAP gate for qudits, as proposed in ref. [38], allows for a comprehensive and unitary exchange operator across all qudit dimensions, maintaining the full symmetry and universality of quantum state evolution. Although adopting this definition would enhance the rigour and general applicability of the protocol, it introduces additional circuit complexity, including higher gate counts and tighter control over qudit interactions. These trade-offs may impact the fidelity and noise resilience of the protocol in practical implementations. Our current approach simplifies the implementation to focus on demonstrating the feasibility of qudit teleportation and entanglement distribution using quantum switches. However, future work will incorporate the general SWAP gate definition to address the limitations noted and extend the applicability of the proposed protocol. To analytically model a quantum switch, we extend the concept of switch for one-qubit system to that for a one-qudit system $$\mathcal{P}\left(\mathcal{A},\mathcal{B},{\rho }_{c}\right)(\rho )$$ as the output function of the two channels $$\mathcal{A}$$ and $$\mathcal{B}$$ along with the control state $${\rho }_{c}=\vert {\varphi }_{c}\rangle \langle {\varphi }_{c}\vert $$ of the control qudit, 10 $$\mathcal{P}\left(\mathcal{A},\mathcal{B},{\rho }_{c}\right)(\rho )=\sum\limits _{s,t}{W}_{st}\left(\rho \otimes {\rho }_{c}\right){W}_{st}^{{\dagger}}$$ where $$\left\{{W}_{st}\right\}$$ denotes the set of Kraus operators associated with the superposed channel trajectories given by the following equation: 11 $${W}_{st}={\mathcal{A}}_{s}{\mathcal{B}}_{t}\otimes \vert k\rangle \langle k\vert $$ with $$\left\{{\mathcal{A}}_{s}\right\}$$ and $$\left\{{\mathcal{B}}_{t}\right\}$$ denoting the Kraus operators associated with the channels $$\mathcal{A}$$ and $$\mathcal{B}$$ respectively. If the control quantum state is a qubit then, 12 $${W}_{st}={\mathcal{A}}_{s}{\mathcal{B}}_{t}\otimes \vert 0\rangle \langle 0{\vert }_{c}+{\mathcal{B}}_{t}{\mathcal{A}}_{s}\otimes \vert 1\rangle \langle 1{\vert }_{c}.$$ If the control quantum state is a qudit then $${\rho }_{c}=\vert {\psi }_{c}\rangle \langle {\psi }_{c}\vert $$, and, 13 $${W}_{st}=\sum\limits _{k,{k}^{\prime }=0}^{d-1}\left[{\mathcal{A}}_{s}{\mathcal{B}}_{t}\otimes \vert k\rangle \langle k\vert +{\mathcal{B}}_{t}{\mathcal{A}}_{s}\otimes \vert {k}^{\prime }\rangle \langle {k}^{\prime }\vert \right].$$ The Kraus operators $${W}_{st}$$ and $${W}_{st}^{+}$$ satisfy the completeness property, 14 $$\sum\limits _{\left\{s,t\right\}{\vert }_{s,t=0}^{d-1}}{W}_{st}{W}_{st}^{{\dagger}}={I}_{T}\otimes {I}_{c}.$$ where $${I}_{T}$$ and $${I}_{c}$$ are the identity operators in the target and control systems respectively. This check of completeness suggests how the indices $$s$$ and $$t$$ allow for systematic reordering of the sums by isolating or regrouping. It is worth-mentioning here that the entanglement-pair member $${\vert {\Phi }+\rangle }_{1}$$ is already at Alice's side and therefore does not need to go through the communication channel, while the second qudit $${\vert {\Phi }+\rangle }_{2}$$ is distributed to Bob. Therefore, the first qudit basically travels virtually through an ideal channel represented by identity matrix $${I}_{k\times k}$$.

Next we extend quantum switch for one qudit system to the case of a two-qudit system, represented by the density matrix $${\rho }_{e}$$ that is a $${d}^{2}\times {d}^{2}$$ matrix taking into account the cyclic shift and the cyclic clock channels. In order to formulate $${\rho }_{e}$$, we start by defining the Hermitian generators in the orthonormal basis $$\vert j\rangle $$ which takes on the values $$j=1,\text{\ldots },d$$, given by the following: 15 $\begin{align*}\hfill {{\Gamma }}_{j}& \equiv \frac{1}{\sqrt{j(j-1)}}\left(\sum\limits _{u=1}^{j-1}\vert u\rangle \langle u\vert -(j-1)\vert j\rangle \langle j\vert \right)\hfill \\ \hfill {{\Gamma }}_{ju}^{(+)}& \equiv \frac{1}{\sqrt{2}}\left(\vert j\rangle \langle u\vert +\vert u\rangle \langle j\vert \right)\hfill \\ \hfill {{\Gamma }}_{ju}^{(-)}& \equiv \frac{-i}{\sqrt{2}}\left(\vert j\rangle \langle u\vert -\vert u\rangle \langle j\vert \right)\hfill \end{align*}$ for $$1\le j< u\le d$$. It is noteworthy that the generators are unitary matrices. Using orthonormality relation, we can invert the generators to obtain the following: 16 $\begin{align*}\hfill \vert j\rangle \langle j\vert & =\frac{I}{d}+\frac{1}{\sqrt{j(j-1)}}\left(-(j-1){{\Gamma }}_{j}+\sum\limits _{u=j+1}^{d}{{\Gamma }}_{u}\right)\hfill \\ \hfill \vert j\rangle \langle u\vert & =\frac{1}{\sqrt{2}}\left({{\Gamma }}_{ju}^{(+)}+i{{\Gamma }}_{ju}^{(-)}\right)\hfill \\ \hfill \vert u\rangle \langle j\vert & =\frac{1}{\sqrt{2}}\left({{\Gamma }}_{ju}^{(+)}-i{{\Gamma }}_{ju}^{(-)}\right)\hfill \end{align*}$

We consider a class of two qudit states $${\vert {\Phi }\rangle }_{1}$$ and $${\vert {\Phi }\rangle }_{2}$$ with a maximally entangled state, which can be chosen to be $$\vert {{\Phi }}^{+}\rangle =\frac{1}{\sqrt{d}}{\sum }_{j=1}^{d}\vert j\rangle \otimes \vert j\rangle $$. By assuming without any loss of generality $${\rho }_{e}$$ being the $${d}^{2}\times {d}^{2}$$ density matrix associated with the EPR pair $$\vert {{\Phi }}^{+}\rangle $$, we can calculate it from the inverted generators to obtain the following: 17 $\begin{align*}\hfill & {\rho }_{e}=\vert {{\Phi }}^{+}\rangle \langle {{\Phi }}^{+}\vert =\frac{1}{{d}^{2}}(I\otimes I)\hfill \\ \hfill & \quad +\frac{1}{d}\sum\limits _{j}\left[{{\Gamma }}_{j}\otimes {{\Gamma }}_{j}+{{\Gamma }}_{ju}^{(+)}\otimes {{\Gamma }}_{ju}^{(+)}+{{\Gamma }}_{ju}^{(-)}\otimes {{\Gamma }}_{ju}^{(-)}\right].\hfill \end{align*}$ 18 $\begin{align*}\hfill & \mathcal{P}\left(\mathcal{A},\mathcal{B},{\rho }_{c}\right)(\rho )=\left\{(1-p)(1-q){\rho }_{e}+(1-p)q\left[\sum\limits _{s}\sum\limits _{k=0}^{d-1}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]{\rho }_{e}{\left[\sum\limits _{s}\sum\limits _{k=0}^{d-1}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]}^{{\dagger}}+p(1-q)\left[\sum\limits _{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]\right.\hfill \\ \hfill & \left.\times {\rho }_{e}{\left[\sum\limits _{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]}^{{\dagger}}\right\}\otimes \left[\frac{1}{\sqrt{d}}\sum\limits _{j=0}^{d-1}{\omega }^{{\varphi }_{c}j}\vert j\rangle \langle j\vert \right]+pq\left[\sum\limits _{s,t}{\iota }_{t}\vert st\rangle \langle {\lambda }_{st}\vert \right]{\rho }_{e}{\left[\sum\limits _{s,t}{\iota }_{t}\vert st\rangle \langle {\lambda }_{st}\vert \right]}^{{\dagger}}\otimes \left[\frac{1}{\sqrt{d}}\sum\limits _{j=0}^{d-1}{\omega }^{{\varphi }_{c}j}\vert j\rangle \langle j\vert \right].\hfill \end{align*}$

Following the equations for $${\rho }_{e}$$, the global quantum state at the output of the switch can be calculated by exploiting the tensor product properties after some algebraic manipulation to obtain Equation (4), where the control qudit $${\varphi }_{c}$$ is transformed into a mixed state of basis states, $${\lambda }_{sk}={s}_{\oplus (d-1)}k$$, $${\lambda }_{st}=s{t}_{\oplus (d-1)}$$, $${\omega }^{{\varphi }_{c}j}={e}^{i2\pi {\varphi }_{c}j/d}$$ and $${\iota }_{t}={(-1)}^{t\oplus (d-1)}$$, where $$\oplus (d-1)$$ in subscript represents the direct sum of $$(d-1)$$ copies. The global state $$\mathcal{P}\left(\mathcal{A},\mathcal{B},{\rho }_{c}\right)\left({\rho }_{e}\right)$$ at the output of the quantum switch is a mixture of the pure states (i.e. $$\langle {\psi }_{c}\Vert {\psi }_{c}\rangle =1$$ in a complex Hilbert space) of the control qudit. Based on the Hadamard basis, if the measurement outcome is $$\vert -\rangle $$, the global state collapses with probability $$pq$$ to, 19 $${\rho }_{{e}_{\vert -\rangle }}^{QS}=\left[\sum\limits _{s,t}{\iota }_{t}\vert st\rangle \langle {\lambda }_{st}\vert \right]{\rho }_{e}{\left[\sum\limits _{s,t}{\iota }_{t}\vert st\rangle \langle {\lambda }_{st}\vert \right]}^{{\dagger}}.$$ The entanglement distribution in this case is a noiseless process. Bob receives the particle $$\vert {\Phi }\rangle 2$$ without any error if $$\rho eQS={\rho }_{e}$$, thereby yielding to a noiseless entanglement distribution through the $$d$$-dimensional quantum switch with probability $$pq$$. Alternatively, the distribution can be a noisy process degraded by a noisy communication channel with a probability $$(1-pq)$$. In that case, the measurement outcome is $$\vert +\rangle $$ and the global state collapses to, 20 $\begin{align*}\hfill & {\rho }_{{e}_{\vert +\rangle }}^{QS}=\frac{(1-p)(1-q){\rho }_{e}}{1-pq}\hfill \\ \hfill & \quad +\,\frac{p(1-q)\left[{\sum }_{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]{\rho }_{e}{\left[{\sum }_{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]}^{{\dagger}}}{1-pq}\hfill \\ \hfill & \quad +\,\frac{(1-p)q\left[{\sum }_{s,k}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]{\rho }_{e}{\left[{\sum }_{s,k}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]}^{{\dagger}}}{1-pq}.\hfill \end{align*}$ Subsequently, we can derive the expression for the density matrix $${\rho }_{e}^{QS}$$ of the EPR pair distributed between Alice and Bob at the output of the quantum switch which will be equal to $${\rho }_{e}$$ with probability $$pq$$ and otherwise as follows: 21 $\begin{align*}\hfill & {\rho }_{e}^{QS}=\left((1-p)(1-q){\rho }_{e}+p(1-q)\left[\sum\limits _{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]\right.\hfill \\ \hfill & \quad \times \,{\rho }_{e}{\left[\sum\limits _{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]}^{{\dagger}}+(1-p)q\left[\sum\limits _{s}\sum\limits _{k=0}^{d-1}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]\hfill \\ \hfill & \quad \left.\times \,{\rho }_{e}{\left[\sum\limits _{s}\sum\limits _{k=0}^{d-1}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]}^{{\dagger}}\right)/(1-pq).\hfill \end{align*}$ with probability $$pq$$ for the measurement of the control qudit corresponding to the basis states, the entanglement distribution is noiseless. Bob receives the particle $$\vert {\Phi }\rangle 2$$ without any error if $${\rho }_{e}^{QS}={\rho }_{e}$$, while it is degraded by noise with the probability $$(1-pq)$$. Since $$\mathcal{A}$$ and $$\mathcal{B}$$ denote the bit-flip and the phase-flip channels respectively, traversed in a well-defined order $$\mathcal{A}\to \mathcal{B}$$, the density matrix of the entangled pair at Bob's side is given by the following equation: 22 $\begin{align*}\hfill {\rho }_{e}^{CT}& =\mathcal{B}\left[\mathcal{A}\left({\rho }_{e}\right)\right]=\mathcal{B}\left[\sum\limits _{s}{\mathcal{A}}_{s}{\rho }_{e}{\mathcal{A}}_{s}^{{\dagger}}\right]\hfill \\ \hfill & =\sum\limits _{t}{\mathcal{B}}_{t}\left[\sum\limits _{s}{\mathcal{A}}_{s}{\rho }_{e}{\mathcal{A}}_{s}^{{\dagger}}\right]{\mathcal{B}}_{t}^{{\dagger}}.\hfill \end{align*}$ 23 $\begin{align*}\hfill {\rho }_{e}^{CT}& =(1-p)(1-q){\rho }_{e}+p(1-q)\left[\sum\limits _{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]{\rho }_{e}{\left[\sum\limits _{s,t}\vert st\rangle \langle {\lambda }_{st}\vert \right]}^{{\dagger}}+(1-p)q\left[\sum\limits _{s}\sum\limits _{k=0}^{d-1}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]{\rho }_{e}\hfill \\ \hfill & \quad \times {\left[\sum\limits _{s}\sum\limits _{k=0}^{d-1}\vert {\lambda }_{sk}\rangle \langle {\lambda }_{sk}\vert \right]}^{{\dagger}}+pq\left[\sum\limits _{s,t}{\iota }_{t}\vert st\rangle \langle {\lambda }_{st}\vert \right]{\rho }_{e}{\left[\sum\limits _{s,t}{\iota }_{t}\vert st\rangle \langle {\lambda }_{st}\vert \right]}^{{\dagger}}.\hfill \end{align*}$ Consequently, the expression for the density matrix $${\rho }_{e}^{CT}$$ of the EPR pair distributed between Alice and Bob for classical trajectories is given by Equation (23). Using Equations (4, 9, 23), it is possible to calculate the performance gain achievable through the teleportation of qudit over that achievable through the teleportation of qubit. Also it is possible to compute the performance gain achievable through entanglement distribution via quantum switch (for qudits) over that achievable without any entanglement distribution.

QUDIT TELEPORTATION THROUGH QUANTUM SWITCH

Using the expression for the density matrix of the teleported qudit at Bob's side, we can quantify the effect of noise on EPR distribution on the teleported qudit. Towards that end, let us assume $$\rho \psi \triangleq \vert \psi \rangle a\langle \psi \vert a$$ be the $$d\times d$$ density matrix of the unknown pure qudit state $$\vert \psi \rangle a$$ that Alice sends to Bob, such that, 24 $${\rho }_{\psi }=\left[\begin{array}{@{}ccc@{}}\hfill {\rho }_{\psi }^{11}\hfill & \hfill {\cdots}\hfill & \hfill {\rho }_{\psi }^{1d}\hfill \\ \hfill {\vdots}\hfill & \hfill {\ddots}\hfill & \hfill {\vdots}\hfill \\ \hfill {\rho }_{\psi }^{d1}\hfill & \hfill {\cdots}\hfill & \hfill {\rho }_{\psi }^{dd}\hfill \end{array}\right]$$ Let us assume that the expression for the density matrix of the distributed EPR pair at the output of the quantum switch corrupted by noise be equal to $${\tilde{\rho }}_{e}$$. If the entanglement distribution is perfect, $${\tilde{\rho }}_{e}={\rho }_{e}$$. Here $${\tilde{\rho }}_{e}$$ is a $${d}^{2}\times {d}^{2}$$ matrix consisting of $$d\times d$$ block matrices to yield, 25 $${\tilde{\rho }}_{e}=\left[\begin{array}{@{}ccc@{}}\hfill {\tilde{\rho }}_{{e}_{11}}\hfill & \hfill {\cdots}\hfill & \hfill {\tilde{\rho }}_{{e}_{1d}}\hfill \\ \hfill {\vdots}\hfill & \hfill {\ddots}\hfill & \hfill {\vdots}\hfill \\ \hfill {\tilde{\rho }}_{{e}_{d1}}\hfill & \hfill {\cdots}\hfill & \hfill {\tilde{\rho }}_{{e}_{dd}}\hfill \end{array}\right]$$ We also assume that the density matrix of the teleported qudit at Bob's side is equal to $${\rho }_{\tau }$$. Using these notations, we represent the quantum teleportation process of a qudit in terms of density matrices in Figure 3.

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Following the procedure in ref. [26], we first calculate $${\rho }_{\tau }$$ if the measurement outcome is $$\vert kk\rangle $$ (for $$k=0,\text{\ldots },d-1$$) to obtain, 26 $${\rho }_{\tau }^{\vert kk\rangle }=d\left[\sum\limits _{k=1}^{d}{\rho }_{\psi }^{kk}{\tilde{\rho }}_{{e}_{kk}}+\sum\limits _{v=1}^{j}\sum\limits _{u=j+1}^{d}{\rho }_{\psi }^{vu}{\tilde{\rho }}_{{e}_{vu}}\right].$$ If the measurement outcome is $$\vert vu\rangle $$ (for $$v=0,\text{\ldots },j-1$$, $$u=j,\text{\ldots },d-1$$) $${\rho }_{\tau }$$ can be given by the following: 27 $${\rho }_{\tau }^{\vert vu\rangle }=d\left[\sum\limits _{k=1}^{d}{\rho }_{\psi }^{kk}{\tilde{\rho }}_{{e}_{kk}}-\sum\limits _{v=1}^{j}\sum\limits _{u=j+1}^{d}{\rho }_{\psi }^{vu}{\tilde{\rho }}_{{e}_{vu}}\right].$$ Using $${\rho }_{\tau }$$ we can calculate the performance gain achievable through the superposition of causal orders via the quantum switch in terms of quantum fidelity $$F$$. The conditional fidelity $${F}_{\vert +\rangle }^{QS}$$ associated with the teleported qudit at Bob's side can be calculated in terms of $${\rho }_{\tau }$$ to obtain [26], 28 $${F}_{\vert +\rangle }^{QS}=\left[{\rho }_{\tau }{\rho }_{\psi }\right]=\frac{d-dp-dq+pq+1}{(1-pq)(d+1)}$$ while the conditional fidelity $${F}_{\vert -\rangle }^{QS}=1$$ with probability $$pq$$, when the measurement of the control qudit corresponds to the state $$\vert +\rangle $$ and $$\vert -\rangle $$ respectively. In this case, the control qudit $$\vert {\psi }_{c}\rangle =\vert +\rangle \langle +\vert $$ where $$\vert +\rangle \equiv \frac{1}{\sqrt{d}}(\vert 1\rangle +\vert 2\rangle +\text{\ldots }+\vert d\rangle )$$ or $$\vert {\psi }_{c}\rangle =\vert -\rangle \langle -\vert $$ where $$\vert -\rangle \equiv \frac{1}{\sqrt{d}}(\vert 1\rangle -\vert 2\rangle -\text{\ldots }-\vert d\rangle )$$. The average fidelity $${\bar{F}}^{QS}$$ of the teleported qudit on Bob's side using quantum switch can be given by the following: 29 $\begin{align*}\hfill {\bar{F}}^{QS}& =pq{F}_{\vert -\rangle }^{QS}+(1-pq){F}_{\vert +\rangle }^{QS}\hfill \\ \hfill & =\left[d\left(\sum\limits _{k=1}^{d}{\rho }_{\psi }^{kk}{\tilde{\rho }}_{{e}_{kk}}+\sum\limits _{v=1}^{j}\sum\limits _{u=j+1}^{d}{\rho }_{\psi }^{vu}{\tilde{\rho }}_{{e}_{vu}}\right)\right.\hfill \\ \hfill & \quad \left.\times \left[\begin{array}{@{}ccc@{}}\hfill {\rho }_{\psi }^{11}\hfill & \hfill {\cdots}\hfill & \hfill {\rho }_{\psi }^{1d}\hfill \\ \hfill {\vdots}\hfill & \hfill {\ddots}\hfill & \hfill {\vdots}\hfill \\ \hfill {\rho }_{\psi }^{d1}\hfill & \hfill {\cdots}\hfill & \hfill {\rho }_{\psi }^{dd}\hfill \end{array}\right]\right]\hfill \end{align*}$ which can be manipulated using mathematical operations, like matrix multiplication, integration and trace of square matrices, to obtain, 30 $${\bar{F}}^{QS}=\frac{d-dp-dq+{d}^{2}pq+1}{d+1}$$ with $$p$$ and $$q$$ being the error probabilities of the two considered noisy channels $$\mathcal{A}$$ and $$\mathcal{B}$$. It is worth-mentioning that a fidelity of 1 corresponds to the case where the received qudit on Bob's side is identical to the original qudit and the measurement of the control qudit is equal to $$\vert -\rangle $$. The advantage of using a quantum switch for qudit teleportation, compared to using classical methods, can be determined by examining the resulting improvement in quantum fidelity (how closely the teleported qudit matches the original) and the increase in channel capacity (the amount of information that can be sent). We will conduct a detailed analysis of this in a future study.

In Figure 4, we plot the conditional fidelity $$\vert {F}_{\vert +\rangle }^{QS}$$ with respect to $$p$$ and $$q$$, where $$\vert {F}_{\vert +\rangle }^{QS}$$ is the fidelity corresponding to the control qudit $$\vert +\rangle $$. Figure 4 demonstrates that the conditional fidelity deteriorates irreversibly with the increase in error probability $$p$$ of the bit flip channel and $$q$$ of the phase flip channel, if the control qudit $$\vert {\psi }_{c}\rangle $$ is measured to be in state $$\vert +\rangle $$ for any $$p,q > 0$$. This degradation in fidelity results from the degradation in entanglement and thereby in the quality of the received information.

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Figure 5 portrays the average fidelity as a function of the error probabilities $$p$$ and $$q$$ of the noisy channel, as long as the control qudit $$\vert {\psi }_{c}\rangle $$ is measured to be in state $$\vert -\rangle $$. The noise in the channel causes a degradation in fidelity up to a certain threshold and then the fidelity starts increasing with increase in $$p$$ and $$q$$. This happens due to the advantage offered by the qudit switch, which makes it possible to send information even when the noise increases; a task impossible to achieve within the standard quantum Shannon theory. This improvement in average fidelity after a certain point is utilised in our design for quantum switch to yield an overall improvement in fidelity even in a very noisy quantum channel. Highest possible fidelity is achieved for the case when $$p=0$$ and $$q=0$$; that is the case where the channels are technically noiseless. Here the threshold value refers to the values of the error probabilities $$p,q$$. With further increase in $$p$$ and $$q$$ after the threshold value, average fidelity increases. This threshold is actually the limiting condition of the zero-capacity quantum channel, that is, $$p=q=0.5$$. For the values of $$p$$ and $$q$$, lower than the threshold value, fidelity decreases with increase in error probabilities. This means that the advantage in terms of fidelity obtained through employing a quantum switch is only attainable after a certain threshold for $$p$$ and $$q$$ is reached.

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In Figure 6, we compare the average fidelity achievable with a quantum switch where a $$d$$-dimensional qudit is teleported by varying $$d$$, $$d=2,5,10,15,20$$ as a function of the error probabilities $$p$$ and $$q$$ of the bit-flip and the phase-flip channels respectively for the condition when $$q=p$$. When $$p=q=0.5$$ (like a 50% erasure channel), no quantum information can be sent through any classical trajectory traversing the channels $$\mathcal{A}$$ and $$\mathcal{B}$$ [9]. Hence, achievable fidelity increases with the dimension of the quantum particle used for teleportation. Increased dimension refers to an enlarged Hilbert space and a larger Hilbert space offers the advantage of communicating with larger information capacity and higher noise resilience than traditional qubit-based quantum systems. It is evident from Figure 6, average fidelity $${\bar{F}}^{QS}$$ decreases as $$p$$ and $$q$$ increase (i.e. increase in error probabilities), up to a certain threshold like $$p=q=1/2$$ for the case of $$d=2$$. Fidelity starts increasing after the threshold even as noise affecting the channel increases, thereby corroborating the results obtained in ref. [26].

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For the case when $$p=q\to 0.5$$ (the limiting condition of zero-capacity quantum channel), teleportation of qubit over a quantum switch offers a fidelity of approaching 0.5, that is, $${\bar{F}}_{d=2}^{QS}\to 0.5$$. While as we increase $$d$$ to 5, fidelity approaches 0.65. If $$d=10$$, $${\bar{F}}_{d=10}^{QS}\to 0.78$$, $$d=15$$, if $${\bar{F}}_{d=15}^{QS}\to 0.89$$, $$d=20$$, and if $${\bar{F}}_{d=20}^{QS}\to 0.94$$. Therefore, entanglement distribution of a qudit over a quantum switch provides a remarkable gain in fidelity of the teleported qudit at Bob's side with increase in both noise plaguing the communication channel and dimensionality of the quantum particle.

For any scenario of quantum teleportation, the EPR pair while being distributed between transmitter and receiver may lose its coherence, and become a mixed state due to interaction with the environment and noise introduced by the quantum channel. This in turn reduces the fidelity of teleportation, thereby reducing the range of states that can be accurately teleported. However, the results obtained here demonstrate that fidelity can be improved even in presence of noise, if a quantum switch is used. Towards this end, if entanglement is distributed between sender and receiver using a quantum switch, fidelity of teleportation can be enhanced even in presence of noise, thereby offering a wide range of states that can be accurately teleported.

The fidelity improvement observed with increasing qudit dimension $$d$$ can be attributed to the expanded Hilbert space associated with high-dimensional quantum states. As $$d$$ increases, the qudit's Hilbert space grows exponentially, providing a larger set of orthogonal states for encoding and transmitting information. This inherently larger space allows for more precise encoding of quantum states, which enhances the system's resilience to noise and reduces the impact of decoherence. Moreover, the increased dimensionality improves the effectiveness of error correction and noise mitigation. In noisy quantum channels, the larger space enables the quantum switch to leverage the superposition of causal orders more effectively, compensating for errors introduced by bit-flip and phase-flip channels. This is reflected in the observed increase in average fidelity, which transitions from $${F}_{\mathrm{d}=2}=0.5$$ for qubits to $${F}_{\mathrm{d}=20}=0.94$$ for 20-dimensional qudits, even in channels with significant noise (e.g. $$p=q=0.5$$). These improvements underline the advantage of high-dimensional systems in achieving robust quantum communication. The ability to maintain high fidelity despite increasing noise highlights the potential of qudits to outperform traditional qubit systems in practical quantum networks.

CONCLUSION

In this paper, we formulated the first ever theoretical framework for implementing quantum switch for $$d$$-dimensional quantum teleportation system from a communication engineering point of view. With the help of generalised quantum gates, we have designed the basic scheme for teleportation of qudits and then applied it to quantum switch for entanglement distribution. In future, we will use the results obtained in this paper directly to evaluate the performance gain achievable by teleporting and distributing entanglement of qudits over that of qubits in noisy quantum channels, in terms of information carrying capacity and resilience to noise. We will also analyse the impact of designing the generalised SWAP gate (different combinations of higher dimensional quantum gates as demonstrated in Figure 2) on the overall performance of qudit teleportation in our future work.

AUTHOR CONTRIBUTIONS

Indrakshi Dey: Conceptualization; formal analysis; investigation; methodology; resources; validation; visualization; writing—original draft. Nicola Marchetti: Conceptualization; investigation; methodology; validation; writing—original draft; writing—review & editing.

ACKNOWLEDGEMENTS

This work is supported by the HORIZON-MSCA-2022-SE-01-01 project CO- ALESCE under Grant Number 101130739 and the National HEA TU RISE Programme-funded project AIQ-Shield.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

DATA AVAILABILITY STATEMENT

Data sharing is not applicable to this article as no new data were created or analysed in this study.

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