Quantum teleportation and remote sensing through semiconductor quantum dots affected by pure dephasing

Abstract

Quantum teleportation allows the transmission of quantum states over arbitrary distances and is an applied tool in quantum computation and communication. This paper theoretically addresses the feasibility of quantum teleportation based on a single semiconductor quantum dot influenced by pure dephasing through the biexciton cascade decay. We also investigate the idea of remote sensing in quantum teleportation affected by pure dephasing. In particular, we compare the quality of quantum teleportation in single- and two-qubit schemes and show that, within the present model, single-qubit quantum teleportation has a quantum advantage. Finally, to investigate the dynamics of the system, we introduce important witnesses of the non-Markovian dynamics of the system, so that our results may solve outstanding problems in the realization of faithful quantum teleportation over a long time.

Graphical abstract

Keywords

quantum teleportation / semiconductor quantum dots / quantum phase estimation / remote sensing

1 Introduction

Transmitting quantum states and extracting information encoded in them is of paramount importance in quantum communication. During communication, quantum states are necessarily exposed to an external environment [1–3], which can affect information extraction. Often, information must be transmitted securely to protect it from hacking [4]. Furthermore, in many cases, one needs a high capacity to transmit information. Security against hacking and high capacity are considerable quantum advantages of quantum protocols compared to their classical counterparts [5–11]. The implementation of quantum communication protocols can be summarized in five different steps [10, 12]: preparation of the quantum state, sharing of the channel, encoding of information, sending the unknown quantum state, and decoding the information.

Quantum teleportation [13–15] is a well-known quantum communication protocol that was first suggested by Bennett et al. [13], in which Alice and Bob share a classical or non-classical channel such that the sender transmits an unknown quantum state containing quantum information to the receiver via the channel [16–25]. The initial state of the channel plays a crucial role in quantum information teleportation [26]. The quantum information can be encoded in some variables of the system or environment. For instance, information may be encoded either in quantum memory to store quantum states [27], or it may be encoded in the weight parameter of the qubit [28]. Furthermore, quantum information can be encoded in the initial phase of the quantum state [10, 11, 29], so our motivation in this paper is to extract the information that is encoded in the initial phase of the teleportation state. The initial phase can include important encoded information or exhibit the nature of the process that prepared the initial state [10]. Therefore, due to the high importance of the encoded information, we need to somehow access and analyze it to achieve better quantum communication.

There are different methods to extract encoding information in the initial phase. Quantum estimation, one of the most effective among them [30–33], is carried out by sensors whose correct design can greatly accuracy of the measurements. For example, a quantum sensor can be based on a qubit that encodes information in the relative phase of its quantum state by interacting with a weak external field. The extracted information obtained by measuring the qubit can be very useful. Quantum phase estimation is one of the most important concepts in early studies of quantum computing [34]. To estimate the initial phase of the system in quantum teleportation, we can employ the quantum Fisher information (QFI) [31, 35] tool. Then, we can use another powerful and widely used tool called Hilbert−Schmidt speed (HSS) [36, 37] to simplify the calculation. In many cases, the physical presence of the main person (Bob) at the place where a particular estimate is to take place is not possible; or, due to security risks, the conditions are not favorable. Or it may not be possible to transfer the tool to the destination to achieve such a thing, or the person may be present at the desired location, but the tool available to him is not precise enough, although it is connected to a server equipped with precision tools. Under such conditions, performing remote estimation is a way forward in which Alice and Bob carry out the process jointly using classical and quantum communication channels. In this work, we use two QFI and HSS tools to remotely estimate the initial phase of the teleportation state.

In reality, the interaction between the system and its environment is an inevitable phenomenon. Consequently, we can say that we are faced with systems that are not isolated, called open quantum systems in quantum information theory [38–43]. Regarding the interaction of the system with its environment, the dynamics of open quantum systems are divided into two scenarios: Markovian and non-Markovian. If the information flows continuously from the system to the environment, the dynamic is Markovian. But, if at some time intervals, the information flows back to the system from the environment due to quantum memory effects, the dynamic is called non-Markovian [44–46]. In open quantum systems, quantum teleportation depends on the type of system evolution. The significant point here is that QFI can also be considered as the witnesses of entanglement [47, 48] and non-Markovianity of dynamics [37, 49] in quantum systems.

Semiconductor quantum dots have received significant attention due to their applications in quantum information theory [50, 51]. In semiconductor quantum dots electrons and holes can be trapped to form excitonic complexes, which continuously decay via photon emission [52]. The excitonic states due to Coulomb correlation effects have different typical energies and hence can be distinguished in optical experiments [53]. Photon devices based on quantum dots typically use the last or the two last photons emitted in the cascade decay [54]. The last photon which is generated from the decay of the exciton, comprises an electron and a hole in the lowest unoccupied state, and the penultimate photon generated from the decay of the biexciton state, includes two excitons with opposite spin orientations. The uncertainty of the biexciton decay, through either of the two ideally spin-degenerate exciton states, refers to a polarization entanglement of the emitted photons [55, 56]. Quantum dots [57, 58] are quantum heterostructures that are composed of nanoscale domains of one type of material embedded in a second type [59]. Many applications of quantum dot molecules including quantum teleportation, quantum dense coding cryptography, memory, and quantum key distribution have been reported [4, 10, 60–63]. As well, the quantum correlation dynamics for two coupled quantum dots at finite temperatures have been investigated [64]. Furthermore, the thermal Bell state and entanglement properties of electrons within coupled quantum dots molecules have been reported [65, 66]. Besides, the decoherence and entanglement of hybrid qubits related to double quantum dots have been examined [67, 68]. In addition, teleportation on a quantum dot array has been studied [69]. Also, the generation of entangled channels for perfect teleportation using multielectron quantum dots has been reported [70].

In Ref. [52], the influences of pure dephasing on the behavior of quantum entanglement of a single semiconductor quantum dot have been discussed. In this paper, we intend to monitor the effect of pure dephasing through the biexciton cascade decay on quantum teleportation and quantum remote sensing in a single semiconductor quantum dot. We consider that there is a qubit at Alice’s location that encodes the information needed by Bob in its initial state phase. Alice intends to teleport the quantum state of this qubit to Bob, who is equipped with a sensitive sensor for estimation. It can be noted that Bob is affected by pure dephasing through the biexciton cascade decay, which can influence teleportation quality. So, in this work, we plan to probe Bob’s required information, encoded in its initial state phase and its quantum state teleported by Alice, through the attractive tools of QFI and HSS in the presence of pure dephasing. Most significantly, we make a comparison between single-qubit and two-qubit teleportation in the current model, which may be important in applications.

The paper is divided into four parts. After the present introduction, preliminaries of quantum teleportation are defined in Section 2. Section 3 introduces the theoretical model that can be used as a resource for quantum teleportation. Finally, Section 4 summarizes and discusses the most important results. Section 5 gives a conclusion.

2 Preliminaries

2.1 Quantum teleportation

2.1.1 Single-qubit teleportation: Basic concepts

Remote transmission off implies a mixed state of two-qubits {\rho }_{ch}. In the standard protocol [71], which is used as a channel or resource, is formulated by a generalized depolarized quantum channel \mathrm{\Lambda } ({\rho }_{ch} ), according to a single-qubit input state {\rho }_{in}. Alice plans to send her encoded qubit to Bob by this process. We consider the unknown input (initial) state of teleportation for an arbitrary pure single-qubit state as follows:

(1)| {\psi }_{in}⟩=\mathrm{c}\mathrm{o}\mathrm{s}({\displaystyle \frac{\theta }{2}})|0⟩+{\mathrm{e}}^{\mathrm{i}\varphi }\mathrm{s}\mathrm{i}\mathrm{n}({\displaystyle \frac{\theta }{2}})|1⟩,

where \theta and \varphi are the amplitude and phase of the initial state of teleportation, respectively. We use |0⟩=(\begin{array}{c}1\\ 0\end{array} ) and | 1⟩= (\begin{array}{c}0\\ 1\end{array}). In the transmission of an arbitrary single-qubit state (input state {\rho }_{in} =|{\psi }_{in} ⟩⟨{\psi }_{in} |), one can consider the output state of the teleportation as [71]

(2){\rho }_{out}=\mathrm{\Lambda }( {\rho }_{ch}){\rho }_{in}={\displaystyle \sum _{i=0}^3\mathrm{T}\mathrm{r}[{\mathcal{B}}_i{\rho }_{ch}]{\sigma }_i{\rho }_{in}{\sigma }_i,}

in which \mathrm{\Lambda }( {\rho }_{ch}) is a generalized depolarized channel, {\rho }_{ch} represents the state of the channel, and {\mathcal{B}}_i denotes the Bell state corresponds to the Pauli matrix {\sigma }_i and is obtained by

(3){\mathcal{B}}_i= ({\sigma }_0\otimes {\sigma }_i){\mathcal{B}}_0({\sigma }_0\otimes {\sigma }_i), i=1,2,3,

where {\sigma }_0=\mathbb{I},{\sigma }_1={\sigma }_x, {\sigma }_2={\sigma }_y,{\sigma }_3={\sigma }_z and \mathbb{I} is the identity matrix. Furthermore, for any two arbitrary qubits, each defined in basis \{|0⟩ ,|1⟩\}, we have {\mathcal{B}}_0=\frac{1}{2}(|00⟩+|11⟩)(⟨00|+⟨11|).

The quality of the teleported state is specified by the criterion of fidelity f({\rho }_{in} (t),{\rho }_{out}(t)), which is defined as [5, 72]

(4)f({\rho }_{in} (t),{\rho }_{out}(t))= (\mathrm{T}\mathrm{r}(\sqrt{\sqrt{{\rho }_{in}(t)}{\rho }_{out}(t)\sqrt{{\rho }_{in}(t)}}){)}^2,

where the limit for fidelity is 0\le f({\rho }_{in} (t),{\rho }_{out}(t))\le 1. For f=1, the optimum fidelity that leads to optimum teleportation can be achieved. Additionally, averaging on fidelity for all possible input state parameters (\theta and \varphi), one can determine the average fidelity of teleportation f_A as follows:

(5)f_A:={\displaystyle \frac{1}{4\text{π}}}{\displaystyle {\int }_0^{2\pi }\mathrm{d}\varphi {\displaystyle {\int }_0^{\pi }f({\rho }_{in}(t),{\rho }_{out}(t))\sin (\theta )\mathrm{d}\theta .}}

It is worth noting that the threshold of the maximum classical average fidelity appears at f_A=2/3. Beyond, there is the quantum average fidelity. For f_A= 1, the optimum quantum teleportation is reached, meaning that less information is leaked.

2.1.2 Two-qubit teleportation: basic concepts

In addition to single-qubit teleportation, we assume a second scenario where Alice transmits two encoded qubits to Bob. The unknown input (initial) state for an arbitrary two-qubit pure state is described by

(6)| {\psi }_{in}⟩=\mathrm{c}\mathrm{o}\mathrm{s}({\displaystyle \frac{\theta }{2}})|10⟩+{\mathrm{e}}^{\mathrm{i}\varphi }\mathrm{s}\mathrm{i}\mathrm{n}({\displaystyle \frac{\theta }{2}})|01⟩,

where |10⟩= | 1⟩\otimes | 0⟩ and |01⟩= | 0⟩\otimes | 1⟩. Generalizing Eq. (2), we can describe the output state {\rho }_{out} of an arbitrary two-qubit state by [19, 73]

(7){\rho }_{out}={\displaystyle \sum _{i,j=0}^3{p}_{ij}({\sigma }_i\otimes {\sigma }_j){\rho }_{in}({\sigma }_i\otimes {\sigma }_j),}

where \sum p_{ij}=1 and p_{ij}=\mathrm{T}\mathrm{r}[{\mathcal{B}}_i{\rho }_{ch}]\mathrm{T}\mathrm{r}[{\mathcal{B}}_j{\rho }_{ch}].

2.2 Quantum phase estimation

The estimation accuracy is a significant concept in phase estimation. The accuracy limit is specified by a criterion called “Cramér−Rao (CR) inequality”. The difference between the true value and the estimated value of a phase defines the estimation accuracy, which is put on one side of the CR inequality. On the other side of the inequality, there is a quantity called “quantum Fisher information (QFI)” [30, 31, 35, 74] which determines the lowest limit of the accuracy of the estimation corresponding to the number of repeated measurements. Therefore, one can estimate the uncertain true value of a phase with this powerful QFI tool. Hence, one can define the quantum CR inequality [74, 75] as

(8)\mathrm{\Delta }\varphi \ge {\displaystyle \frac{1}{\sqrt{\mathcal{F}(\varphi )}}},

which gives the smallest detectable change of the phase \varphi. Moreover, \mathcal{F}(\varphi ) represents the QFI with respect to \varphi and it is obtained by [31, 76]

(9)\mathcal{F}(\varphi )= {\displaystyle \sum _i{\displaystyle \frac{({\mathrm{\partial }}_{\varphi }{\lambda }_i{)}^2}{{\lambda }_i}}+4{\displaystyle \sum _{i\ne j}{\displaystyle \frac{({\lambda }_i-{\lambda }_j{)}^2}{{\lambda }_i+{\lambda }_j}}|⟨{\phi }_i|{\mathrm{\partial }}_{\varphi }{\phi }_j⟩{|}^2,}}

in which |{\phi }_i⟩ and {\lambda }_i are eigenvectors and eigenvalues of the density matrix, respectively. Within the quantum estimation theory, the larger the QFI, the greater the accuracy of the estimate.

2.3 Hilbert−Schmidt speed

Quantum statistical speed (QSS) quantifies the sensitivity of an initial state with respect to variations of the unknown parameter of dynamical evolution [36]. More sensitivity denotes that an unknown parameter, which may be an initial phase of the system, can be estimated with more accuracy [74, 76]. Hence, the criteria of QSS in Hilbert space can be associated with quantum estimation theory for example quantum phase estimation [36]. Hilbert–Schmidt speed (HSS) is known as a simple and powerful tool to improve quantum phase estimation in open quantum systems. Assuming the quantum state \rho (\varphi ), one can write HSS [36, 37, 77] as

(10)HS{S}_{\varphi }=\sqrt{{\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}[{\displaystyle \frac{\mathrm{d}\rho (\varphi )}{\mathrm{d}\varphi }}{]}^2},

which can be computed without diagonalizing \mathrm{d}\rho (\varphi )/\mathrm{d}\varphi. The QFI and HSS are QSSs corresponding, respectively, to the Hilbert−Schmidt and Bures distances [36]. It is interesting to examine the relationship between these two criteria in different models.

2.4 Von Neumann entropy

Due to the importance of investigating the correlation of a quantum system with its environment, in this subsection, we introduce a measure for investigating such correlation. Von Neumann’s entropy is the main measure of entanglement between the components of a bipartite system [78] if the total system is in a pure state [79]. More precisely, Bennett et al. showed in Ref. [80] that for any bipartite system in its pure state, the Von Neumann entropy calculated for the reduced density matrix of each of its components can be used as a measure of the entanglement. The same discussion can be extended to the situation in which a system interacts with its environment. In fact, the state of the system+environment can be considered a pure state according to the principle of purification [5], where entropy, as a measure of the entanglement, can be calculated for the reduced density matrix of the system. In other words, if |\mathrm{\Psi }⟩ is the state of the system+environment, the entanglement between the system and the environment can be defined by

(11)E(\mathrm{\Psi })=S (\rho )=-\mathrm{T}\mathrm{r}[\rho {\log }_2(\rho )]=-{\displaystyle \sum _i{\lambda }_i{\log }_2({\lambda }_i),}

where {\lambda }_i are the eigenvalues of the density matrix. In addition, S(\rho ) is the Von Neumann entropy.

2.5 Success probability

The distinguishability between two evolving states of the quantum system {\rho }_{in} and {\rho }_{out}, can be done via a widely recognized method proposed by Breuer et al. [44, 81] the capable of recognizing the non-Markovianity of the dynamics of the system through the trace distance (TD) [44, 45, 82]:

(12)D\left({\rho }_{in},{\rho }_{out}\right)= {\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}|{\rho }_{in}-{\rho }_{out}|,

where the modulus of the operator is given by |A|=\sqrt{A^{†}A}. The minimum and maximum thresholds of TD are 0⩽D\left({\rho }_{in},{\rho }_{out}\right)⩽1, where D\left({\rho }_{in},{\rho }_{out}\right)=0 if and only if {\rho }_{in}={\rho }_{out}, and D\left({\rho }_{in},{\rho }_{out}\right)=1 if and only if {\rho }_{in} and {\rho }_{out} are orthogonal. One of the crucial properties of TD is that it supplies a clear physical explanation for the distinguishability between two quantum states. Coming back to the subject of quantum teleportation, we consider two imaginary actors Alice and Bob, and suppose that Alice prepares a quantum system in one of two states, {\rho }_{in} or {\rho }_{out}, with a probability of \frac{1}{2} each, and transfers the quantum state to Bob. Bob’s task is to determine, through a quantum measurement, whether the system is in state {\rho }_{in} or {\rho }_{out}. It can be proven that the maximum probability of success (P_S) that Bob can achieve through an optimal strategy that is directly related to the TD and is given by [45]

(13)P_{S,max}={\displaystyle \frac{1}{2}}\left[1+D\left({\rho }_{in},{\rho }_{out}\right)\right].

If the states prepared by Alice are orthogonal such that D\left({\rho }_{in},{\rho }_{out}\right)=1, the maximum success probability can be obtained as P_{S,max}=1 so that orthogonal states can be distinguished with certainty by a single measurement. It should be noted that we can calculate the success probability of single-qubit quantum teleportation by using Eq. (13) with the input [Eq. (1)] and output [Eq. (2)] states of the channel and for two-qubit quantum teleportation with the input [Eq. (6)] and output [Eq. (7)] states of the channel.

3 Theoretical model

Consider the quantum dot level scheme displayed in Fig.1, which includes the biexciton state u, the two exciton states with polarizations (H and V) along x and y, and the ground state g. The system is initially prepared in the biexciton state u by optical pumping [55, 56] or electrical injection of carriers [83]. Then, it decays radiatively in a cascade action emitting two photons. Due to the biexciton binding \mathrm{\Delta }, the two photons have different energies and may be spectrally distinguished. The quantum dot dynamics is considered as an open quantum system interacting with the environment. Therefore, the free propagation of the quantum-dot states is described by the following Hamiltonian [52]:

(14)\hat{H}={\displaystyle \sum _{i=x,y}{E}_i|i⟩⟨i|+E_u|u⟩⟨u|,}

where i represents the two exciton states x and y with energies E_x= h{\nu }_x and E_y=h{\nu }_y, respectively, and u denotes the biexciton state with energy E_u= h{\nu }_u. Transitions between the various quantum dot states and dephasing are due to interaction with the environment. Note that we assume radiative decay and pure dephasing, which are considered to be the main scattering channels for excitons in quantum dots. Pure dephasing is due to phonon couplings while spectral diffusion is due to charging centers in the vicinity of the dot. Here we take the straightforward explanation of radiative decay and pure dephasing with respect to scattering and dephasing rates {\gamma }_r and {\gamma }_d, respectively, to compute the density operator elements analytically, similar to the method used in Ref. [52].

We use a master equation approach of Lindblad form as follows [84, 85]:

(15)\mathrm{i}{\displaystyle \frac{\mathrm{d}\hat{\rho }}{\mathrm{d}t}}=[ \hat{H},\hat{\rho }(t )]-{\displaystyle \frac{\mathrm{i}}{2}}{\displaystyle \sum _i({\hat{L}}_i^{†}{\hat{L}}_i\hat{\rho }+\hat{\rho }{\hat{L}}_i^{†}{\hat{L}}_i-2{\hat{L}}_i\hat{\rho }{\hat{L}}_i^{†}),}

where \hat{\rho } denotes the density operator of the quantum dot states and \hat{L} represents the Lindblad operators corresponding to the various scattering channels given in Tab.1 in Appendix A. Defining the short-hand notation \mathcal{L}\hat{\rho } on the right-hand side of Eq. (15), where \mathcal{L} represents the Liouville superoperator [38], one can describe the formal solution of the master equation in the quantum jump approach [86] by the following form:

(16)\hat{\rho }(t)= \mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{i}\mathcal{L}t ){\hat{\rho }}_0,

in which {\hat{\rho }}_0 is the initial density operator. The important point is that we do not consider any emission of photons which does not change the number of excitons. If the system is initially in the state {\hat{\rho }}_0= | p⟩⟨q|, with p and q labeling the quantum dot states g, x, y, and u, after straightforward calculations following the method of Ref. [52], the final time-dependent density operator of the current system in rotating wave approximation can be written as

(17)\hat{\rho }(t)= \left(\begin{array}{llll}1& {\mathrm{e}}^{\left(-\mathrm{i}\frac{\delta }{2}t-\frac{{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right)}& {\mathrm{e}}^{\left(\mathrm{i}\frac{\delta }{2}t-\frac{{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right)}& {\mathrm{e}}^{\left(\mathrm{i}\mathrm{\Delta }t-{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}t\right)}\\ {\mathrm{e}}^{\left(\mathrm{i}\frac{\delta }{2}t-\frac{{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right)}& {\mathrm{e}}^{\left(-{\gamma }_{r}t\right)}& {\mathrm{e}}^{\left[-\mathrm{i}\delta t-\left({\gamma }_r+{\gamma }_1\right)t\right]}& {\mathrm{e}}^{\left[\mathrm{i}\left(\mathrm{\Delta }-\frac{\delta }{2}\right)t-\frac{3{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right]}\\ {\mathrm{e}}^{\left(-\mathrm{i}\frac{\delta }{2}t-\frac{{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right)}& {\mathrm{e}}^{\left[\mathrm{i}\delta t-\left({\gamma }_r+{\gamma }_1\right)t\right]}& {\mathrm{e}}^{\left(-{\gamma }_{r}t\right)}& {\mathrm{e}}^{\left[\mathrm{i}\left(\mathrm{\Delta }+\frac{\delta }{2}\right)t-\frac{3{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right]}\\ {\mathrm{e}}^{\left(-\mathrm{i}\mathrm{\Delta }t-{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}t\right)}& {\mathrm{e}}^{\left[-\mathrm{i}\left(\mathrm{\Delta }-\frac{\delta }{2}\right)t-\frac{3{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right]}& {\mathrm{e}}^{\left[-\mathrm{i}\left(\mathrm{\Delta }+\frac{\delta }{2}\right)t-\frac{3{\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}}{2}t\right]}& {\mathrm{e}}^{\left(-2{\gamma }_{r}t\right)}\end{array}\right),

where {\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}={\gamma }_r+{\gamma }_d is the sum of radiative and dephasing rates, {\gamma }_1 denotes cross dephasing rate, \delta denotes the fine-structure splitting, and \mathrm{\Delta } represents the biexciton binding energy. In this paper, we consider Eq. (16) when it is normalized {\hat{\rho }}_N(t)=\hat{\rho }(t)/\mathrm{T}\mathrm{r}[\hat{\rho }(t) ], as a resource for quantum teleportation. Furthermore, according to the method described in Appendix B, all parameters are considered nondimensionalized (see Tab.2), for plotting the figures throughout the paper.

4 Discussion and results

4.1 Single-qubit teleportation scenario

Here, we study single-qubit teleportation through the theoretical model presented above. Using Eqs. (1)−(3) and (17), one can calculate the output state of the channel in the single-qubit teleportation as

(18){\rho }_{out}^{SQ}(t)=\left(\begin{array}{cc}{\rho }_{11}^{SQ}(t)& {\rho }_{12}^{SQ}(t)\\ {\rho }_{21}^{SQ}(t)& {\rho }_{22}^{SQ}(t)\end{array}\right),

where the superscript SQ denotes the single-qubit teleportation. Besides, the elements of the output density matrix are given by

(19)\begin{array}{rl}& {\rho }_{11}^{SQ}(t)=\frac{1}{2}\left[\cos (\theta ){\tanh }^2(\frac{{\gamma }_{r}t}{2})+1\right],\\ & {\rho }_{12}^{SQ}(t)=\frac{1}{2({\mathrm{e}}^{{\gamma }_{r}t}+1{)}^2}\sin (\theta ){\mathrm{e}}^{-t({\gamma }_d+{\gamma }_r+\mathrm{i}\mathrm{\Delta })-\mathrm{i}\varphi }[(1+2{\mathrm{e}}^{2\mathrm{i}\delta t}){\mathrm{e}}^{t({\gamma }_d-{\gamma }_1+2{\gamma }_r-\mathrm{i}\delta +\mathrm{i}\mathrm{\Delta })+2\mathrm{i}\varphi }+{\mathrm{e}}^{2t({\gamma }_r+\mathrm{i}\mathrm{\Delta })}+{\mathrm{e}}^{2{\gamma }_{r}t}],\\ & {\rho }_{21}^{SQ}(t)=\frac{1}{2({\mathrm{e}}^{{\gamma }_{r}t}+1{)}^2}\sin (\theta ){\mathrm{e}}^{-t({\gamma }_d+{\gamma }_r+\mathrm{i}\mathrm{\Delta })-\mathrm{i}\varphi }[{\mathrm{e}}^{t({\gamma }_d-{\gamma }_1+2{\gamma }_r-\mathrm{i}\delta +\mathrm{i}\mathrm{\Delta })}+{\mathrm{e}}^{t({\gamma }_d-{\gamma }_1+2{\gamma }_r+\mathrm{i}(\delta +\mathrm{\Delta }))}+(1+2{\mathrm{e}}^{2\mathrm{i}\mathrm{\Delta }t}){\mathrm{e}}^{2{\gamma }_{r}t+2\mathrm{i}\varphi }],\\ & {\rho }_{22}^{SQ}(t)=\frac{1}{2}\left[1-\cos (\theta ){\tanh }^2(\frac{{\gamma }_{r}t}{2})\right].\end{array}

As mentioned, the quality of teleportation is specified by the criterion of fidelity Eq. (4). Thus, using Eqs. (1)−(4) and Eq. (18), the fidelity of single-qubit teleportation in the current model is given by

(20)f_1={\displaystyle \frac{4{\mathrm{e}}^{{\gamma }_{r}t}\left[2{\sin }^2(\theta )\left({\mathrm{e}}^{-{\gamma }_{d}t}\cos (\mathrm{\Delta }t)+{\mathrm{e}}^{-{\gamma }_{l}t}\cos (2\varphi )\cos (\delta t)\right)+3\cosh ({\gamma }_{r}t)+1\right]+2\cos (2\theta ){\left({\mathrm{e}}^{{\gamma }_{r}t}-1\right)}^2}{8{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^2}}.

Then, using Eq. (5) the average fidelity for single-qubit teleportation is

(21)f_1^{av}= {\displaystyle \frac{2\left[{\mathrm{e}}^{t({\gamma }_r-{\gamma }_d)}\cos (\mathrm{\Delta }t)+{\mathrm{e}}^{t({\gamma }_r-{\gamma }_1)}\cos (\delta t)+{\mathrm{e}}^{{\gamma }_{r}t}+{\mathrm{e}}^{2{\gamma }_{r}t}+1\right]}{3{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^2}}.

Furthermore, by using Eq. (9) the initial phase estimation of the output teleported state via a single teleported qubit is given by

(22)\begin{array}{rl}{\mathcal{F}}_1(\varphi )=& \frac{1}{{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4}\\ & \cdot {\sin }^2(\theta ){\mathrm{e}}^{-2t({\gamma }_d+{\gamma }_r+\mathrm{i}\mathrm{\Delta })-2\mathrm{i}\varphi }\left[\left(1+{\mathrm{e}}^{2\mathrm{i}\delta t}\right)\left(-{\mathrm{e}}^{t({\gamma }_d-{\gamma }_l+2{\gamma }_r-\mathrm{i}\delta +\mathrm{i}\mathrm{\Delta })+2\mathrm{i}\varphi }\right)+{\mathrm{e}}^{2t({\gamma }_r+\mathrm{i}\mathrm{\Delta })}+{\mathrm{e}}^{2{\gamma }_{r}t}\right]\\ & \cdot \left[-{\mathrm{e}}^{t({\gamma }_d-{\gamma }_l+2{\gamma }_r-\mathrm{i}\delta +\mathrm{i}\mathrm{\Delta })}-{\mathrm{e}}^{t({\gamma }_d-{\gamma }_l+2{\gamma }_r+\mathrm{i}(\delta +\mathrm{\Delta }))}+\left(1+{\mathrm{e}}^{2\mathrm{i}\mathrm{\Delta }t}\right){\mathrm{e}}^{2{\gamma }_{r}t+2\mathrm{i}\varphi }\right].\end{array}

Moreover, to probe the initial phase, the HSS [Eq. (10)] for the output state of a single teleported qubit can be obtained by

(23)\begin{array}{rl}HS{S}_1(\varphi )=& {\displaystyle \frac{1}{2}}\sin (\theta )\\ & \cdot \sqrt{{\displaystyle \frac{{\mathrm{e}}^{-2t({\gamma }_d+{\gamma }_r+\mathrm{i}\mathrm{\Delta })-2\mathrm{i}\varphi }\left[\left(1+{\mathrm{e}}^{2\mathrm{i}\delta t}\right){\mathrm{e}}^{t({\gamma }_d-{\gamma }_l+2{\gamma }_r-\mathrm{i}\delta +\mathrm{i}\mathrm{\Delta })+2\mathrm{i}\varphi }-{\mathrm{e}}^{2t({\gamma }_r+\mathrm{i}\mathrm{\Delta })}-{\mathrm{e}}^{2{\gamma }_{r}t}\right]\left[{\mathrm{e}}^{t({\gamma }_d-{\gamma }_l+2{\gamma }_r-\mathrm{i}\delta +\mathrm{i}\mathrm{\Delta })}+{\mathrm{e}}^{t({\gamma }_d-{\gamma }_l+2{\gamma }_r+\mathrm{i}(\delta +\mathrm{\Delta }))}+\left(1+{\mathrm{e}}^{2\mathrm{i}\mathrm{\Delta }t}\right)\left(-{\mathrm{e}}^{2{\gamma }_{r}t+2\mathrm{i}\varphi }\right)\right]}{{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4}}}.\end{array}

Now, taking Eqs. (20)−(23), we analyze the qualitative behaviors of fidelity, average fidelity, entropy, HSS, and QFI in the single-qubit teleportation. This analysis helps probe the quality of quantum teleportation. Fig.2 describes the qualitative behavior of average fidelity f_1^{av} in the single-qubit quantum teleportation versus fine-structure splitting \delta and time t at the same time [Fig.2(a)], and versus biexciton binding energy \mathrm{\Delta } and time t at the same time [Fig.2(b)]. The important result seen in this figure is that the average fidelity is higher than the classical threshold which is equal to CT=2/3, meaning that the single-qubit teleportation based on the current model has a quantum advantage. Besides, when biexciton binding energy \mathrm{\Delta } is equal to zero, the average fidelity f_1^{av} is maximum. Furthermore, it can be seen that when the fine-structure splitting \delta is zero, there is no quantum advantage. Instead, for \delta > 0, the quantum advantage is always achieved. Hence, in this article, we set \delta =1 and \mathrm{\Delta } =0 to maximize the quantum advantage.

In Fig.3, the time evolution of average fidelity f_1^{av} in the single-qubit teleportation with increasing dephasing rate {\gamma }_d [Fig.3(a)], cross dephasing rate {\gamma }_1 [Fig.3(b)], decay rate {\gamma }_r [Fig.3(c)], and fine-structure splitting \delta [Fig.3(d)] is investigated. The analysis of these figures is one of the main goals of this paper, in such a way that the quality of quantum teleportation in the present model is examined under the influence of pure dephasing. Fig.3(a), shows that, with increasing dephasing rate {\gamma }_d, the average fidelity f_1^{av} decreases over time. A very important point is that in the initial times of the beginning of the teleportation process, this decrease is smaller. Consequently, over time, the average fidelity goes below the classical threshold \mathrm{C}\mathrm{T}, meaning that the quantum advantage in quantum teleportation is lost. Moreover, in Fig.3(b), with increasing cross-dephasing rate {\gamma }_1, the average fidelity f_1^{av} decreases over time. The amplitude of oscillations for average fidelity decreases over time with increasing cross-dephasing rate {\gamma }_1. Further, in Fig.3(c), with increasing decay rate {\gamma }_r, the same results as in Fig.3(b) can be obtained, with the difference that in Fig.3(b), the points of maximum average fidelity go below the classical threshold over time, while in Fig.3(c), this is not the case and a stable quantum teleportation occurs, which still has a quantum advantage. Besides, Fig.3(d) shows that with increasing fine-structure splitting \delta, the qualitative behavior of average fidelity oscillates more and improves the quality of teleportation.

4.1.1 Comparison between results in the single-qubit teleportation scenario

In Fig.4, the comparison between dynamics of the fidelity f_1, average fidelity f_1^{av}, entropy S_1, QFI with respect to initial phase {\mathcal{F}}_1(\varphi ), HSS with respect to initial phase HS{S}_1( \varphi ), and success probability P_{S1} for the single-qubit teleportation are represented. When the dephasing rate, cross-dephasing rate, and decay rate are equal i.e. {\gamma }_d= {\gamma }_1={\gamma }_r, as depicted in Fig.4(a), we see that the qualitative behaviors of {\mathcal{F}}_1(\varphi ), HS{S}_1(\varphi ), entropy S_1, and success probability P_{S1} are the same. It means that the maxima and minima of {\mathcal{F}}_1(\varphi ), HS{S}_1(\varphi ), entropy S_1, and success probability P_{S1} are coincident. This result indicates that, when the entanglement between the system and environment, and the success probability of teleportation are maximum, the maximum extraction of encoded information can be obtained, that is, the best initial phase estimation occurs. Another significant result of the present model is that the qualitative behaviors of fidelity and mean fidelity are opposite to those of QFI, HSS, S, and P_S. That is, when the amount of information extraction (phase estimation) and entanglement is minimal, fidelity and average fidelity are maximum. This result is valid for other cases [Fig.4(b)−(d)], i.e., when {\gamma }_d>({\gamma }_1={\gamma }_r), {\gamma }_1> ({\gamma }_d and {\gamma }_r), and {\gamma }_r> ({\gamma }_d= {\gamma }_1), respectively. In addition, in Fig.4(b)−(d), we observe that, when one of the decay and dephasing rates is greater than the other rates, the oscillations of the qualitative behavior of {\mathcal{F}}_1(\varphi ), HS{S}_1(\varphi ), S_1, f_1, and f_1^{av} decreases with time, due to increasing decoherence effects of the environment. This leads to the suppression of the phase estimation, meaning that the extraction of information can be decreased. Recently, there has been a non-Markovianity witness in [87], that a flow of fidelity is defined as \mathrm{d}{f}_1/\mathrm{d}t, that if we have \mathrm{d}{f}_1/\mathrm{d}t>0 for some interval t, thus the time evolution of the system is non-Markovian. Therefore, it can be seen in Fig.4(a)−(d) that at some times, the non-Markovian behavior of the system dynamics can be obvious. As expressed in the introduction, this behavior occurs due to the backflow of information from the environment to the system. As suggested in Refs. [37, 49], a flow of QFI is determined as I_{\varphi }(t) =\mathrm{d}{\mathcal{F}}_1(\varphi (t) )/\mathrm{d}t, that if we have I_{\varphi }(t) >0 for some interval t, thus the time evolution is non-Markovian. Hence, the QFI can be employed as a witness of non-Markovianity of the dynamics. Besides, the flow of HSS is specified as \mathrm{d}HS{S}_1(\varphi (t))/\mathrm{d}t >0, which was recently proposed as a faithful witness of the non-Markovianity of the dynamics [37, 77, 88]. Thus, the non-Markovian dynamics of the system can be witnessed with the HSS, QFI, Fidelity, and average fidelity. Hence, we have established the non-Markovianity of the system dynamics at some time with the tools introduced in this paper.

4.1.2 Comparison between results in the single and two-qubit teleportation scenarios

Teleportations of higher than single qubit are of particular importance or practical experiments and their application in quantum communication. However, eventually, there are many problems in some models, such as the reduction of the quality of teleportation of quantum states [10]. Here, we investigate two-qubit teleportation based on a single semiconductor quantum dot influenced by decay and dephasing rates. Using Eqs. (7) and (17), we calculate, within the present model, the output state of the channel in two-qubit teleportation as

(24){\rho }_{out}^{TQ}(t)=\left(\begin{array}{cccc}{\rho }_{11}^{TQ}(t)& 0& 0& {\rho }_{14}^{TQ}(t)\\ 0& {\rho }_{22}^{TQ}(t)& {\rho }_{23}^{TQ}(t)& 0\\ 0& {\rho }_{32}^{TQ}(t)& {\rho }_{33}^{TQ}(t)& 0\\ {\rho }_{41}^{TQ}(t)& 0& 0& {\rho }_{44}^{TQ}(t)\end{array}\right),

where the non-vanish elements of the output state are given by

(25)\begin{array}{rl}& {\rho }_{11}^{TQ}(t)=\frac{1}{4}\cosh ({\gamma }_{r}t){\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^4\left(\frac{{\gamma }_{r}t}{2}\right),\\ & {\rho }_{14}^{TQ}(t)=\frac{4\sin (\theta )\cos (\varphi )\cos (\delta t)\cos (\mathrm{\Delta }t){\mathrm{e}}^{-t({\gamma }_d+{\gamma }_l-2{\gamma }_r)}}{{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4},\\ & {\rho }_{22}^{TQ}(t)=\frac{-\cos (\theta ){\left({\mathrm{e}}^{2{\gamma }_{r}t}-1\right)}^2+6{\mathrm{e}}^{2{\gamma }_{r}t}+{\mathrm{e}}^{4{\gamma }_{r}t}+1}{2{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4},\\ & {\rho }_{23}^{TQ}(t)=\frac{\sin (\theta ){\mathrm{e}}^{4{\gamma }_{r}t-\mathrm{i}\varphi }\left({\mathrm{e}}^{2\mathrm{i}\varphi -2t({\gamma }_d+{\gamma }_r-\mathrm{i}\mathrm{\Delta })}+{\mathrm{e}}^{2\mathrm{i}\varphi -2t({\gamma }_d+{\gamma }_r+\mathrm{i}\mathrm{\Delta })}+2{\mathrm{e}}^{-2t({\gamma }_d+{\gamma }_r)+2\mathrm{i}\varphi }+{\mathrm{e}}^{-2t({\gamma }_l+{\gamma }_r-\mathrm{i}\delta )}+{\mathrm{e}}^{-2t({\gamma }_l+{\gamma }_r+\mathrm{i}\delta )}+2{\mathrm{e}}^{-2t({\gamma }_l+{\gamma }_r)}\right)}{2{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4},\\ & {\rho }_{32}^{TQ}(t)=\frac{\sin (\theta ){\mathrm{e}}^{4{\gamma }_{r}t-\mathrm{i}\varphi }\left({\mathrm{e}}^{-2t({\gamma }_d+{\gamma }_r-\mathrm{i}\mathrm{\Delta })}+{\mathrm{e}}^{-2t({\gamma }_d+{\gamma }_r+\mathrm{i}\mathrm{\Delta })}+2{\mathrm{e}}^{-2t({\gamma }_d+{\gamma }_r)}+{\mathrm{e}}^{2\mathrm{i}\varphi -2t({\gamma }_l+{\gamma }_r-\mathrm{i}\delta )}+{\mathrm{e}}^{2\mathrm{i}\varphi -2t({\gamma }_l+{\gamma }_r+\mathrm{i}\delta )}+2{\mathrm{e}}^{-2t({\gamma }_l+{\gamma }_r)+2\mathrm{i}\varphi }\right)}{2{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4},\\ & {\rho }_{33}^{TQ}(t)=\frac{\cos (\theta )-2(\cos (\theta )-3){\mathrm{e}}^{2{\gamma }_{r}t}+(\cos (\theta )+1){\mathrm{e}}^{4{\gamma }_{r}t}+1}{2{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4},\\ & {\rho }_{41}^{TQ}(t)=\frac{4\sin (\theta )\cos (\varphi )\cos (\delta t)\cos (\mathrm{\Delta }t){\mathrm{e}}^{-t({\gamma }_d+{\gamma }_l-2{\gamma }_r)}}{{\left({\mathrm{e}}^{{\gamma }_{r}t}+1\right)}^4},\\ & {\rho }_{44}^{TQ}(t)=\frac{1}{4}\cosh ({\gamma }_{r}t){\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^4\left(\frac{{\gamma }_{r}t}{2}\right).\end{array}

Before starting the comparison between the quality of single and two-qubit teleportation scenarios, it is necessary to state that because of the cumbersome form of the expressions calculated for the two-qubit teleportation scheme, we will refrain from reporting them here.

In Fig.5, the comparison between the dynamics of fidelity and average fidelity in single and two-qubit teleportation is illustrated. It can be seen that because the fidelity and the average fidelity in the two-qubit teleportation are below the classical threshold (CT), the single-qubit teleportation in the present channel is of better quality. Moreover, the qualitative behaviors of fidelity and average fidelity in the two-qubit teleportation have more oscillations. The quality of quantum teleportation in two-qubit teleportation is lower than in single-qubit teleportation with respect to QFI, HSS, and entropy. Therefore, the single-qubit teleportation scheme is successful in the current model, but the two-qubit teleportation fails. This result is significant in practical quantum teleportation plans, so it should be considered for successful teleportation.

5 Conclusion

The examination of different models as a resource in the realization of quantum teleportation, besides being particularly attractive, can be very beneficial. In addition, the investigation of efficient schemes in quantum remote sensing is also of practical value. In this paper, we investigated theoretically the feasibility of quantum teleportation based on a single semiconductor quantum dot affected by pure dephasing through the biexciton cascade decay. We also studied the idea of quantum remote sensing in quantum teleportation influenced by pure dephasing. To check the quality of quantum teleportation, we employed the criteria of fidelity and average fidelity.

Specifically, we compared the quality of quantum teleportation in single- and two-qubit schemes and showed that, within the present model, single-qubit quantum teleportation has a quantum advantage. Finally, to investigate the dynamics of the system, we introduce important witnesses of the non-Markovian dynamics of the system, so that our results may solve outstanding problems in the realization of faithful quantum teleportation over a long time.

Often, Bob cannot be physically present at the location where a special estimation needs to occur due to security risks, unfavorable conditions, or logistical issues related to transporting the instrument. Even if Bob is at the desired location, the tool he has may lack the necessary accuracy, despite being connected to a server equipped with precision instruments. In such conditions, performing remote estimation is useful. The quantum estimation process is carried out by sensors whose correct design can greatly increase the accuracy of the measurement. For example, a quantum sensor can be according to a qubit that encodes information in the relative phase of its quantum state by interacting with a weak external field. The extracted information obtained by measuring the qubit can be very useful. Quantum phase estimation is one of the most significant concepts in quantum computing. For example, the way to prepare the initial state or important information can be deduced from the initial phase estimate. In this work, to estimate the initial phase of the system at the output state in quantum teleportation, we used the quantum Fisher information (QFI) and Hilbert−Schmidt speed (HSS) tools.

A very important result that can be widely used in quantum communication is the comparison between the relations of fidelity, average fidelity, HSS, QFI, and entropy, in both single- and two-qubit teleportation scenarios, which shows the superiority of single-qubit teleportation. Moreover, we explored the success probability of the single-qubit quantum teleportation and compared it with the qualitative behaviors of the other tools. Using the aforementioned tool, we evaluated the feasibility of a single- and two-qubit quantum teleportation, which can be very useful in quantum communication and computing. The quantum teleportation based on semiconductor quantum dots influenced by high decay and dephasing rates is disturbed. Furthermore, the quality of quantum teleportation based on the current model, decreases under low fine-structure splitting and high biexciton binding energy. We managed to achieve successful quantum teleportation in this article by adjusting the parameters of the system effectively based on the current model. Our demonstration highlights the superior quantum advantage of single-qubit teleportation.

Our results may solve outstanding problems in the realization of faithful quantum teleportation over a long time, for example, improving quantum remote sensing [89] in navigation [90], radar [91], lidar [92], imaging [93], antenna design [94], and information security [95].

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