A one-way quantum amplifier for long-distance quantum communication

Article
  • 5.6k Downloads

Abstract

In this paper, a model for single photon amplification based on cluster-state quantum computation is proposed. A rescaling of the probability amplitudes of a deteriorated qubit in favor of the one-photon component will define the amplifier’s gain. Unlike the heralded quantum amplifiers, the probabilistic success of the whole process will not depend on the successful detection of a heralding signal. Instead, the whole procedure will rely upon a single-qubit measurement, which is simpler compared to any two-qubit interaction gate in the heralded quantum amplifiers. The proposed model can be used as a qubit protector against propagation losses in long-distance quantum communication networks.

Keywords

Quantum communication Linear optics Noiseless linear amplifiers Cluster computation Entanglement 

1 Introduction

Information carriers in quantum optical networks will eventually suffer from transmission loss. A propagating single photon \(\vert {1}\rangle \) upon which the success of many quantum communication protocols relies, always ends up in a mixed state due to its interaction with the surrounding environment [1]. This resulting statistical mixture can be represented as \(\gamma _{t}\) \(\vert {1}\rangle \langle {1}\vert +(1-\gamma _{t})\vert {0}\rangle \langle {0}\vert \) where \(\vert {0}\rangle \langle {0}\vert \) means that the photon is totally lost, while \(\gamma _{t}\) is the channel transmission efficiency.

Amplifiers seem to be a reasonable solution. However, a successful, deterministic and noiseless amplification is not possible [2]. Besides, the laws of quantum physics prohibit us from creating exact replicas of our input qubit [3]. As a result, we will always be obligated to include a noise source in order to save the commutation relation between the creation and annihilation operators of our mode of interest \([a_{m},a_{n}^{\dagger }]=\delta _{mn}\).

An amplifying medium can always be regarded as a bath of 2-level atoms, where the larger population dwells in the excited state (population inversion). In this case, the noise source will be the spontaneously emitted photons, while stimulated emission will provide amplification. This limitation proved to be one of the major challenges in the face of long-distance quantum communication.

Alternatively, we can perform the whole process in a non-deterministic manner [4, 5, 6, 7]; this class of amplifiers is called noiseless linear amplifiers (NLA) where the success of the amplification process is heralded by a single-photon detection.

What makes NLAs really promising is that they can be realized using only linear optical elements which is a huge step towards actualizing efficient quantum communication with linear optics [8].

Another interesting idea that will define our model along side the NLA is cluster-state quantum computation [9, 10, 11].

Dissimilar to conventional circuit-based quantum computation [12], cluster-state quantum computing starts from an entanglement resource and then proceeds with single-qubit measurements performed in any desired Pauli-basis.

Quantum entanglement [13]—the workhorse of our proposed model— has been highly regarded as the defining characteristic of mostly all recent quantum communication protocols. Secure quantum communication [14, 15, 16, 17], quantum teleportation [18, 19], noiseless quantum amplifiers [4, 5, 6, 7] and quantum repeaters [20, 21, 22] are all among the new technologies that made possible thanks to this striking phenomenon.

Surprisingly, most of the aforementioned techniques are achieved using the simplest type of entanglement that can be created, this specific type of entanglement is called mode entanglement [23].

Using only a beamsplitter, mode entanglement can be attained successfully. Mode entanglement is a highly valuable resource and will play an important role when constructing our model.

These were the two main ideas that will shape our model . We will also focus our attention on a specific NLA model Ref. [5] and draw some comparisons concerning the gain curve. Then, we will prove that we can achieve similar results using our measurement-based model which is simpler and fault tolerant compared to the NLA.

So, this paper will be organized as follows: in Sect. 2, we will describe the underlying principles of our model, after that, in Sect. 3 we shall propose a setup and a possible application of our protocol, and then in the final section we will have our conclusion.

2 The one-way quantum amplifier

2.1 Concept

2.1.1 Background

The main goal of our proposed model is to redistribute the weightings of the probability amplitudes of a degraded single photon. This signal degradation is a direct consequence of the interaction between our single photon \(\hat{\rho }\) (System) and the propagation channel \(\mathcal {\hat{T}}\)(Environment) [24]. This propagation channel will be a map that takes states from one Hilbert space to another. Specifically, \(\mathcal {T}: \mathcal {B(\mathcal {H_{A}})} \rightarrow \mathcal {B(\mathcal {H_{B}})}\) [25] will be a map from the set of bounded linear operators on the Hilbert space \(\mathcal {H_{A}}\) to those residing in \(\mathcal {H_{B}}\) . So by adopting this formalism, we opted to represent our system in question by a density operator, which is the most convenient way to describe open quantum systems.

These density operators belong to \(\in \) \(\mathcal {B(H)}\) such that \(\mathcal {B(H)}\) is the set of bounded linear operators on the Hilbert space, where a linear operator \(\rho : \mathcal {V_{X}} \rightarrow \mathcal {V_{Y}}\) is bounded if there exists a constant \(K \ge 0\) such thatwhere the smallest possible value of K that can be used in (1) is called the operator norm.

Density operators are Hermitian (self-adjoint), positive semi-definite \(\hat{\rho } \ge 0\) with unit trace \(Tr(\hat{\rho }) = 1\). Therefore, a valid propagation channel is the one that maps density operators to valid density operators, as a result, our channel must have the following properties: Mapping Hermitian operators to Hermitian operators, trace preservation and complete positivity.

Under the previous requirements, these mathematical objects will be called CPTP maps (completely positive trace preserving maps).

In order to observe the action of the propagation channel on our system, we will describe the evolution of the whole state combining our system and the propagation channel \(\hat{\rho }_{\mathrm{SE}}\) as the following
$$\begin{aligned} \hat{\rho }_\mathrm{SE}(t)=\hat{U(t)}\hat{\rho }_\mathrm{S} \otimes \hat{\rho }_\mathrm{E}\hat{U^{\dagger (t)}} \end{aligned}$$
(2)
where \(\hat{\rho }_\mathrm{S} , \hat{\rho }_\mathrm{E}\) are the density matrices of the system and the channel (environment), respectively, while \(\hat{U(t)}\) is a unitary evolution.
A situation like this hinders our ability to gain maximum information about the exact dynamics of our system, leaving our single photon in a mixed state upon tracing out the environment degrees of freedom
$$\begin{aligned} \begin{aligned} \mathcal {T}(\hat{\rho }_\mathrm{s}) = Tr_{\mathrm{E}}\{\hat{U(t)}\hat{\rho }_{\mathrm{S}}\otimes \hat{\rho _{\mathrm{E}}} \hat{U^{\dagger }(t)\}} = \sum _{m} \mathcal {\hat{K}}_{m} {\hat{\rho }_\mathrm{s}} \mathcal {\hat{K}}_{m}^{\dagger } \end{aligned} \end{aligned}$$
(3)
such that \(\mathcal {\hat{K}}_{m}, \mathcal {\hat{K}}_{m}^{\dagger }\) are the famous Kraus operators, satisfying the completeness relation \(\sum _{m} \mathcal {\hat{K}}_{m}^{\dagger } \mathcal {\hat{K}}_{m} = \mathcal {\hat{I}}\), where in our particular case these Kraus operators will be \(\alpha \hat{\sigma _{x}}\) and \(\beta \mathcal {\hat{I}}\).
$$\begin{aligned} \mathcal {T}(\hat{\rho _{s}}) = \beta ^{2} \vert {1}\rangle \langle {1}\vert + \alpha ^{2} \hat{\sigma _{x}} \vert {1}\rangle \langle {1}\vert \hat{\sigma }^{\dagger }_{x} \end{aligned}$$
(4)
Then, our final state will be
$$\begin{aligned} \mathcal {T}(\hat{\rho _{s}}) = \alpha ^2 \vert {0}\rangle \langle {0}\vert +\beta ^2 \vert {1}\rangle \langle {1}\vert \end{aligned}$$
(5)
where \(\alpha ^2\) and \(\beta ^2\) are the vacuum and single-photon weightings, respectively, such that both are assumed to be real and obey the normalization condition \(|\alpha |^{2} + |\beta |^{2} = 1\). This previous formalism is what we call operator sum, or Kraus representation.
Fig. 1

The beamsplitter model simulating a noisy quantum channel

Therefore, we can always regard our system as a part of a larger composite one, such that upon tracing out the degrees of freedom of the other part of the composite system we get the density matrix description of our system. This other constitute of the composite system is usually called the environment. Thus, the resulting mixed, noise-corrupted state is always a direct consequence of this partial tracing.

We could have arrived at the same result, if we had used a beamsplitter model (Fig. 1) to simulate the previous dynamics [27, 28]. The quantum treatment of a beamsplitter accounts for the insertion of vacuum modes into our calculations, where these modes will flow through the unused port of our beamsplitter. The states of these vacuum modes are unchanged and uncorrelated, obeying the Born–Markov approximation.

Then, our single photon will either be captured by one of the environment modes or not, thus leaving the state of our system in exactly the same statistical mixture described before.

Mathematically speaking, this beamsplitter will represent a transfer matrix \(\begin{pmatrix} \alpha &{} \beta \\ -\beta &{} \alpha \end{pmatrix}\) that relates input modes to output ones, such that this matrix is unitary (since a beamsplitter is a passive element) with real entries1 and a unit determinant. Then, the output of this beamsplitter will be an entangled state which takes the following form \(\vert {\psi _\mathrm{SE}}\rangle = \alpha \vert {0_\mathrm{S}1_\mathrm{E}}\rangle + \beta \vert {1_\mathrm{S}0_\mathrm{E}}\rangle \), where this special type of entanglement is what we call mode entanglement. (We have to note that a maximally entangled state can be achieved by choosing \(\alpha = \beta =\frac{1}{\sqrt{2}}\)). After that, tracing over the environment complete orthonormal basis {ONB} will give us the same expression in Eq. (5).

Consequently, we can use this simple beamsplitter in Fig. 1 to efficiently model our noisy quantum channel. In conclusion, the entanglement built up between our system and environment will always be reflected in the impossibility of assigning a definite state for the system by itself.

Not only our beamsplitter model is capable of simulating efficiently a noisy quantum channel, but also it can be used to model an amplification medium [27], as the evolution of a single mode field encountering a gain medium is governed by the quantum Langevin equation which is a simple input–output relation plus a noise source [29, 30, 31], and as we have shown before, the beamsplitter model is capable of simulating successfully input–output relations, where the noise term will be accounted for through the beamsplitter dark port.

We now proceed with the second foundational step that characterizes our model, one-way quantum computation. One-way quantum computing (cluster-state computation or measurement-based quantum computation) is an alternative approach to the circuit scheme of quantum computation where it assumes the existence of a collection of qubits (vertices, such that all of them are in the \(\vert {+}\rangle \)) that are initially entangled with each other by the application of a controlled-Z operation (edges connecting these vertices) where this combination is what we call a cluster state, and then the computation proceeds as single-qubit measurements applied to qubits forming this cluster state. Some of these qubits are measured in particular basis, while others are measured in different sets. This powerful idea can simulate efficiently any quantum circuit.
Fig. 2

A simple qubit teleportation circuit [32, 33]

Fig. 3

A 2-qubit measurement-based cluster which is equivalent to the above circuit

To elaborate more on the previous idea, we will consider the simple protocol of teleporting a single qubit (Fig. 2) [33] as an illustrious example, and see how it can be simulated using a cluster-state model. As shown in Fig. 3, we first begin with a pair of qubits in the states \(\vert {\psi }\rangle \) 2 and \(\vert {+}\rangle \), respectively, and then we entangle them through the action of a controlled-Z gate where the output state will be \(\vert {\phi }\rangle _\mathrm{out} = \alpha \vert {0}\rangle \otimes \vert {+}\rangle +\beta \vert {1}\rangle \otimes \vert {-}\rangle \). After that, we perform a single-qubit operation on the first qubit, where this operation is just the composition of a Hadamard transformation and a Pauli-Z basis measurement. The result of this measurement will give us either a 0 or 1, and then depending on this result we either apply a Pauli correction to the whole output transformed state or not, this Pauli correction is simply a Pauli \(\sigma _{z}^{m}\) which also can be written as \(Z^{m}\).

So the output transformed state will be either \(\vert {\phi }\rangle _\mathrm{out}=\alpha \vert {+}\rangle + \beta \vert {-}\rangle \) or \(\vert {\phi }\rangle _\mathrm{out}=\alpha \vert {+}\rangle - \beta \vert {-}\rangle \) where the first expression is a consequence of a zero-measurement outcome, while the other indicates a measurement outcome that is equal to one, after that we apply a Hadamard transformation to transform our qubit back to the computational basis; then after applying the Pauli correction, both expressions will be exactly the same, and this completes our protocol.

Obviously, we would have came up with the same result if we had done a straightforward calculation using the circuit-based model in Fig. 2, thus proving equivalency between the two approaches.

One final remark, the evolution of qubits can always be modeled by arbitrary rotations on the Bloch sphere, where this fact can be absorbed into the previous model by composing a rotation operator along with the Hadamard and the measurement operators, such that this new composition will act on our system in exactly the same manner described before.

After gathering all the needed tools to tackle our main problem, we now proceed with describing our proposed model. As previously mentioned, our model will consider specifically Ref. [5] as a building block; then, we will introduce our modification in the language of one-way cluster computation.

2.1.2 Noiseless linear amplifiers (NLAs)

Fig. 4

A heralded single-photon amplifier [5]

Figure 4 describes the NLA that will be our reference model when building our one-way quantum amplifier.

A single photon in a mixed state is the input to this device. This statistical mixture is the result of the encountering between our single photon and the first beamsplitter, where such a situation is analogous to the propagation of a single photon through a communication channel, just like what we did in the previous section. So, our photon will be in the following mode entangled state \( \alpha a^{\dagger } \vert {0_{a}0_\mathrm{in}}\rangle +\beta \mathrm{in} ^{\dagger }\vert {0_{a}0_\mathrm{in}}\rangle \), then after tracing out the environment degrees of freedom the final state will be \({\hat{\rho }}_\mathrm{in} = \alpha ^{2}\vert {0}\rangle \langle {0}\vert + \beta ^{2} \hat{\mathrm{in}^{\dagger }} \vert {0}\rangle \langle {0}\vert \hat{\mathrm{in}}\).

After that, an auxiliary photon sent through the second beamsplitter which has a transmission coefficient of t will entangle the beamsplitter’s output modes \(\hat{c}\) and \(\hat{\mathrm{out}}\), and hence the output entangled state will be \(\vert {\psi _\mathrm{out}}\rangle = \sqrt{1-t} \hat{c^{\dagger }}\vert {0_{c}0_\mathrm{out}}\rangle + \sqrt{t} \hat{\mathrm{out}^{\dagger }}\vert {0_{c}0_\mathrm{out}}\rangle \). Then, a 50/50 beamsplitter will combine \(\hat{c}\) and \(\hat{\mathrm{in}}\) producing the modes \(\hat{d_{+}}=\frac{1}{\sqrt{2}}(\hat{c} + \hat{\mathrm{in}})\) and \(\hat{d_{-}}=\frac{1}{\sqrt{2}}(\hat{c} - \hat{\mathrm{in}})\). As a result, the overall state of the system will be
$$\begin{aligned} \begin{aligned} \vert {\theta }\rangle&= \alpha \sqrt{1-t} (a^{\dagger }c^{\dagger }\vert {0_{a}0_\mathrm{in}0_{c}0_\mathrm{out}}\rangle )_\mathrm{single \_ click}\\&\quad + \alpha \sqrt{t}(a^{\dagger } out^{\dagger }\vert {0_{a}0_\mathrm{in}0_{c}0_\mathrm{out}}\rangle )_\mathrm{no \_ clicks}\\&\quad + \beta \sqrt{1-t} (in^{\dagger }c^{\dagger }\vert {0_{a}0_\mathrm{in}0_{c}0_\mathrm{out}}\rangle )_\mathrm{double \_ clicks}\\&\quad + \beta \sqrt{t}(in^{\dagger }out^{\dagger }\vert {0_{a}0_\mathrm{in}0_{c}0_\mathrm{out}}\rangle )_\mathrm{single \_ click} \end{aligned} \end{aligned}$$
(6)
Using the previously stated mode relations, we can substitute \(\hat{d_{+}}\) 3 and \(\hat{d_{-}}\) for \(\hat{c}\) and \(\hat{\mathrm{in}}\). So, if the auxiliary and the input photon are indistinguishable,4 the detection of a single photon in one of the previous modes will act like a Bell-state measurement and will project the output state \(\hat{\mathrm{out}}\) to
$$\begin{aligned} \begin{aligned}&\frac{\alpha \sqrt{1-t}\vert {1_{a}0_\mathrm{out}}\rangle +\beta \sqrt{t}\vert {0_{a}1_\mathrm{out}}\rangle }{\sqrt{N}}\\&\quad \dfrac{1}{N}[(1-t)\alpha ^{2}\vert {0}\rangle \langle {0}\vert + t\beta ^{2}\hat{\mathrm{out}^{\dagger }}\vert {0}\rangle \langle {0}\vert \hat{\mathrm{out}}] \end{aligned} \end{aligned}$$
(7)
where \(N = (1-t)\alpha ^{2} + t\beta ^{2}\) is a normalization constant.5 This whole operation leads to a change in the weightings of both the vacuum and the single-photon components in favor of the single-photon component as can be seen from Eq. 7. The gain of the whole process identifies this change, where this gain will be defined as the ratio of the single-photon component weighting before and after the process
$$\begin{aligned} G = \frac{t}{t+\alpha ^{2}(1-2t)} \end{aligned}$$
(8)
We should note that when t takes the value \(\frac{1}{2}\), the gain G will be equal to 1, which means that the output state will be mapped to the input one; in another words, the whole process reduces to mere qubit teleportation. So noiseless amplification will take place when t is \({>}\frac{1}{2}\); consequently, G will be \({>}1\) and the desired amplification is achieved.

Now, we will describe how we can achieve similar results using our proposed cluster model.

2.1.3 Cluster-based linear amplifier

As mentioned earlier, cluster states are a set of entangled qubits (nodes) by the action of a controlled-Z gate (edges), where these nodes are initially in the \(\vert {+}\rangle \) state and this simply means that the ground state of our environment is the \(\vert {+}\rangle \), so our deteriorated photon that is corrupted with environment noise can be modeled using the two-node cluster in Fig. 5.

Again we should always be aware of the fact that any system can be considered as a part of a larger composite one; then by taking the partial trace over the degrees of freedom of the environment, we get a density matrix description of our system alone. Another important remark concerning cluster-state computing is that a cluster of unentangled \(\vert {+}\rangle \) states is an unchanging uncorrelated environment under the Born–Markov approximation, and then the process of entanglement imposed on these cluster nodes sets the stage for the realization of any type of computation. This specific role was performed in exactly the same manner by the beamsplitter in the previous section, where the beamsplitter coupled our input state with that of the vacuum. Thus, vacuum modes represented the environment in the beamsplitter model, while the unconnected \(\vert {+}\rangle \) states will represent the environment in the cluster one.

Now, we get back to our protocol, as we can see from Fig. 5, an arbitrary single photon is entangled with the superposition state \(\vert {+}\rangle \) by a controlled-Z gate such that the mode ‘in’ is the control, while the mode ‘a’ is the target. This entanglement is analogous to that which occurs during the propagation of a single photon through a noisy channel. So the total state of this cluster will be \(\vert {\psi _{1}}\rangle = \alpha \vert {0_\mathrm{in}0_{a}}\rangle + \beta \vert {1_\mathrm{in}1_{a}}\rangle \) where this state can be put in the standard cluster form by applying a Hadamard transformation to any of the two qubits. As a result, \(\hat{\rho _\mathrm{in}} = Tr_{a}[\hat{\rho _\mathrm{in}} \otimes \hat{\rho _{a}}] = \alpha ^{2}\vert {0}\rangle _\mathrm{in}\langle {0}\vert + \beta ^{2} \vert {1}\rangle _\mathrm{in}\langle {1}\vert \)

Next, we create a supplementary system which will also be a 2-node cluster state, as shown in Fig. 6
Fig. 5

Two-node cluster model of a deteriorated photon

Fig. 6

The supplementary system which is also a two-node cluster state

The above cluster state will be constructed by applying a controlled-Z operation between the following two qubits \(\vert {\chi }\rangle = \sqrt{1-t}\) 6 \(\vert {0_{c}}\rangle + \sqrt{t} \vert {1_{c}}\rangle \) and \(\vert {+}\rangle _\mathrm{out}\) such that ‘c’ is the control while ‘out’ is the target. So the overall output state will be \(\vert {\psi _{2}}\rangle = \sqrt{1-t} \vert {0_{c}0_\mathrm{out}}\rangle + \sqrt{t} \vert {1_{c}1_\mathrm{out}}\rangle \). Again this state can also be put in the standard form as suggested before.

Now we push our analysis one step further by combining the previous two cluster states in a single 4-qubit cluster as depicted in Fig. 7.
Fig. 7

The combined 4-qubit cluster state, where we applied a controlled-Z between the second and third qubits to achieve this graph

This graph is a direct consequence of the application of a controlled-Z between the qubits a and c, such that a is the control qubit, while c is the target one. As a result, we end up with the following 4-qubit entangled state which can be put in the standard form by simply applying a Hadamard transformation to the first and fourth qubits.
$$\begin{aligned} \begin{aligned} \vert {\lambda }\rangle&= \alpha \sqrt{1-t}\vert {0_\mathrm{in}0_{a}0_{c}0_\mathrm{out}}\rangle + \alpha \sqrt{t}\vert {0_\mathrm{in}0_{a}1_{c}1_\mathrm{out}}\rangle \\&\quad + \beta \sqrt{1-t}\vert {1_\mathrm{in}1_{a}0_{c}0_\mathrm{out}}\rangle - \beta \sqrt{t}\vert {1_\mathrm{in}1_{a}1_{c}1_\mathrm{out}}\rangle \end{aligned} \end{aligned}$$
(9)
After that we apply a Hadamard gate to qubits ‘a’ and ‘c’ in order to transform both qubits to the Pauli-X basis. Therefore, our state becomes
$$\begin{aligned} \begin{aligned} \vert {\lambda }\rangle&= \alpha \sqrt{1-t}\vert {0_\mathrm{in}+_{a}+_{c}0_\mathrm{out}}\rangle + \alpha \sqrt{t}\vert {0_\mathrm{in}+_{a}-_{c}1_\mathrm{out}}\rangle \\&\quad + \beta \sqrt{1-t}\vert {1_\mathrm{in}-_{a}+_{c}0_\mathrm{out}}\rangle - \beta \sqrt{t}\vert {1_\mathrm{in}-_{a}-_{c}1_\mathrm{out}}\rangle \end{aligned} \end{aligned}$$
(10)
Then, we perform an X-basis measurement7 on the qubits a and c; this measurement will remove the respective qubits from the cluster and will directly entangle their neighborhoods—the ‘in’ and ‘out’ qubits.

The previous argument is better set up under the stabilizer formalism [34, 35], where the manipulation of any cluster state is most efficient.

2.1.4 Stabilizer formalism

Suppose that \(\vert {{\psi }}\rangle \) is an arbitrary qubit state. A stabilizer G is a set of operators \(\hat{S_{i}}\) \(\in \) to the Pauli group,8 that leaves the state \(\vert {\psi }\rangle \) invariant, \(\hat{S_{i}}\vert {\psi }\rangle =\vert {\psi }\rangle \). When two operators \(\hat{S_{i}}\) and \(\hat{S_{j}}\) are members of a stabilizer, their products \(\hat{S_{i}}\) \(\hat{S_{j}}\) and \(\hat{S_{j}}\) \(\hat{S_{i}}\) are also members of the stabilizer, forming what we call an abelian group, such that all of the group members commute with each other \([\hat{S_{i}},\hat{S_{j}}]=0\). In the stabilizer formalism, there is a class of unitary operators called the Clifford operators. A Clifford operator is an operator \(\hat{U}\) that transforms a Pauli product of operators into another Pauli product, and hence the action of a Clifford operator on a stabilizer state can be described as follows
$$\begin{aligned} {\hat{U}}\vert {\psi }\rangle = {\hat{U}}{\hat{S_{i}}}\vert {\psi }\rangle = {\hat{U}}{\hat{S_{i}}}{\hat{U^{\dagger }}} {\hat{U}}\vert {\psi }\rangle ={{\hat{S_{i}}{}^{'}}}\vert {\psi }\rangle \end{aligned}$$
(11)
Since \(\hat{U}\) is a Clifford unitary operator, the group \(\{{\hat{S_{i}}{}^{'}}\}\) is also an abelian subgroup of the Pauli group.

The Clifford group consists of the following operators {\(H_{i}\),\(\phi _{i}\),\(CZ_{ij}\)}where the Hadamard and phase gates are single-qubit transformation, while the controlled-Z is a two-qubit entangling gate.

Conclusion: The Stabilizer formalism corresponds to the Heisenberg picture of quantum mechanics, where states are stationary while operators evolve with time.

An interesting question that poses itself right now is what are the stabilizer generators of a cluster state of N-qubits? A cluster without any edges \(\vert {+_{j}}\rangle \) has the following set of stabilizers \({S_{j}}={X_{j}}\).9

We need then to find stabilizers for less trivial clusters just like the one in Fig. 7.

Since \(\hat{S_{j}}\vert {+}\rangle = \vert {+}\rangle \), we can write \(\hat{S_{j}}\) as the following projector \(\vert {+}\rangle \langle {+}\vert \) which looks like a density operator (a rank-1 projector to be precise) in the Pauli X-basis. Having this in mind, plus the rule of evolution for density operators under unitary transformation,10 we can easily derive an expression for the evolution of our stabilizers under the action of the controlled-Z operator.

Therefore, the set of stabilizers of an N-qubit cluster state will be
$$\begin{aligned} {S_{j}} = {X_{j}} \prod _{k\in n(j)} {Z_{k}} \end{aligned}$$
(12)
where k belongs to the direct neighbors of our node.

After this brief introduction about Stabilizers, we return back to our model and see how we can use this formalism to describe our measurement procedure.

Using the aforementioned rules, the set of stabilizer operators defining our cluster in Fig. 7 will be as follows
$$\begin{aligned} S_\mathrm{in}&= X_\mathrm{in} Z_{a}\\ S_{a}&= Z_\mathrm{in} X_{a} Z_{c}\\ S_{c}&= Z_{a} X_{c} Z_\mathrm{out}\\ S_\mathrm{out}&= Z_{c} X_\mathrm{out} \end{aligned}$$
Then, after performing an X-basis measurement on qubits a and c, we get the following set of stabilizers as a consequence of some Clifford operation-based manipulations.
$$\begin{aligned} S_\mathrm{in}&= X_\mathrm{in} Z_\mathrm{out}\\ S_{a}&= X_{a} \\ S_{c}&= X_{c} \\ S_\mathrm{out}&= Z_\mathrm{in} X_\mathrm{out} \end{aligned}$$
As a result, our graph will look like the one depicted in Fig. 8.
Fig. 8

Our cluster state after performing X-basis measurement on qubits a and c

So, this output entangled state is either
$$\begin{aligned} \vert {\lambda _\mathrm{out}}\rangle = \frac{\alpha \sqrt{1-t}\vert {0_\mathrm{in}0_\mathrm{out}}\rangle - \beta \sqrt{t}\vert {1_\mathrm{in}1_\mathrm{out}}\rangle }{\sqrt{N}} \end{aligned}$$
(13)
or
$$\begin{aligned} \vert {\lambda _\mathrm{out}}\rangle = \frac{\alpha \sqrt{t}\vert {0_\mathrm{in}1_\mathrm{out}}\rangle + \beta \sqrt{1-t}\vert {1_\mathrm{in}0_\mathrm{out}}\rangle }{\sqrt{N}} \end{aligned}$$
(14)
where \(N = (1-t)\alpha ^{2}+t\beta ^{2}\) is a normalization constant. Obviously by inspecting the probability amplitudes, we can see that a successful amplification will only occur when we get similar outcomes upon measuring the ‘a’ and ‘c’ qubits. Therefore, Eq. 13 will be the equation defining our one-way quantum amplifier, accordingly our output state will be
$$\begin{aligned} \hat{{\rho }_\mathrm{out}} = \dfrac{1}{N}[(1-t)\alpha ^{2}\vert {0}\rangle \langle {0}\vert + t\beta ^{2}\hat{\mathrm{out}^{\dagger }}\vert {0}\rangle \langle {0}\vert \hat{\mathrm{out}}] \end{aligned}$$
(15)
with a gain equal to
$$\begin{aligned} G_\mathrm{cluster} = \frac{t}{t+\alpha ^{2}(1-2t)} \end{aligned}$$
(16)
which is similar to that of Eq. 8.

Another important point worth mentioning here is the fact that all the participating qubits in our one-way quantum amplifier are indistinguishable, thus satisfying the requirement set by the NLA. This specific attribute is very crucial to the success of the amplification protocol for both of the two approaches and also is considered to be an important security check [36].

One last remark, due to the symmetry of cluster states we get a similar expression for the input state \(\hat{\rho _\mathrm{in}}\) upon tracing out the output qubit. And this leads us to the conclusion that the act of measurement itself is a quantum operation and shall be formulated under the Kraus formalism as well.

2.2 Discussion

So far, we were able to recreate the results in Ref. [5] by using a cluster-based model, where we got a similar gain expression.

An important remark is the equivalence between the Bell-state measurement in Ref. [5] and our measurement technique. A Bell-state measurement in our protocol was realized by the composition of a controlled-Z based entanglement followed by a two X-basis measurement on the qubits ‘a’ and ‘c’ , where we were able to post-select an output state based on our measurement result, just like the circuit-based Bell-state measurement.

While a Bell-state measurement in Ref. [5] needed a photo-detector that was able to resolve photonic number states, all we needed in our model is single-qubit measurements, thus reducing the complexity and eliminating the need for two-qubit circuits during detection. Therefore, our model is more efficient when calculating the success probability \(P_\mathrm{success} = (\alpha ^{2}(1-t)+\beta ^{2}t) \eta _\mathrm{detector}\) as a function of the detector efficiency \(\eta _\mathrm{detector}\) , since we will not have to resolve photon-numbers to signify the success of our protocol.
Fig. 9

The gain curve for different values of \(\alpha ^{2}\)

We now return back to our gain expression and analyze it a bit further. The gain expression defining our amplifier is a function of ‘t’ for a fixed \(\alpha ^{2}\) as stated in Eq. 16, where ‘t’ is the transmission coefficient of the beamsplitter that created the superposition state \(\vert {\chi }\rangle = \sqrt{1-t}\vert {0}\rangle + \sqrt{t} \vert {1}\rangle \).

As shown in Fig. 9, when \(t = 1/2\) the gain G is equal to 1 regardless of the value of \(\alpha ^{2}\) and this means that our protocol reduces to mere qubit teleportation. In the case of a very noisy channel such that \(\alpha ^{2}\) is equal to 0.9, gain is achieved for values of \(t> 1/2\). Similarly, for other values of \(\alpha ^{2}\) included in Fig. 9 gain is achieved for the same values of t, except for the final case where \(\alpha ^{2}\) is equal to 0.1. In this specific case, we achieved a very small gain, since the weightings are already in favor of the single-photon component. Another important remark in Ref. [5] is that the gain expression will be modified to
$$\begin{aligned} G = \frac{t}{1-\alpha ^{2}t} \end{aligned}$$
(17)
in the case of a non-photon-number resolving detector, as a result, more restrictions will be applied upon gain calculations. For example, if \(\alpha ^{2}\) is equal to 0.5, amplification will only take place for t> 2/3. In our model, we will not face such difficulty since our model is based on single-qubit measurements, where there is no need for photon-number resolving detectors to distinguish between different output states.

Now we want to see how close the output of our amplifier to the original input mixed state. A suitable measure to estimate the distance between two different density operators is Fidelity [37]. Fidelity measures the probability of getting an output state that resembles exactly the input one.

For pure states, the fidelity will be \({{\varvec{F}}} = \langle {\psi }\vert _\mathrm{in}\hat{\rho _\mathrm{out}}\vert {\psi }\rangle _\mathrm{in}\), while for the convex set of mixed states \(\mathcal {M}_{N} := \{\hat{\rho } : \mathcal {H}_{N} \rightarrow \mathcal {H}_{N};\hat{\rho } = \hat{\rho }^{\dagger }, \hat{\rho } \ge 0, Tr\hat{\rho } = 1\}\) where any state is expressible as a convex combination of other states belonging to the same set, the fidelity will be
$$\begin{aligned} F(\hat{\rho _\mathrm{in}},\hat{\rho _\mathrm{out}}) = \left[ Tr|\sqrt{\hat{\rho _\mathrm{in}}} \sqrt{\hat{\rho _\mathrm{out}}}|\right] ^{2} \end{aligned}$$
(18)
So now we want to investigate the effect of the amplifier gain on the fidelity of the output states, the fidelity of our amplifier as a function of the gain will be
$$\begin{aligned} F = \left[ \frac{\sqrt{1-G+\alpha ^{2}G} \alpha ^{2} + \sqrt{\alpha ^{2}G} \beta ^{2}}{\alpha }\right] ^{2} \end{aligned}$$
(19)
As a result, we can plot the fidelity as a function of the gain provided that we have a fixed \(\alpha ^{2}\).

So by investigating Fig. 10, we can see that at unit gain all of the three curves coincide, such that we have a fidelity of 1, which is the case of qubit teleportation that we discussed before, while at zero gain the fidelity is equal to the corresponding \(\alpha ^{2}\).

Also we can see that we achieved the minimum fidelity that corresponds to a significant gain in the \(\alpha ^{2} = 0.5\) case, where we had equal vacuum and single-photon weightings to begin with. In this case, a gain of 2 totally eliminates the vacuum component from the output mixed state, as a result the fidelity as a measure of the distance between the two density operators will be 0.5, which makes perfect sense, since as we previously mentioned the input state is an equal mixture of the vacuum and single-photon states.
Fig. 10

The fidelity as a function of our amplifier gain

One final key feature of our proposed model yet the most significant is the ability of our device to correct errors. As we have shown earlier, the resulting output states upon the two X-basis measurement performed on the two middle qubits in our 4-node cluster are either
$$\begin{aligned} \vert {\lambda _\mathrm{out}}\rangle = \frac{\alpha \sqrt{1-t}\vert {0_\mathrm{in}0_\mathrm{out}}\rangle - \beta \sqrt{t}\vert {1_\mathrm{in}1_\mathrm{out}}\rangle }{\sqrt{N}} \end{aligned}$$
or
$$\begin{aligned} \vert {\lambda _\mathrm{out}}\rangle = \frac{\alpha \sqrt{t}\vert {0_\mathrm{in}1_\mathrm{out}}\rangle + \beta \sqrt{1-t}\vert {1_\mathrm{in}0_\mathrm{out}}\rangle }{\sqrt{N}} \end{aligned}$$
The second expression that signals the failure of the amplification process is the consequence of getting different outcomes when measuring qubits ‘a’ and ‘c’ (the two middle qubits). By then we can apply a pauli correction to the first qubit, particularly a Pauli X and a correction factor when measuring the second one (to put the probability amplitude in the right order), which will act like a quantum error correction ancilla [38]. As a result, we get the following corrected state
$$\begin{aligned} \vert {\lambda _\mathrm{out}}\rangle = \frac{\alpha \sqrt{t}\hat{\mathcal {X}_{1}}\vert {0_\mathrm{in}1_\mathrm{out}}\rangle + \beta \sqrt{1-t}\hat{\mathcal {X}_{1}}\vert {1_\mathrm{in}0_\mathrm{out}}\rangle }{\sqrt{N}} \end{aligned}$$
This feature is not possible in the NLA since we can not operate on an annihilated vacuum state. Therefore, our model is fault tolerant compared with NLAs, which justifies the increased number of used photons during the amplification process.

In the upcoming section, we will discuss a possible realization of our device; this proposal will only be composed of linear optical elements, since modern quantum communication protocols are currently embracing this trend due to its high potentiality and simplicity.

3 Setup

So far, all of our calculations and analyses were made under the assumption that our qubits are represented by photon-number kets, such that \(\vert {0}\rangle \) represents a ground state, while \(\vert {1}\rangle \) represents an excited state. But since detectors cannot detect zero photon states and we can not operate on the vacuum, we will translate all of our previous calculations to polarization space where all the manipulations of the last section can be realized physically.

A single photon with an arbitrary polarization angle \(\theta \) with respect to the horizontal direction can be expanded as a linear combination (superposition) of a vertically polarized photon \(\vert {V}\rangle \) and a horizontally polarized one \(\vert {H}\rangle \) [39], such that \(\vert {V}\rangle \) and \(\vert {H}\rangle \) are the basis of our polarization space, thus \(\langle H |H \rangle = \langle V |V\rangle \) = 1 while \(\langle H |V \rangle = \langle V |H\rangle \) = 0. So, any arbitrary polarization state can be written as
$$\begin{aligned} \vert {\psi _{\theta }}\rangle = \alpha \vert {H}\rangle + \beta \vert {V}\rangle \end{aligned}$$
(20)
where \(\alpha \) = \(\cos {\theta }\), while \(\beta \) = \(\sin {\theta }\).

But the polarization state of our propagating photon cannot be simply described in the previous manner, since the transmission of photons through optical fibers is always vulnerable to random polarization transformation. Consequently, the polarization state of our propagating photon is best described as a statistical mixture \(\hat{\rho }_\mathrm{polarization} = \sum _{i}{P_{i}\hat{\rho _{i}}}\). Now, imagine this randomly polarized photon entering a polarizing beamsplitter (PBS) where such a device transmits horizontally polarized photons while reflects vertically polarized ones, so the output polarization state after encountering the PBS is either horizontal or vertical with respect to the desired output port; therefore, at either outputs of the PBS we can expect a pure polarization state, by then we can place a polarization rotator (wave plate) at the horizontal polarization output port, for example, to create a superposition state just like the one in Eq. 20.

So, the composition of the PBS and the polarization rotator will be the constitute of our polarization encoder as shown in Fig. 11.
Fig. 11

Encoding polarization information into single photons

Therefore, all of our qubits that are taking part in our protocol can be translated into polarization space in exactly the same manner.

We can always modify the parameters of our linear optical devices (beamsplitters, polarizers, etc., ) to recreate the probability amplitudes of the linear cluster in Fig. 7. Assuming we did so, we will resume our analysis using the same probability amplitudes of the previous section. Now to create the two-node cluster state in Fig. 5, we will use the polarizing beamsplitter as an entangling device [40, 41], but we have to encode first each qubit in our cluster into a two-qubit state \(\alpha \vert {H}\rangle +\beta \vert {V}\rangle \) \(\rightarrow \) \(\alpha \vert {HH}\rangle +\beta \vert {VV}\rangle \) 11 [42]12 such that one to be detected, while the other qubit will complete the computational steps. So our state before the PBS will be the following tensor product \((\alpha \vert {H_\mathrm{in}H_\mathrm{in}}\rangle + \beta \vert {V_\mathrm{in}V_\mathrm{in}}\rangle ) \otimes \frac{1}{\sqrt{2}}(\vert {H_{a}H_{a}}\rangle + \vert {V_{a}V_{a}}\rangle )\), and as we said before the polarizing beamsplitter transmits horizontally polarized photons while reflects vertically polarized ones. Then, a coincidence count will only happen if both input modes have the same polarization, thus detecting a pair and leaving the other in an entangled superposition to complete the computation; therefore, the PBS performs a projective measurement described by the operator \(P = \vert {HH}\rangle \langle {HH}\vert + \vert {VV}\rangle \langle {VV}\vert \) on the tensor product state of the whole system. In another words, the action of the PBS is to post-select a Hilbert space where only equal polarization states can exist.

Particularly, a coincidence count assigns a definite position to the detected photons, while leaving the polarization (spin angular momentum) totally uncertain, thus making use of the famous uncertainty principle governing those two conjugate observables.

Then, after post-selection our state becomes \(\alpha \vert {H_\mathrm{in}H_{a}}\rangle + \beta \vert {V_\mathrm{in}V_{a}}\rangle \), such that our input photon will be in the following mixed state \(\hat{\rho }_\mathrm{in} = \alpha ^{2}\vert {H}\rangle \langle {H}\vert +\beta ^{2}\vert {V}\rangle \langle {V}\vert \). Figure 12 describes the previous analysis.
Fig. 12

Creation of 2-node cluster state

In order to get our state in the standard cluster form, we can simply apply a Hadamard transformation to any of the two qubits.

Similarly, we create the supplementary system in Fig. 6, with the following output state \(\sqrt{1-t}\vert {H_{c}H_\mathrm{out}}\rangle +\sqrt{t}\vert {V_{c}V_\mathrm{out}}\rangle \). Now, we want to assemble our full system in Fig. 7. So, we again use a polarizing beamsplitter to combine the modes ‘a’ and ‘c’ to get the following 4-qubit state13 [43]
$$\begin{aligned} \begin{aligned} \vert {\lambda }\rangle = \alpha \sqrt{1-t}\vert {H_\mathrm{in}H_{a}H_{c}H_\mathrm{out}}\rangle + \alpha \sqrt{t}\vert {H_\mathrm{in}H_{a}V_{c}V_\mathrm{out}}\rangle \\ +\, \beta \sqrt{1-t}\vert {V_\mathrm{in}V_{a}H_{c}H_\mathrm{out}}\rangle - \beta \sqrt{t}\vert {V_\mathrm{in}V_{a}V_{c}V_\mathrm{out}}\rangle \end{aligned} \end{aligned}$$
(21)
after that, a coincidence count will act like a bell-state measurement that gives the desired amplified output
$$\begin{aligned} \vert {\lambda _\mathrm{out}}\rangle= & {} \frac{\alpha \sqrt{1-t}\vert {H_\mathrm{in}H_\mathrm{out}}\rangle + \beta \sqrt{t}\vert {V_\mathrm{in}V_\mathrm{out}}\rangle }{\sqrt{N}} \end{aligned}$$
(22)
$$\begin{aligned} \hat{\rho _\mathrm{out}}= & {} \dfrac{1}{N}[(1-t)\alpha ^{2}\vert {H}\rangle \langle {H}\vert + t\beta ^{2}\vert {V}\rangle \langle {V}\vert ] \end{aligned}$$
(23)
where the very act of projective measurement will include a decoding map that takes us back to photon-number basis. The full setup is shown in Fig. 13.
Fig. 13

One-way quantum amplifier where each ‘polarization encoding’ block outputs the superposition state \(\vert {HH}\rangle +\vert {VV}\rangle \) with a corresponding specific probability amplitudes, thus playing the same role of the EPR sources in Ref. [43]. So a setup with EPR sources instead should give similar results

Fig. 14

One-way quantum amplification as a full communication system [44]

Abstractly speaking, we can reinterpret the whole amplification process as shown in Fig. 14 such that we first encode our input mixed state into polarization basis, and then the formation of the 4-node cluster state by entangling our system with the supplementary one is equivalent to the entrance of a single-qubit mixed state into an amplification channel, where the amplifier will act like a map that takes deteriorated inputs to amplified outputs. Finally, our projective measurement will be our decoding map that produces our amplified output in the number basis.
Fig. 15

The role of the one-way quantum amplifier in a quantum repeater network, where it will be used to increase the distance covered by heralding signals

This was a suggested implementation based on linear optical elements to realize our protocol. We should also note that this implementation will be limited by the scope of current technology such as single-photon sources, linear optical elements and precise photon counters.

4 Application

We now suggest a possible application of our protocol. Our device can be implemented in quantum repeater networks to increase the separating distance between repeater nodes, thus allowing us to distribute entanglement over a larger distance, since the success of these protocols depend on the successful detection of a heralding signal that travels from the repeater nodes to a detecting mid-way station.

The probability of finding a photon after propagating a distance of \(L_{0}\) through a communication link is given by \(P(L_{0}) = e^{-L_{0}/2L_\mathrm{att}}\) [21], where we considered a fiber communication link of 0.2 db/km attenuation, while the corresponding attenuation length will be \(L_\mathrm{att}\) = 22 km. So a propagation length of 88 km which is 4 times the attenuation length gives rise to a success probability of \(\approx \) 0.1; this means that \(\beta ^{2}\) will be equal to 0.1, while \(\alpha ^{2}\) will be 0.9. So by another quick look at the gain curve of Fig. 9, we can see that the case in hands was exactly considered in the aforementioned figure, where we were able to achieve gain for values of t that are greater than 1/2. Consequently, we can implement our device after the previously stated distance to regenerate our heralding signal, thus increasing the distance between the two repeater nodes. Figure 15 illustrates the idea.

Of course an accurate treatment should also consider other important parameters such as: the success probability of the linear optical elements participating in our protocol, besides the successful generation of a heralding signal in first place.

5 Conclusion

In this paper, we have presented a model of a linear optical qubit amplifier built upon an entanglement resource often called a cluster state. Our device showed an ability to change the ratio between vacuum and single-photon components, thus introducing a gain factor. Unlike other approaches, our model depends on single-qubit measurements which is way simpler than any two-qubit gate interactions used by the other NLA models, another important advantage of our model is that it creates entanglement first before proceeding with any computational step, which is more stable than other protocols that generate entanglement during the calculation process. Our device also showed a great ability to tolerate errors, where such a feature is not possible in the NLA since we cannot operate on vacuum states. To make our simple idea more rigorous, we aided our model with some proof of concept calculations; we first began with calculating the gain expression that defines our device where we found that our protocol reduces to mere qubit teleportation when G = 1, while for the values of \(G > 1\) our model works as an amplifier. We also used the fidelity measure to calculate the distance between our output state and the input one, and the result was also consistent with the previous conclusion, where we got a unit fidelity in the case of \(G = 1\) which simply means that our output state is just a teleported version of the input one.

A possible direct application of our device is to protect heralding signals against transmission losses in quantum repeater networks. Transmission losses always limit the range between repeater nodes, so we can use our device to extend the distance between repeater nodes since it enhances the weighting of the single-photon component over the vacuum one, thus allowing the heralding signal to travel further than the expected distance.

Footnotes

  1. 1.

    This requirement is specific to our analysis, where in general the transfer matrix is Hermitian.

  2. 2.

    Where \(\vert {\psi }\rangle \) = \(\alpha \vert {0}\rangle +\beta \vert {1}\rangle \) is an arbitrary qubit state.

  3. 3.

    The \(\frac{1}{\sqrt{2}}\)factor will be taken care of when normalizing the whole state.

  4. 4.

    This indistinguishability is crucial to the whole amplification process.

  5. 5.

    Since the detection of one photon in either \(\hat{d_{+}}\) or \(\hat{d_{-}}\) leads to Eq. 7, the probability to successfully produce that state is given by twice its norm.

  6. 6.

    All of the probability amplitudes can be adjusted as a function of the linear optical elements that will be used during experimentation. This will be clear in Sect. 3.

  7. 7.

    \(\vert {+}\rangle \) and \(\vert {-}\rangle \) are the eigenstates of the Pauli-X operator with eigenvalues +1, −1, respectively.

  8. 8.

    The Pauli group is the group of the famous Pauli \(\sigma \) operators.

  9. 9.

    Where X is the Pauli \(\sigma _{x}, \sigma _{x} \vert {+}\rangle =\vert {+}\rangle \).

  10. 10.

    \(\rho ^{\prime }={U}\rho {U^{\dagger }}\).

  11. 11.

    Where this step to be included in the ‘polarization encoding’ block in our diagrams from now on.

  12. 12.

    This simple encoder consists of: an input in a general polarization state to be encoded, a PBS and an entanglement resource in the state \(\vert {\phi ^{+}}\rangle \). At the PBS, the general input state will be mixed with one member of the entangled resource, and then the detection of exactly one photon at the output of the PBS means that the other two are exiting the device.

  13. 13.

    Again this can be put in the standard form by applying a Hadamard to qubits 1 and 4.

References

  1. 1.
    Preskill, J.: Physics 219. Lecture notes. http://www.theory.caltech.edu/people/preskill/ph229/
  2. 2.
    Caves, C.M.: Quantum limits on noise in linear amplifiers. Phys. Rev. D 26, 1817 (1982)ADSCrossRefGoogle Scholar
  3. 3.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)ADSCrossRefGoogle Scholar
  4. 4.
    Ralph, T.C., Lund, A.P.: Nondeterministic noiseless linear amplification of quantum systems. In: Lvovsky, A. (eds.) Quantum Communication Measurement and Computing Proceedings of 9th International Conference, pp. 155–160 (2009)Google Scholar
  5. 5.
    Osorio, C.I., Bruno, N., Sangouard, N., Zbinden, H., Gisin, N., Thew, R.T.: Heralded photon amplification for quantum communication. Phys. Rev. A 86, 023815 (2012)ADSCrossRefGoogle Scholar
  6. 6.
    Jeffers, J.: Nondeterministic amplifier for two-photon superpositions. Phys. Rev. A 82, 063828 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Meyer-Scott, E., Bula, M., Bartkiewicz, K., Ĉernoch, A., Soubusta, J., Jennewein, T., Lemr, K.: Entanglement-based linear-optical qubit amplifier. Phys. Rev. A 88, 012327 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    Knill, E., Laflamme, R., Milburn, G.J.: A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001)ADSMATHCrossRefGoogle Scholar
  9. 9.
    Raussendorf, R., Briegel, H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    Browne, D.E., Briegel, H. J.: One-way quantum computation—a tutorial introduction. arXiv:quant-ph/0603226
  11. 11.
    Nielsen, M. A.: Cluster-state quantum computation. arXiv:quant-ph/0504097
  12. 12.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information, 1st edn. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  13. 13.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSMathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Bennett, C.H., Brassard, G., Mermin, N.D.: Quantum cryptography without Bells theorem. Phys. Rev. Lett. 68, 557 (1992)ADSMathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ekert, A.K.: Quantum cryptography based on Bells theorem. Phys. Rev. Lett. 67, 661 (1991)ADSMathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Zbinden, H., Bechmann-Pasquinucci, H., Gisin, N., Ribordy, G.: Quantum cryptography. Appl. Phys. B 67, 743748 (1998)CrossRefGoogle Scholar
  17. 17.
    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002)ADSCrossRefGoogle Scholar
  18. 18.
    Bennett, C.H., Brassard, G., Crpeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)ADSMathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Bouwmeester, D., Pan, J.-W., Mattle, K., Eibl, M., Weinfurter, H., Zeilinger, A.: Experimental quantum teleportation. Nature 390, 575–579 (1997)ADSCrossRefGoogle Scholar
  20. 20.
    Duan, L.-M., Lukin, M.D., Cirac, J.I., Zoller, P.: Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413–418 (2001)ADSCrossRefGoogle Scholar
  21. 21.
    Sangouard, N., Simon, C., de Riedmatten, H., Gisin, N.: Quantum repeaters based on atomic ensembles and linear optics. Rev. Mod. Phys. 83, 33 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    Azuma, K., Tamaki, K., Lo, H.-K.: All-photonic quantum repeaters. Nat. Commun. 6, Article number: 6787 (2015)Google Scholar
  23. 23.
    Gerry, C., Knight, P.: Introductory Quantum Optics, Chapter 6, 1st edn. Cambridge University Press, Cambridge (2005)Google Scholar
  24. 24.
    Barnum, H., Nielsen, M.A., Schumacher, B.: Information transmission through a noisy quantum channel. Phys. Rev. A 57, 4153 (1997)ADSCrossRefGoogle Scholar
  25. 25.
    Gupta, V.P., Mandayam, P., Sunder, V.S.: The Functional Analysis of Quantum Information Theory. Springer, Berlin (2015)MATHCrossRefGoogle Scholar
  26. 26.
    Christensen, O.: Functions, Spaces, and Expansions. Birkhuser Mathematics, Basel (2010)MATHCrossRefGoogle Scholar
  27. 27.
    Desurvire, E.: A three-dimensional quantum vacuum-noise/signal beamsplitter model for nonideal linear optical amplifiers. Opt. Fiber Technol. 5(1), 81 (1999)ADSGoogle Scholar
  28. 28.
    Ketterle, W.: 8.422 Atomic and Optical Physics II, Spring 2013 (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu/courses/physics/8-422-atomic-and-optical-physics-ii-spring-2013/
  29. 29.
    Lambropoulos, P., Petrosyan, D.: Fundamentals of Quantum Optics and Quantum Information. Springer, Berlin (2007)Google Scholar
  30. 30.
    Agarwal, G.S.: Quantum Optics, Chapter 10, 1st edn. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  31. 31.
    Deutsch, I.H.: Physics 581 Quantum Optics II. http://info.phys.unm.edu/~ideutsch/Classes/Phys581F16/index.htm
  32. 32.
    Eastin, B., Flammia, S.T.: Q-circuit tutorial. arXiv:quant-ph/0406003v2 (This diagram was plotted using this tool)
  33. 33.
    Kok, P., Lovett, B.W.: Introduction to Optical Quantum Information Processing. Cambridge University Press, Cambridge (2010)MATHCrossRefGoogle Scholar
  34. 34.
    Gottesman, D.: The Heisenberg Representation of Quantum Computers. arXiv:quant-ph/9807006
  35. 35.
    Gottesman, D.: Stabilizer Codes and Quantum Error Correction. arXiv:quant-ph/9705052
  36. 36.
    Korzh, B., et al.: Provably secure and practical quantum key distribution over 307 km of optical fibre. Nat. Photon. 9, 163168 (2015)CrossRefGoogle Scholar
  37. 37.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41(12), 2315–2323 (1994)ADSMathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Gottesman, D.: Quantum Error Correction QECC2007. Course http://perimeterinstitute.ca/personal/dgottesman/QECC2007/
  39. 39.
    Vazirani, U.: CS294-2: quantum computation. Lecture notes. https://people.eecs.berkeley.edu/~vazirani/s09quantum.html
  40. 40.
    Chao-Yang, L., Zhou, X.-Q., Ghne, O., Gao, W.-B., Zhang, J., Yuan, Z.-S., Goebe, A., Yang, T., Pan, J.-W.: Experimental entanglement of six photons in graph states. Nat. Phys. 3, 91–95 (2007)CrossRefGoogle Scholar
  41. 41.
    Browne, D.E., Rudolph, T.: Resource-efficient linear optical quantum computation. Phys. Rev. Lett. 95, 010501 (2005)ADSCrossRefGoogle Scholar
  42. 42.
    Pittman, T.B., Jacobs, B.C., Franson, J.D.: Probabilistic Quantum Encoder for Single-Photon Qubits. arXiv:quant-ph/0312097
  43. 43.
    Walther, P., Resch, K.J., Rudolph, T., Schenck, E., Weinfurter, H., Vedral, V., Aspelmeyer, M., Zeilinger, A.: Experimental one-way quantum computing. Nature 434, 169–176 (2005)ADSCrossRefGoogle Scholar
  44. 44.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Faculty of EngineeringAin-Shams UniversityCairoEgypt

Personalised recommendations