## Abstract

Based on the circuit including linear optical elements, a fault-tolerant distribution of GHZ states against collective noise among three parties is proposed. Additionally, two controlled DSQC protocols using the shared GHZ states as quantum channels are also presented under the charge of the controller. The first controlled DSQC protocol applies single parity analysis based on weak cross-Kerr nonlinearities. The receiver Bob performs single-photon measurement to obtain the secret information after the outcome publication of the single parity analysis executed by the sender Alice. The second protocol applies dense coding to double information transmission capacity, and the double parity analyses based on weak cross-Kerr nonlinearities are performed to obtain the secret information.

© 2017 Optical Society of America

## 1. Introduction

Combining the techniques of classical cryptography and quantum mechanics, quantum cryptography [1] is one of the most striking and charming developments in the communication field. Employing the principle of quantum mechanics, the legitimate participants can communicate with each other in a private and secret way.

Referring to classical key distribution, quantum key distribution (QKD) was proposed and has made a great progress. Due to the different generation modes of the secret key, there are two kinds of quantum key distribution. One is the random quantum key distribution, such as the protocols of the individual information carriers (BB84-QKD protocol [2] and B92-QKD protocol [3]) and the protocols using quantum entanglement (E91-QKD protocol [4] and BBM92-QKD protocol [5]). The other is deterministic quantum key distribution, for instance, the protocol using order rearrangement of the photons [6].

Long and Liu proposed another kind of quantum communication methods [7], quantum secure deterministic communication (QSDC), which is actually the direct quantum communication without using the key, and the secret information are sent directly through the quantum channel. Following this idea, Deng et al. presented two QSDC protocols using the methods of two-step [8] and one-time pad [9] respectively. Recently, the entanglement based QSDC protocols [7] and the two-step QSDC protocol [8] have been experimentally demonstrated [10]. And the one-time pad QSDC protocol [9] is experimentally realized [11] in a noisy and lossy environment.

Deterministic secure quantum communication (DSQC) can send information deterministically and be read out by the receiver after receiving another classical communication from the sender. DSQC is actually a variant of deterministic QKD. Normally, one first establishes the key, and then encrypts the information into ciphertext, and sends the ciphertext through a classical communication. In DSQC, one can send the ciphertext through the quantum channel first, and if the transmission is secure, then the key can be sent through a classical channel to the receiver. Essentially, DSQC is a combined process of deterministic QKD plus a classical communication [12–16].

In some scenarios, information transmission and communication between two legitimate participants should be in the charge of a controller. In different physical environment, a large amount of controlled deterministic secure quantum communication (Controlled DSQC) protocols were proposed [17–20]. As to the realization of Controlled DSQC, entanglement is a useful resource. Recently, there are many researches about the entanglement generation [21–25], the entanglement concentration [26], the entanglement distribution [26] and transmission [21], and the applications of entanglement [27], such as teleportation [28], quantum state sharing [29], quantum operation sharing [30], and hyperentanglement applications [27].

Noise does great harm to communication. It reduces the accuracy and security of the communication protocols. There are many useful methods to alleviate the noise effect, such as error correction [31], entanglement concentration [32–34], and self-stabilizing algorithm [35]. In practice, if we select photons as information carriers and the free space or the optical fiber as the transmitting paths of photons, the collective noise channel will be a reasonable assumption, which means that the transmitted photons are sufficient near to each other in the space or time interval so that the noise can not judge their difference and the photons suffer from the same noise.

Based on this reasonable assumption and the corresponding elaborate processing and operations, some communication protocols can be realized [36–39].

Taking the collective noise into account, we propose two controlled DSQC protocols exploiting GHZ states in the optical systems. At first, the secure distribution is presented to tolerantly distribute two photons of three-photon GHZ states from the controller to two legitimate participants via collective noise channels. Based on weak cross-Kerr nonlinearities, two explicit controlled DSQC protocols with GHZ states employing single parity analysis and double parity analyses are proposed.

## 2. Controlled DSQC protocols based on fault-tolerant distribution of GHZ states

It needs to note that the realization of two controlled DSQC protocols are based on the secure distribution of GHZ states. Based on the secure distribution of GHZ states, Hillery et al. [29] presented the first secret sharing protocol, which is considered in the ideal environment. Here we consider the secure distribution of the quantum channel comprised of GHZ states against collective noise.

#### 2.1. The secure distribution of GHZ states as the quantum channel

Suppose that the controller Charlie possesses GHZ states expressed as

*H*〉 and |

*V*〉 denote the horizontal polarization mode and vertical polarization mode of the photons.

Illustrated in Fig. 1, Charlie distributes photon *A _{i}* and photon

*B*from each GHZ polarization-entangled state to two communicators Alice and Bob through path

_{i}*P*(channel

_{A}*C*

_{A1}and channel

*C*

_{A2}) and path

*P*(channel

_{B}*C*

_{B1}and channel

*C*

_{B2}) respectively.

Here, we consider the collective noise channel. Under the assumption of collective noise, the photon *A _{i}* sent to Alice via the channel

*C*

_{A1}and the channel

*C*

_{A2}suffers from the identical noise, and the photon

*B*sent to Bob via the channel

_{i}*C*

_{B1}and the channel

*C*

_{B2}suffers from the identical noise respectively.

For the requirement of communication security and lesser communication cost, communication sequence is transmitted by photon blocks [7, 8]. If one photon block encounters trouble, it will be discarded. However, its failure can not affect other blocks. Furthermore, before sending *A*-sequence (is composed of photons *A _{i}*) and

*B*-sequence (is composed of photons

*B*), Charlie changes the order of the transmitted photons [6,14,15], which can also be used to prevent from the Toffoli gate attack [40–42] and guarantee his controlled function.

_{i}After the photon *A _{i}* and the photon

*B*pass through polarization beam splitters, PBS

_{i}_{A1}and PBS

_{B1}, the GHZ state to be distributed can be denoted as

*C*

_{A1}

*C*

_{B1}〉 and |

*C*

_{A2}

*C*

_{B2}〉 stand for the spatial modes of the photons (

*A*,

_{i}*B*), which are corresponding to their polarization modes |

_{i}*HH*〉

_{Ai,Bi}and |

*VV*〉

_{Ai,Bi}.

To articulate the fault-tolerant distribution, here we adopt the representation of the product state of the polarization modes and the spatial modes of the photons (*A _{i}*,

*B*). As for the photon

_{i}*C*, we express their states by only using the polarization modes, that is, |

_{i}*H*〉

_{Ci}and |

*V*〉

_{Ci}.

Half wave plates ${\text{HWP}45}_{A11}^{\xb0}$ and ${\text{HWP}45}_{B11}^{\xb0}$ functioning as a *NOT* gate change the state denoted as Eq. (2) to

Before entering into the collective noise channel, the initial state of the whole system is

Suffered from the collective noise, the polarization mode of the transmitted photons *A _{i}*,

*B*evolves as

_{i}*α*(

_{klmn}*k*,

*l*,

*m*,

*n*= 1 or 2) denote the noise parameters which are only corresponding to the different paths (

*P*,

_{A}*P*), rather than the different channels ({

_{B}*C*

_{A1},

*C*

_{A2}}, {

*C*

_{B1},

*C*

_{B2}}) in the same path, and we adopt the symbols |1〉 ≡ |

*H*〉, |2〉 ≡ |

*V*〉 for simplification.

Leaving from the collective noise channels, the photon *A _{i}* and the photon

*B*are allowed to pass through ${\text{HWP}45}_{x12}^{\xb0}$, and their state evolves as

_{i}Subsequently, the photon *A _{i}* and the photon

*B*go through PBS

_{i}_{x2},

And then, the photon *A _{i}* and the photon

*B*get through ${\text{HWP}45}_{{x}^{\prime}2}^{\xb0}$ (

_{i}*x*=

*A*,

*B*), and the state can be denoted as

Finally, Alice and Bob obtain the photon *A _{i}* and the photon

*B*at the output ports (${O}_{{A}_{1}^{\prime}}$, ${O}_{{B}_{1}^{\prime}}$), ports (${O}_{{A}_{1}^{\prime}}$, ${O}_{{B}_{2}^{\prime}}$), ports (${O}_{{A}_{2}^{\prime}}$, ${O}_{{B}_{1}^{\prime}}$), or ports (${O}_{{A}_{2}^{\prime}}$, ${O}_{{B}_{2}^{\prime}}$) with probabilities

_{i}*α*

_{1111},

*α*

_{1212},

*α*

_{2121}, or

*α*

_{2222}respectively. That is to say, the photon

*A*and the photon

_{i}*B*in GHZ states can be distributed deterministically from Charlie to Alice and Bob via the collective noise channels.

_{i}After photon blocks are distributed, Alice and Bob perform the measurement along two nonorthogonal sets of orthogonal bases {|*H*〉, |*V*〉} and $\{|+\u3009=\frac{1}{\sqrt{2}}\left(|H\u3009+|V\u3009\right)$, $|-\u3009=\frac{1}{\sqrt{2}}\left(|H\u3009-|V\u3009\right)$} at random to check security of the distribution procedure, by which the legitimate participants can confirm the fact whether there is an eavesdropper on the line or not. Taking the imperfect factors into account, Alice and Bob evaluate the disadvantageous influence and set the threshold value of communication security. That is, the error rate beyonds the value will be reviewed as a dangerous signal and the legitimate participants discard this block and check other blocks.

Confirming no eavesdropping, the communication between Alice and Bob can be processed by employing the other photons of this photon block under Charlie’s agreement and cooperation.

Here, we consider two controlled DSQC protocols on the basis of the above preparatory work.

Cooperating with Alice and Bob, Charlie needs to perform single-photon measurements under the basis of {|+〉, |−〉} on the communication photon group *C _{c}*, and publicizes the measurement outcomes and the corresponding correct order of communication photons. The publicized information and the measurement outcomes are subject to the rule that ‘0’ corresponds to |+〉 and ‘1’ corresponds to |−〉.

#### 2.2. The first controlled DSQC protocol with single parity analysis

At first, we propose the first controlled DSQC protocol with local nondemoliton single parity analysis. Local single parity analysis is exploited to distinguish parity states, while the discrimination of Bell states is not necessary. Moreover, the resulted states after nondemoliton parity analyses can be continued to fulfil other tasks of quantum information processing.

Alice possesses the encoding photons (photons *E _{i}*), called as

*E*-sequence, which are encoded based on the rules: a photon in the state |+〉 is corresponding to secret information ‘0’ and a photon in the state |−〉 corresponds to secret information ‘1’. Alice wants to send the secret information to Bob.

After two communicators obtain Charlie’s permission, Alice performs single parity analysis on the photon *E _{i}* and the photon

*A*along the basis of {|+〉, |−〉}. Based on weak cross-Kerr nonlinearities, the parity analysis can be realized nearly deterministically [43–49], illustrated in Fig. 2, and as a consequence Alice achieves the parity states which can be denoted as

_{i}After that, Alice publicizes the measurement outcomes. Explicitly, if her measurement outcome is |*Even*〉_{+−}, Alice sends the information ‘0’ to Bob, and if her measurement outcome is |*Odd*〉_{+−}, Alice sends the information ‘1’ to Bob.

After he receives the parity analysis information from Alice, Bob performs single-photon measurement along the basis of {|+〉, |−〉}, and registers the measurement outcomes according to the following rules. If his single-photon measurement outcome is |+〉, he records ‘0’, while if his measurement outcome is |−〉, he records ‘1’.

Finally, on the basis of the above information, Bob can obtain the secret information sent from Alice according to the addition rule of binary systems, that is, secret information=Charlie’s measurement outcome ⊕ Alice’s parity analysis information ⊕ Bob’s measurement outcome, which can be referred to Table 1.

#### 2.3. The second controlled DSQC protocol with double parity analyses and dense coding

Next, we describe the second controlled DSQC protocol with local double nondemoliton parity analyses and dense coding.

Firstly, Alice performs single-photon unitary transformations *I*, *X*, *Y*, and *Z* denoted as

*I*, 01 :

*Z*, 10 :

*X*, and 11 :

*Y*.

Subsequently, Alice sends the encoding photons to Bob by fault-tolerant distribution, which is similar to the transmitting process of the photon A* _{i}* from Charlie to her.

After reception of the photons sent from Alice, Bob performs the first parity analysis on photons (*A _{c}*,

*B*) to check security. If the security check is passed, Bob can obtain the secret information by performing the second parity analysis to distinguish Bell states. That is called as double parity analyses which mean that two continuous parity analyses are performed along two different measurement bases. The double parity analyses can also be realized with the circuit illustrated in Fig. 2. Explicitly, if the first parity analysis is performed under the basis of {|

_{c}*H*〉, |

*V*〉}({|+ 〉, |−〉}), the second parity analysis should be performed on the basis of {|+〉, |−〉}({|

*H*〉, |

*V*〉}).

The detailed relations among Charlie’s single-photon measurement outcomes, Alice’s encoding operations, the resulted states, the outcomes of double nondemolition parity analyses, and the secret information sent from Alice can be found in Table 2, where Bell states along the basis of {|+〉, |−〉} can be denoted as

The photons carried secret information are transmitted from Alice to Bob, so Trojan horse attacks need to be considered. To counterattack Trojan horse attacks, the corresponding countermeasures should be adopted, that is, a wavelength filter before Alice’s reception device to hold back the entry of the photons with the illegitimate frequencies, accurately controlling the operation time of reception window to prevent from delay-time photons, and counting the photon number or putting a photon number splitter to check the trace of multi-photon signals [1,50,51].

## 3. Error rate and efficiency

In the above two protocols, the measurement error occurs due to imperfect measurement methods, that is, the odd parity and even parity can not be fully distinguished if X Homodyne measurement [43, 52, 53] is applied. When displacement measurement [54–56] or photon number measurement based on double cross-Kerr nonlinearities [45, 46] is applied, the error rates are lowered largely but they need more complicated operations. Here, taking X Homodyne measurement as an example, the error rate of the first Controlled DSQC protocol can be denoted as [43,53]

Taking the effect of photon losses into accounted, a model of a beam splitter [57–59] is exploited, where the photon dissipation changes the amplitude *α* of the coherent state to the *η _{r}*

*α*(0 ≤

*η*≤ 1). So the error rate is increased to ${P}_{\mathit{err}1}^{\prime}=\frac{1}{2}\text{erfc}\left[\sqrt{{\eta}_{r}{\eta}_{d}}\alpha (1-\text{cos}\theta )/\sqrt{2}\right]$ and ${P}_{\mathit{err}2}^{\prime}=\text{erfc}\left[\sqrt{{\eta}_{r}{\eta}_{d}}\alpha (1-\text{cos}\theta )/\sqrt{2}\right]$, where

_{r}*η*denotes the efficiency of detectors.

_{d}Illustrated in Fig. 3, if the sufficient large phase shift and large amplitude of the coherent state are available, two protocols can be realized with nearly deterministic probabilities. Moreover, due to double parity analyses, the error rate of the second Controlled DSQC protocol is higher than that of the first Controlled DSQC protocol.

As for the efficiencies of two protocols, the classical transmission information needs to be considered [62–64]. In the first Controlled DSQC protocol, one-bit secret information can be obtained per two-bit classical information (one-bit is Charlie’s classical information and one-bit is Alice’s parity analysis outcome). In the second Controlled DSQC protocol, two-bit secret information can be obtained per one-bit classical information (Charlie’s classical information). Furthermore, the obtained states after nondemolition double parity analyses are not destroyed and can be repeatedly exploited in other protocols, and consequently the communication cost is reduced.

## 4. Conclusion

Based on the fault-tolerant distribution of GHZ states via the collective noise channels, we propose two controlled DSQC protocols to realize that two participants transmit secret information under the control of the controller. The security of communication and the controlled function of the controller can be confirmed by the security check and the order rearrangement of transmitted photons. The data block transmission reduces effectively communication cost.

In the first protocol, two participates apply two nonorthogonal sets of orthogonal bases measurements to check the security. Then Alice performs the single parity measurement to encode the secret information, and Bob performs the single-photon measurement to decode the secret information. So the first protocol is simple and easy to operate.

In the second protocol, employing dense coding, the information capacity is doubled. Nondemolition double parity analyses are applied, and as a consequence the obtained states can be repeatedly exploited in other protocols after measurement, so the cost is reduced in the second protocol.

Based on cross-Kerr nonlinearities, some tasks of quantum information processing have been proposed [49,65–74], so two controlled DSQC protocols presented here are expectable.

## Funding

National Natural Science Foundation of China (NSFC) (11674037, 11544013, 11305016, 61301133, 11271055) and Natural Science Foundation of Liaoning Province (20170540010).

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