One-step hyperentanglement purification and hyperdistillation with linear optics

Tie-Jun Wang; Lei-Lei Liu; Ru Zhang; Cong Cao; Chuan Wang

doi:10.1364/OE.23.009284

1. Introduction

Quantum entanglement is a fundamental resource for quantum information processing, and also plays a significant role in various fascinating applications, such as quantum computation [1], quantum secret sharing [2, 3], quantum dense coding [4, 5], and quantum cryptography [6–11]. Entangled photons are generally considered as an ideal information carriers in quantum communication for its high-speed transmission and conspicuous low-noise properties. Hence, they can act as the flying qubits and are ordinarily used to connect distant quantum nodes in long distance communication. Theoretically, entanglement is realized under local operations and classical communications (LOCC), however, photon loss and decoherence caused by the interaction between the quantum system and the environment will inevitably occur during the practical entanglement distribution and long distance transmission.

The consequences of photon loss will cause the signal distortion. To overcome the signal distortion, one can increase the channel capacity to prevent signal from distorting or missing. One of the effective ways to increase capacity of channel is commonly known as hyperentanglement [12] which encodes information on one qubit in multiple degree of freedoms (DOFs). Hyperentanglment state refers to the entangled state of particle simultaneously on multiple DOFs. Besides improving the capacity of channel, hyperentanglement are also employed to increase the security of the communication [13, 14], eliminate the influence of the noise [15–18], and entanglement analysis [19,20]. Moreover, the security of the quantum communication will be reduced by photon loss. For example, the violation of Bell inequalities can rigorously guarantee the presence of entanglement and the quantum nonlocal correlations, which is of fundamental importance in demonstrating entanglement-assisted reduction of communication complexity [21] and device-independent quantum communication. However, due to the low detection efficiency caused by photon losses in quantum communications (not all entangled photons are detected and the overall detection efficiency is below a certain threshold value), the optical loophole-free violation of Bell inequalities over long distance [22] can not be satisfied [23]. Therefore, signal losses in an optical Bell test can open the so-called detection efficiency loophole [24] which allows for local-hidden-variable theories that are able to reproduce the observed data in the experiments. Eliminating the vacuum component and improving the detection efficiency are the main requirements in defending the attacks based on the detection loophole [25, 26].

Meanwhile, decoherence will also decrease the entanglement of the quantum system. After transmitted over a noisy channel, the maximally (hyper)entangled photon states decay into less entangled pure states or mixed states, leading to a destruction of the fidelity and the security of long-distance quantum communication protocols. In order to eliminate the decoherence effect in entangled systems, two interesting quantum techniques could be exploited, entanglement purification and entanglement concentration, to obtain high-fidelity entangled photon systems. Specifically, entanglement purification is used to distill high-fidelity entangled photon systems from a less entangled states ensemble in a mixed state [27–36] while entanglement concentration [37–39] is to achieve the maximally entangled states from partially entangled pure states. Due to the fact that the maximally (hyper)entangled photon states usually decay into a mixed state after being transmitted over a noisy channel, entanglement purification is more general than entanglement concentration in practical applications. In 2013, Ren et al. [40] theoretically proposed a hyperentanglement concentration protocol for photons by performing parity-check operations on both spatial-mode and polarization DOFs independently resorting to linear optics, and they point out that it is impossible to accomplish hyperentanglement purification only using linear optics. However, they did not considered the influence of photon loss, and it has been proved that the relative weight of the vacuum component will increase largely after the linear-optical parity-check operations [23]. Therefore, the detection efficiency in the quantum communication protocols decreases sharply.

In this paper, we investigate the possibility of achieving hyperdistillation and hyperentanglement purification operations simultaneously on two-photon systems in nonlocal hyperentangled Bell states on both the spatial mode and polarization DOFs. The key operations of our scheme are the quantum nondemolition (QND) parity-checking measurement and the heralded two-qubit amplification which can be completed by one step exploiting linear optics and local entanglement resource. With these operations, the bit-flip errors and phase-flip errors caused by decoherence in noisy channels and the vacuum errors caused by the transmission losses could all be corrected, and also our scheme provides a new solution to overcome the problem of photon losses and decoherence simultaneously.

This paper is organized as follows: In Sec. II, we describe the approach for QND parity-checking operation which could accomplish parity-check measurement on both spatial-mode DOF and polarization DOF, and the heralded two-qubit amplification which can remove the vacuum components of the photons assisted by linear optical elements and local entanglement resources. In Sec. III, we present the hyperdistillation and hyperentanglement purification model in spatial mode and polarization DOFs. And Sec. IV is the discussion and summary.

2. Implementation of QND parity-check measurement and qubit amplification

The basic model of the present hyperdistillation and hyperentanglement purification scheme is depicted in Fig. 1 which consists of linear optical elements and single-photon detectors. The polarizing beam splitters (PBSs) transmit a horizontally polarized photon (|H〉) and reflect a vertically polarized photon (|V〉). There are four half-wave plates (HWPs) which can flip the polarization state of the photons, and eight quarter-wave plates (QWPs) which perform Hadamard operation [ $| H 〉 \to \frac{1}{\sqrt{2}} (| H 〉 + | V 〉)$ , $| V 〉 \to \frac{1}{\sqrt{2}} (| H 〉 - | V 〉)$ ] on the polarizations DOF of the photons. To demonstrate the principle of the setup, we first describe the QND parity-check measurement process in both polarization DOF and spatial-mode DOF and implement qubit amplification that can distill expected information from a mixed entangled state ensemble.

Fig. 1 The schematic diagram of the setup for QND parity-check measurement and qubit amplification. The inset shows the structure of detectors D. The setup consists of four input ports (marked with a₁, a₂, c₁, c₂), eight auxiliary input ports (marked with b₁, b₂, b₃, b₄, e₁, e₂, e₃, e₄), four output ports (marked with a′₁, a′₂, c′₁, c′₂), two polarizing beam splitters (PBS) which transmit a horizontally polarized photon |H〉 and reflect a vertically polarized photon |V〉. Successful operation of the distilling is heralded by eight-photon coincidence detection on the eight detectors D₁, D₂, D₃, D₄, D₅, D₆, D₇ and D₈.

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Assume that the initial state of the two photons A and C are

{| ψ_{0} 〉}_{AC} = [(1 - P_{1}) {| vac 〉}_{A} 〈 vac | + P_{1} | ψ_{A} 〉 〈 ψ_{A} |] \otimes [(1 - P_{2}) {| vac 〉}_{C} 〈 vac | + P_{2} {| ψ 〉}_{C} 〈 ψ_{C} |],

where

\begin{array}{l} | ψ_{A} 〉 = {(α_{1} | a_{1} 〉 + β_{1} | a_{2} 〉)}_{A}^{S} \otimes {(γ_{1} | H 〉 + ξ_{1} | V 〉)}_{A}^{P}, \\ | ψ_{C} 〉 = {(α_{2} | c_{1} 〉 + β_{2} | c_{2} 〉)}_{C}^{S} \otimes {(γ_{2} | H 〉 + ξ_{2} | V 〉)}_{C}^{P} . \end{array}

Here, |vac〉 denotes vacuum state, and P₁ (P₂) represents the relative weight of the qubit components of the photon A(C). P denotes polarized state of the photons, S represents the photons’ spatial-mode state, and the coefficients satisfy |α₁|² + |β₁|² = |α₂|² + |β₂|² = |γ₁|² + |ξ₁|² = |γ₂|² + |ξ₂|² = 1. The even-parity component of the input photons AC is

{| ψ 〉}_{AC}^{even} = χ (α_{1} α_{2} | a_{1} c_{1} 〉 + β_{1} β_{2} | a_{2} c_{2} 〉) (γ_{1} γ_{2} | H H 〉 + ξ_{1} ξ_{2} | V V 〉)

, where χ denotes the normalization coefficient. The locally prepared ancillary 8-photon state can be written as

\begin{array}{l} | ω 〉 = & \frac{1}{2} [{(| H V H V 〉 + | V H V H 〉)}_{b_{1} b_{2} b_{3} b_{4}} \otimes ({| H H H H 〉}_{e_{1} e_{2} e_{3} e_{4}}) \\ + {(| V V V V 〉)}_{b_{1} b_{2} b_{3} b_{4}} \otimes {(| H V H V 〉 + | V H V H 〉)}_{e_{1} e_{2} e_{3} e_{4}}], \end{array}

where b₁, b₂, b₃, and b₄ indicate the input-ports for 4 ancillary photons B₁, B₂, B₃, and B₄, respectively, and e₁, e₂, e₃, e₄ indicate the input-ports for the other 4 ancillary photons E₁, E₂, E₃, E₄, respectively. Successful operation of the distilling is heralded by 8-photon coincidence detection on the eight detectors D₁, D₂, D₃, D₄, D₅, D₆, D₇, and D₈. If the input state contains vacuum component, it will not cause the eight detectors response simultaneously, therefore, we can discard vacuum errors through the coincidence detection events.

After PBS₁ and PBS₂, the state of the ten-photon ACB₁B₂B₃B₄E₁E₂E₃E₄ system can be written as

\begin{array}{l} | ψ_{A} 〉 \otimes | ψ_{C} 〉 \otimes | w 〉 \\ \overset{{PBS}_{1, 2}}{\to} (α_{1} γ_{1} | H_{D_{1}} 〉 + β_{1} γ_{1} | H_{D_{2}} 〉 + α_{1} ξ_{1} | V_{D_{8}} 〉 + β_{1} ξ_{1} | V_{D_{7}} 〉) \otimes (α_{2} γ_{2} | H_{D_{3}} 〉 + β_{2} γ_{2} | H_{D_{4}} 〉 \\ + α_{2} ξ_{2} | V_{D_{6}} 〉 + β_{2} ξ_{2} | V_{D_{5}} 〉) \otimes \frac{1}{2} [(| H_{a_{1}^{'}} V_{D_{2}} H_{c_{1}^{'}} V_{D_{4}} 〉 + | V_{D_{1}} H_{a_{2}^{'}} V_{D_{3}} H_{c_{2}^{'}} 〉) \\ \otimes | H_{D_{5}} H_{D_{6}} H_{D_{7}} H_{D_{8}} 〉 + | V_{D_{1}} V_{D_{2}} V_{D_{3}} V_{D_{4}} 〉 \otimes (| H_{D_{5}} V_{c_{1}^{'}} H_{D_{7}} V_{a_{1}^{'}} 〉 + | V_{c_{2}^{'}} H_{D_{6}} V_{a_{2}^{'}} H_{D_{8}} 〉)] . \end{array}

From Eq. (4), one can easily observes that there are only four circumstances which can cause the eight detectors response simultaneously. So utilizing eight-photon coincidence detection, the postselection process only picks up the case of the state

\begin{array}{l} χ (α_{1} α_{2} γ_{1} γ_{2} | H_{a_{1}^{'}} H_{c_{1}^{'}} 〉 | H_{D_{1}} H_{D_{2}} H_{D_{3}} V_{D_{4}} H_{D_{5}} H_{D_{6}} H_{D_{7}} H_{D_{8}} 〉 \\ + α_{1} α_{2} ξ_{1} ξ_{2} | V_{a_{1}^{'}} V_{c_{1}^{'}} 〉 | V_{D_{1}} V_{D_{2}} V_{D_{3}} V_{D_{4}} H_{D_{5}} V_{D_{6}} H_{D_{7}} V_{D_{8}} 〉 \\ + β_{1} β_{2} γ_{1} γ_{2} | H_{a_{2}^{'}} H_{c_{2}^{'}} 〉 | V_{D_{1}} H_{D_{2}} V_{D_{3}} H_{D_{4}} H_{D_{5}} H_{D_{6}} H_{D_{7}} H_{D_{8}} 〉 \\ + β_{1} β_{2} ξ_{1} ξ_{2} | V_{a_{2}^{'}} V_{c_{2}^{'}} 〉 | V_{D_{1}} V_{D_{2}} V_{D_{3}} V_{D_{4}} V_{D_{5}} H_{D_{6}} V_{D_{7}} H_{D_{8}} 〉) . \end{array}

After the flipping operations of the HWPs and the rotation operations of the QWPs at output-ports, the single-photon detections on the eight detectors are performed on the bases {|H〉,|V〉}. Then, we can rewrite the system in the corresponding subspace as

\begin{array}{l} \frac{1}{8} {| ψ 〉}_{A^{'} C^{'}}^{even} \otimes ({| L 〉}_{D_{1} D_{2} D_{3} D_{4}} {| L 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| M 〉}_{D_{1} D_{2} D_{3} D_{4}} {| M 〉}_{D_{5} D_{6} D_{7} D_{8}}) \\ + \frac{χ}{8} [(| A 〉 + | B 〉 - | C 〉 + | D 〉) \otimes ({| L 〉}_{D_{1} D_{2} D_{3} D_{4}} {| M 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| M 〉}_{D_{1} D_{2} D_{3} D_{4}} {| L 〉}_{D_{5} D_{6} D_{7} D_{8}}) \\ + (| A 〉 - | B 〉 + | C 〉 - | D 〉) \otimes ({| N 〉}_{D_{1} D_{2} D_{3} D_{4}} {| N 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| K 〉}_{D_{1} D_{2} D_{3} D_{4}} {| K 〉}_{D_{5} D_{6} D_{7} D_{8}}) \\ + (| A 〉 - | B 〉 - | C 〉 + | D 〉) \otimes ({| N 〉}_{D_{1} D_{2} D_{3} D_{4}} {| K 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| K 〉}_{D_{1} D_{2} D_{3} D_{4}} {| N 〉}_{D_{5} D_{6} D_{7} D_{8}}) \\ + (| A 〉 + | B 〉 + | C 〉 - | D 〉) \otimes ({| L 〉}_{D_{1} D_{2} D_{3} D_{4}} {| N 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| M 〉}_{D_{1} D_{2} D_{3} D_{4}} {| K 〉}_{D_{5} D_{6} D_{7} D_{8}}) \\ + (| A 〉 + | B 〉 - | C 〉 + | D 〉) \otimes ({| L 〉}_{D_{1} D_{2} D_{3} D_{4}} {| K 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| M 〉}_{D_{1} D_{2} D_{3} D_{4}} {| N 〉}_{D_{5} D_{6} D_{7} D_{8}}) \\ + (| A 〉 - | B 〉 + | C 〉 + | D 〉) \otimes ({| N 〉}_{D_{1} D_{2} D_{3} D_{4}} {| L 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| K 〉}_{D_{1} D_{2} D_{3} D_{4}} {| M 〉}_{D_{5} D_{6} D_{7} D_{8}}) \\ + (| A 〉 - | B 〉 - | C 〉 - | D 〉) \otimes ({| N 〉}_{D_{1} D_{2} D_{3} D_{4}} {| M 〉}_{D_{5} D_{6} D_{7} D_{8}} - {| K 〉}_{D_{1} D_{2} D_{3} D_{4}} {| L 〉}_{D_{5} D_{6} D_{7} D_{8}})], \end{array}

where

\begin{array}{l} {| ψ 〉}_{A^{'} C^{'}}^{even} = χ {(α_{1} α_{2} | a_{1}^{'} c_{1}^{'} 〉 + β_{1} β_{2} | a_{2}^{'} c_{2}^{'} 〉)}_{A^{'} C^{'}} \otimes {(γ_{1} γ_{2} | H H 〉 + ξ_{1} ξ_{2} | V V 〉)}_{A^{'} C^{'}}, \\ | A 〉 = α_{1} α_{2} γ_{1} γ_{2} {| H H 〉}_{A^{'} C^{'}} {| a_{1}^{'} c_{1}^{'} 〉}_{A^{'} C^{'}}, | B 〉 = β_{1} β_{2} γ_{1} γ_{2} {| H H 〉}_{A^{'} C^{'}} {| a_{2}^{'} c_{2}^{'} 〉}_{A^{'} C^{'}}, \\ | C 〉 = α_{1} α_{2} ξ_{1} ξ_{2} {| V V 〉}_{A^{'} C^{'}} {| a_{1}^{'} c_{1}^{'} 〉}_{A^{'} C^{'}}, | D 〉 = β_{1} β_{2} ξ_{1} ξ_{2} {| V V 〉}_{A^{'} C^{'}} {| a_{2}^{'} c_{2}^{'} 〉}_{A^{'} C^{'}}, \\ | L 〉 = \frac{1}{2} (| H H H H 〉 + | H V H V 〉 + | V H V H 〉 + | V V V V 〉), \\ | M 〉 = \frac{1}{2} (| H H V V 〉 + | H V V H 〉 + | V H H V 〉 + | V V H H 〉), \\ | N 〉 = \frac{1}{2} (| H H V H 〉 + | H V V V 〉 + | V H H H 〉 + | V V H V 〉), \\ | K 〉 = \frac{1}{2} (| H H H V 〉 + | H V H H 〉 + | V H V V 〉 + | V V V H 〉) . \end{array}

From Eq. (6), we can conclude that the state of output photons A′C′ have even parity in both polarization DOF and spatial-mode DOF. If the results on D₁D₂D₃D₄ (D₅D₆D₇D₈) is |HHHH〉, |VVVV〉, |HVHV〉 or |VHVH〉, one dose not need to do any operation on the spatial-mode DOF of the photon C′ when its polarized state is |H〉_C′ (|V〉_C′), that is,

I_{H (V)}^{S} = (| c_{1}^{'} 〉 〈 c_{1}^{'} | + | c_{2}^{'} 〉 〈 c_{2}^{'} |) {| H (V) 〉}_{C^{'}} 〈 H (V) |

. Meanwhile, if the measurement result on D₁D₂D₃D₄ (D₅D₆D₇D₈) is |HHVV〉, |VVHH〉, |HVVH〉 or |VHHV〉, a single-photon gate

- I_{H (V)}^{S} = (| c_{1}^{'} 〉 〈 c_{1}^{'} | + | c_{2}^{'} 〉 〈 c_{2}^{'} |) {| H (V) 〉}_{C^{'}} 〈 H (V) |

should be used on the photon C′ at the output-port. If the results on D₁D₂D₃D₄ (D₅D₆D₇D₈) reveal |HHVH〉, |VVHV〉, |HVVV〉 or |VHHH〉, a single qubit operation

σ_{Z, H (V)}^{S} = (| c_{1}^{'} 〉 〈 c_{1}^{'} | - | c_{2}^{'} 〉 〈 c_{2}^{'} |) {| H (V) 〉}_{C^{'}} 〈 H (V) |

on the photon C′ is needed to recover the same state as

{| ψ 〉}_{A^{'} C^{'}}^{even}

at output port. For the last case, a single qubit operation

- σ_{Z, H (V)}^{S}

on the photon C′ is required at output-port when the results of D₁D₂D₃D₄ (D₅D₆D₇D₈) is |HHHV〉, |VVVH〉, |HVHH〉 or |VHVV〉.

Based on the above discussion, the process of the QND parity-check measurement and qubit amplification is heralded by the detector clicks, simultaneously. The relation between the final measuring results at the detctors D₁D₂D₃D₄ (D₅D₆D₇D₈) and the single-photon operations on the photon C′ is shown in Table I. On condition that the detectors D₁, D₂, D₃, D₄, D₅, D₆, D₇, and D₈ are simultaneously registered, the relative probability of the qubit component in the output signal mode increases to 100%, which leads to a success probability $P_{suc} = \frac{1}{4} P_{1} P_{2} ({| α_{1} α_{2} |}^{2} + {| β_{1} β_{2} |}^{2}) ({| γ_{1} γ_{2} |}^{2} + {| ξ_{1} ξ_{2} |}^{2})$ to achieve the even-parity states in both polarization DOF and spatial-mode DOF in a heralded strategy.

Table 1. Relation between the final states of D₁D₂D₃D₄(D₅D₆D₇D₈) and the corresponding single qubit operations on the photon C′.

View Table

3. Hyperentanglement purification in spatial-mode and polarization DOFs

So far, we have constructed the QND parity-check measurement and qubit amplification using linear optics. Based on the above results, the hyperdistillation and the hyperentanglement purification could work simultaneously. Suppose a hyperentangled photon source is located at the middle point between Alice and Bob, and the hyperentangled state can be written as $\frac{1}{2} {(| H H 〉 + | V V 〉)}_{AB} {(| a_{1} b_{1} 〉 + | a_{2} b_{2} 〉)}_{AB}$ . In practical transmission, the channel noise will inevitably induce the signal losses and decoherence, and the state becomes a mixture of the vacuum component with probability (1 − τ₁) and a mixed hyperentangled component with probability τ₁. The density matrix of the system AB is deteriorated into

\begin{array}{l} ρ_{AB} = & τ_{1} {[F_{1} {| ϕ^{+} 〉}^{P} 〈 ϕ^{+} | + (1 - F_{1}) {| ψ^{+} 〉}^{P} 〈 ψ^{+} |]}_{AB} \otimes {[F_{2} {| ϕ^{+} 〉}^{S} 〈 ϕ^{+} | + (1 - F_{1}) {| ψ^{+} 〉}^{S} 〈 ψ^{+} |]}_{AB} \\ + τ_{2} {| + 〉}_{A}^{S} 〈 + | \otimes {| + 〉}_{A}^{P} 〈 + | \otimes {| vac 〉}_{B} 〈 vac | + τ_{3} {| vac 〉}_{A} 〈 vac | \otimes {| + 〉}_{B}^{S} 〈 + | \otimes {| + 〉}_{B}^{P} 〈 + | \\ + τ_{4} {| vac 〉}_{AB} 〈 vac |, \end{array}

where

{| ϕ^{+} 〉}_{AB}^{P} = \frac{1}{\sqrt{2}} {(| H H 〉 + | V V 〉)}_{AB}

,

{| ψ^{+} 〉}_{AB}^{P} = \frac{1}{\sqrt{2}} {(| H V 〉 + | V H 〉)}_{AB}

,

{| ϕ^{+} 〉}_{AB}^{S} = \frac{1}{\sqrt{2}} {(| a_{1} b_{1} 〉 + | a_{2} b_{2} 〉)}_{AB}

, and

{| ψ^{+} 〉}_{AB}^{S} = \frac{1}{\sqrt{2}} {(| a_{1} b_{2} 〉 + | a_{2} b_{1} 〉)}_{AB}

. And F₁ and F₂ refer to the probabilities of |ϕ⁺〉_P, and |ϕ⁺〉_S in the mixed hyperentangled component, respectively. The coefficients satisfy the normalization relation as τ₁ + τ₂ + τ₃ + τ₄ = 1. Thus the fidelity of the two-photon system is defined as F = τ₁F₁F₂.

Assume that there exists another identical two-photon system CD which has the same form as the photons AB,

\begin{array}{l} ρ_{CD} = & τ_{1} {[F_{1} {| ϕ^{+} 〉}^{P} 〈 ϕ^{+} | + (1 - F_{1}) {| ψ^{+} 〉}^{P} 〈 ψ^{+} |]}_{CD} \otimes {[F_{2} {| ϕ^{+} 〉}^{S} 〈 ϕ^{+} | + (1 - F_{2}) {| ψ^{+} 〉}^{S} 〈 ψ^{+} |]}_{CD} \\ + τ_{2} {| + 〉}_{C}^{S} 〈 + | \otimes {| + 〉}_{C}^{P} 〈 + | \otimes {| vac 〉}_{D} 〈 vac | + τ_{3} {| vac 〉}_{C} 〈 vac | \otimes {| + 〉}_{D}^{S} 〈 + | \otimes {| + 〉}_{D}^{P} 〈 + | \\ + τ_{4} {| vac 〉}_{CD} 〈 vac |, \end{array}

where

{| ϕ^{+} 〉}_{CD}^{S} = \frac{1}{\sqrt{2}} {(| c_{1} d_{1} 〉 + | c_{2} c_{2} 〉)}_{CD}

,

{| ψ^{+} 〉}_{CD}^{S} = \frac{1}{\sqrt{2}} {(| c_{1} d_{2} 〉 + | c_{2} d_{1} 〉)}_{CD}

. The photons A and C, B and D belong to Alice and Bob, respectively, and the four-photon system can be described by density operators ρ₀ = ρ_AB ⊗ ρ_CD.

Exploiting the devices shown in Fig. 1, both Alice and Bob perform QND parity-checking measurement on spatial-mode DOF and on polarization DOF of their own photons simultaneously, and remove the vacuum component of the photons. The four-photon systems in the two DOFs can be amplified and purified independently. Alice and Bob perform qubit amplification and pick up the even-parity cases in two DOFs on the photon-pairs AC and CD, respectively. According to Eq. (6), when the is finished on both sides in the desired way, the density operator of the system ABCD becomes

\begin{array}{l} ρ_{ABCD}^{'} & = {[F_{1}^{'} {| Φ^{+} 〉}^{P} 〈 Φ^{+} | + (1 - F_{1}^{'}) {| Ψ^{+} 〉}^{P} 〈 Ψ^{+} |]}_{ABCD} \\ \otimes {[F_{2}^{'} {| Φ^{+} 〉}^{S} 〈 Φ^{+} | + (1 - F_{2}^{'}) {| Ψ^{+} 〉}^{S} 〈 Ψ^{+} |]}_{ABCD}, \end{array}

where the entangled state in different DOFs could be described as

\begin{array}{l} {| Φ^{+} 〉}_{ABCD}^{P} = \frac{1}{\sqrt{2}} {(| H H H H 〉 + | V V V V 〉)}_{ABCD}, \\ {| Ψ^{+} 〉}_{ABCD}^{P} = \frac{1}{\sqrt{2}} {(| H V H V 〉 + | V H V H 〉)}_{ABCD}, \\ {| Φ^{+} 〉}_{ABCD}^{S} = \frac{1}{\sqrt{2}} {(| a_{1} b_{1} c_{1} d_{1} 〉 + | a_{2} b_{2} c_{2} d_{2} 〉)}_{ABCD}, \\ {| Ψ^{+} 〉}_{ABCD}^{S} = \frac{1}{\sqrt{2}} {(| a_{1} b_{2} c_{1} d_{2} 〉 + | a_{2} b_{1} c_{2} d_{1} 〉)}_{ABCD} . \end{array}

Here,

F_{1}^{'} = \frac{F_{1}^{2}}{F_{1}^{2} + {(1 - F_{1})}^{2}}

and

F_{2}^{'} = \frac{F_{2}^{2}}{F_{2}^{2} + {(1 - F_{2})}^{2}}

refer to the probabilities of |Φ⁺〉_P and |Φ⁺〉_S in the mixed hyperentangled state, respectively.

Then Alice and Bob each measure the photons C and D in the polarization basis { ${| + 〉}^{P} = \frac{1}{\sqrt{2}} (| H 〉 + | V 〉)$ , ${| - 〉}^{P} = \frac{1}{\sqrt{2}} (| H 〉 - | V 〉)$ } and in the spatial-mode basis { ${| + 〉}^{S} = \frac{1}{\sqrt{2}} (| c_{1} (d_{1}) 〉 + | c_{2} (d_{2}) 〉)$ , ${| - 〉}^{S} = \frac{1}{\sqrt{2}} (| c_{1} (d_{1}) 〉 - | c_{2} (d_{2}) 〉)$ }, respectively. The density operator of the system ABCD can be rewritten as

\begin{array}{l} ρ_{ABCD}^{″} = & \frac{1}{16} {{[F_{1}^{'} {| ϕ^{+} 〉}^{P} 〈 ϕ^{+} | + (1 - F_{1}^{'}) {| ψ^{+} 〉}^{P} 〈 ψ^{+} |]}_{AB} {| + + 〉}_{CD}^{P} \\ + {[F_{1}^{'} {| ϕ^{+} 〉}^{P} 〈 ϕ^{+} | - (1 - F_{1}^{'}) {| ψ^{+} 〉}^{P} 〈 ψ^{+} |]}_{AB} {| - - 〉}_{CD}^{P} \\ + {[F_{1}^{'} {| ϕ^{-} 〉}^{P} 〈 ϕ^{-} | + (1 - F_{1}^{'}) {| ψ^{-} 〉}^{P} 〈 ψ^{-} |]}_{AB} {| - + 〉}_{CD}^{P} \\ + {[F_{1}^{'} {| ϕ^{-} 〉}^{P} 〈 ϕ^{-} | - (1 - F_{1}^{'}) {| ψ^{-} 〉}^{P} 〈 ψ^{-} |]}_{AB} {| + - 〉}_{CD}^{P}} \\ \otimes {{[F_{2}^{'} {| ϕ^{+} 〉}^{S} 〈 ϕ^{+} | + (1 - F_{2}^{'}) {| ψ^{+} 〉}^{S} 〈 ψ^{+} |]}_{AB} {| + + 〉}_{CD}^{S} \\ + {[F_{2}^{'} {| ϕ^{+} 〉}^{S} 〈 ϕ^{+} | - (1 - F_{2}^{'}) {| ψ^{+} 〉}^{S} 〈 ψ^{+} |]}_{AB} {| - - 〉}_{CD}^{S} \\ + {[F_{2}^{'} {| ϕ^{-} 〉}^{S} 〈 ϕ^{-} | - (1 - F_{2}^{'}) {| ψ^{-} 〉}^{S} 〈 ψ^{-} |]}_{AB} {| - + 〉}_{CD}^{S} \\ + {[F_{2}^{'} {| ϕ^{-} 〉}^{S} 〈 ϕ^{-} | - (1 - F_{2}^{'}) {| ψ^{-} 〉}^{S} 〈 ψ^{-} |]}_{AB} {| + - 〉}_{CD}^{S}}, \end{array}

where

\begin{array}{l} {| ϕ^{-} 〉}_{AB}^{P} = \frac{1}{\sqrt{2}} {(| H H 〉 - | V V 〉)}_{A B,} {| ψ^{-} 〉}_{AB}^{P} = \frac{1}{\sqrt{2}} {(| H V 〉 - | V H 〉)}_{A B,} \\ {| ϕ^{-} 〉}_{AB}^{S} = \frac{1}{\sqrt{2}} {(| a_{1} b_{1} 〉 - | a_{2} b_{2} 〉)}_{A B,} {| ψ^{-} 〉}_{AB}^{S} = \frac{1}{\sqrt{2}} {(| a_{1} b_{2} 〉 - | a_{2} b_{1} 〉)}_{AB} . \end{array}

If the two clicked photon detectors of photons C and D are in the even-parity polarization(spatial) mode ${| + + 〉}_{CD}^{P (S)}$ , the two-photon system AB is projected into [F′₁|ϕ⁺〉^P(S) 〈ϕ⁺| + (1 − F′₁)|ψ⁺〉^P(S) 〈ψ⁺|]_AB. However, if the two clicked photon detectors of photons C and D are in the other even-parity polarization(spatial) mode ${| - - 〉}_{CD}^{P (S)}$ , the two-photon system AB is projected into [F′₁|ϕ⁺〉^P(S) 〈ϕ⁺| − (1 − F′₁)|ψ⁺〉^P(S) 〈ψ⁺|]_AB. Similarly, if the outcome of two clicked detectors is in the odd-parity spatial mode ${| + - 〉}_{CD}^{P (S)}$ ( ${| - + 〉}_{CD}^{P (S)}$ ), a phase-flip operation $σ_{z}^{P (S)}$ performed on the photon B is required to obtain the state [F′₁|ϕ⁺〉^P(S) 〈ϕ⁺| − (1 − F′₁)|ψ⁺〉^P(S) 〈ψ⁺|]_AB([F′₁|ϕ⁺〉^P(S) 〈ϕ⁺|+(1 − F′₁)|ψ⁺〉|^P(S) 〈ψ⁺]_AB). The fidelity of the two-photon system AB is F′ = F′₁F′₂. The condition for F′ > F is that F₁ > 0.5 and F₂ > 0.5. The fidelity of the final hyperentangled states in polarization DOF and spatial-mode DOF increases rapidly after the hyperentanglement purification operation (see in Fig. 2 for the cases with F₁ = F₂ = f).

Fig. 2 Fidelity of the hyperentangled photon-pairs in polarization DOF and spatial-mode DOF for the cases with F₁ = F₂ = f. F’ and F represent the fidelity after and before hyperentanglement purification operation, respectively.

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According to the above discussion, by picking up the case that Alice and Bob have the same parity in both polarization DOF and spatial-mode DOF mentioned above, the bit-flip errors and the vacuum errors can be eliminated successfully. However, in a practical system, there exists not only bit-flip errors but also phase-flip errors. In principle, the phase-flip errors can be transformed into bit-flip errors with local Hadamard operations in both the spatial-mode and the polarization DOFs. Therefore, our scheme can also be used to purify the mixed hyperentangled Bell states with phase-flip errors in two DOFs.

4. Discussion and summary

In this section, we will prove that, without the hyperdistillation operation, the relative weight of the vacuum component P_vac largely increases after general entanglement purification protocols, and the detection efficiency for quantum communication schemes decreases sharply. Suppose a entangled-photon source is located at middle point between Alice and Bob, and the entangled state can be written as ${| ϕ^{+} 〉}_{AB}^{P}$ . The photons A and B are sent to Alice and Bob through two lossy channels a₁ and b₁ with two independent transmission coefficients T₁ and T₂, respectively (1 > T₁ > 0 and 1 > T₂ > 0). These noisy channels would deteriorate the pure entangled state AB to

\begin{array}{l} ρ_{AB}^{P} = T_{1} T_{2} {[f {| ϕ^{+} 〉}^{P} 〈 ϕ^{+} | + (1 - f) {| ψ^{+} 〉}^{P} 〈 ψ^{+} |]}_{AB} + (1 - T_{1}) T_{2} {| v a c 〉}_{A} 〈 v a c | \otimes {| + 〉}_{B}^{P} 〈 + | \\ + T_{1} (1 - T_{2}) {| + 〉}_{A}^{P} 〈 + | \otimes {| v a c 〉}_{B} 〈 v a c | + (1 - T_{1}) (1 - T_{2}) {| v a c 〉}_{A} 〈 v a c | \otimes {| v a c 〉}_{B} 〈 v a c | . \end{array}

Here, the relative weight of the vacuum components of the photons AB is

P_{vac}^{0} = 1 - T_{1} T_{2}

, and the fidelity of the qubit-components of the photons AB is f.

In order to increase the fidelity f of mixed entangled system nonlocally, Alice and Bob perform an entanglement purification operation on this resource. Suppose there are four identical two-photon systems AB, A₁B₁, CD and C₁D₁ in the mixed entangled Bell-class state as described in Eq. (13), Alice first performs the most simple parity-checking measurements described in [29] on the polarization DOF of the photon pairs AA₁ and CC₁, respectively, with only linear optics. Simultaneously, Bob perform the same parity-checking measurements directly on the photon pair BB₁ and DD₁, respectively. Without the distilling operation, after the entanglement purification, the relative weight of the vacuum component of the new mixed entangled four-qubit system is P_vac = 1 − ξ², where the value ξ could be described as [41]

ξ = \frac{T_{1} T_{2} [f^{2} + {(1 - f)}^{2}]}{T_{1} T_{2} [f^{2} + {(1 - f)}^{2} + (1 - T_{1}) T_{2} + (1 - T_{2}) T_{1} + (1 - T_{1}) (1 - T_{2}) (2 f + 1)]} .

Figure 3 depicts the relative weight of the vacuum components $P_{vac}^{0}$ and P_vac as a function of the transmission T in the case of T₁ = T₂ = T for different f. From Fig. 3, we see that without the distillation operation, the relative weight of the vacuum component P_vac increases largely, and the detection efficiency for Alice and Bob decreases sharply. The low detection efficiency will lead to the detection loophole of quantum commutation. However, by using the setup described in Fig. 1, the same improved four-qubit entangled component is distilled in a heralded way with a certain possibility. The heralding signal allows one party to introduce a new input in his detectors only when a hyperentangled state is shared between the other. Hence the overall detection efficiency required to close the detection loophole does not depend on the transmission efficiency, but reduces to the intrinsic detection efficiency of Alices and Bobs detectors. That is, the effective transmission efficiency of photons can been boosted up to 100% in theory even where channel loss exists.

Fig. 3 Relative weight of the vacuum component of the hyperentangled photon-pairs (P_vac or P′_vac) as a function of the transmission coefficient T in the case T₁ = T₂ = T for different initial real parameters f.

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To mirror a likely experimental scenario, we numerically simulated P′_vac with the detector efficiency η_d = 95%, the dark count probability D = 1 dark count/s, and the coupling efficiency between the photon-pair source and the optical fibers η_c = 0.9. We assume that the photon sources are excited with a repetition rate of 10 GHz, and the initial source of entanglement is the spontaneous-parametric down-conversion process with photon pair generation rate 2 × 10⁻³ [23]. The final relative weight of the vacuum component can be calculated as P′_vac = 1 − R/(R + D), where R denotes the final key rate (per second) of hyperentanglement-based device-independent quantum key distribution (DIQKD) [42] after entanglement distribution, hyperdistillation and hyperentanglement purification operations. From Fig. 3, the relative weight of the vacuum component of the final signals decreases sharply with the hyperdistillation operation, and when T > 0.1, P′_vac is below 0.05. Thus, our scheme for hyerdistilling and hyperentanglement purification in a heralded strategy provides us a powerful tool to overcome the problem of losses and decoherence, and it can be applied to all hyperentanglement-based quantum communication protocols.

In this study, we have proposed a hyperdistillation and hyperentanglement purification protocol using linear optics. The parity-check measurement in two DOFs could be realized by only one-step operation. Compared with the previous works using linear optics, our protocol can overcome the difficulties posed by inherent channel losses during photon transmission, simplify the complex measurement procedure, and optimise quantum communication process. With the feasible technologies, this scheme may be widely used in quantum communication and quantum networks.

Acknowledgments

This work was supported by the National Natural Science Foundation of China through Grants (No. 11404031 and No. 61471050), Beijing Higher Education Young Elite Teacher Project (No. YETP0456), the Fundamental Research Funds for the Central Universities (No. 2014RC0903). The project was supported by Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P. R. China.

References and links

1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

2. M. Hillery and V. Bužek, “Quantum secret sharing,” Phys. Rev. A 59, 1829 (1999). [CrossRef]

3. L. Xiao, G. L. Long, F.-G. Deng, and J. -W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004). [CrossRef]

4. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881 (1992). [CrossRef] [PubMed]

5. X. S. Liu, G. L. Long, D. M. Tong, and Feng Li, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002). [CrossRef]

6. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991). [CrossRef] [PubMed]

7. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys. Rev. Lett. 68, 557 (1992). [CrossRef] [PubMed]

8. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002). [CrossRef]

9. G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002). [CrossRef]

10. X. -H. Li, F. -G. Deng, and H. -Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006). [CrossRef]

11. W. Y. Wang, C. Wang, G. Y. Zhang, and G. L. Long, “Arbitrarily long distance quantum communication using inspection and power insertion,” Chin. Sci. Bull. 54, 158 (2009). [CrossRef]

12. R. Heilmann, M. Gräfe, S Nolte, and A Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96(2015). [CrossRef]

13. D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002). [CrossRef] [PubMed]

14. N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002). [CrossRef] [PubMed]

15. M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009). [CrossRef]

16. Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010). [CrossRef]

17. B. C. Ren and G. L. Long, “General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities, ” Optics express 22, 6547 (2014). [CrossRef] [PubMed]

18. Y. B. Sheng, J. Liu, S. Y. Zhao, and L. Zhou, “Multipartite entanglement concentration for nitrogen-vacancy center and microtoroidal resonator system, ” Chin. Sci. Bull. 59, 3507 (2013). [CrossRef]

19. C. Wang, L. Y. He, Y. Zhang, H. Q. Ma, and R. Zhang, “Complete entanglement analysis on electron spins using quantum dot and microcavity coupled system, ” Sci. China- Phys Mech Astron 56, 2054 (2013). [CrossRef]

20. T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities, ” Phys. Rev. A 86, 042337 (2012). [CrossRef]

21. H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665 (2010). [CrossRef]

22. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).

23. T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014). [CrossRef]

24. P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418 (1970). [CrossRef]

25. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969). [CrossRef]

26. A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).

27. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722 (1996). [CrossRef] [PubMed]

28. D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818(1996). [CrossRef] [PubMed]

29. J. W. Pan, C. Simon, C. Brukner, and A. Zelinger, “Entanglement purification,” Nature(London). 410, 1067 (2001). [CrossRef]

30. C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).

31. W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91, 107903 (2003). [CrossRef] [PubMed]

32. M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206 (1999). [CrossRef]

33. A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005). [CrossRef] [PubMed]

34. Y. W. Cheong, S. W. Lee, J. Lee, and H. -W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007). [CrossRef]

35. S. Y. Hou, Y. B. Sheng, G. R. Feng, and G. L. Long, “Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor,” Scientific reports 4, 6857(2014). [CrossRef] [PubMed]

36. Y. Liu and J. X. Cui, “Realization of Kraus operators and POVM measurements using a duality quantum computer,” Chin. Sci. Bull. 59, 2298(2014). [CrossRef]

37. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996). [CrossRef] [PubMed]

38. H. K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A 63, 022301 (2001). [CrossRef]

39. T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069 (2013). [CrossRef]

40. B. C. Ren, F. F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013). [CrossRef]

41. After the linear optical entanglement purification operation [23] in polarization DOF on the photon-pairs AB and A₁B₁, the final state of the resived photon-pair A′B′ is $\begin{array}{l} ρ_{A^{'} B^{'}} = \\ \frac{1}{T_{1} T_{2} [f^{2} + {(1 - f)}^{2}] + (1 - T_{1}) T_{2} + (1 - T_{2}) T_{1} + (1 - T_{1}) (1 - T_{2}) (2 f + 1)} {\frac{(1 - T_{1}) (1 - T_{2}) (2 f + 1)}{2} {| vac, vac 〉}_{A^{'} B^{'}} 〈 vac | + \\ \frac{(1 - T_{1}) T_{2}}{4} {| vac 〉}_{A^{'}} 〈 vac | \otimes [f {| H 〉}_{B^{'}} 〈 H | + (1 - f) {| V 〉}_{B^{'}} 〈 V |] + \frac{T_{1} (1 - T_{2})}{4} {| vac 〉}_{A^{'}} 〈 vac | \otimes [f {| V 〉}_{B^{'}} 〈 V | + (1 - f) {| H 〉}_{B^{'}} 〈 H |] + \\ \frac{(1 - T_{1}) T_{2}}{4} {| vac 〉}_{B^{'}} 〈 vac | \otimes [f {| V 〉}_{A^{'}} 〈 V | + (1 - f) {| H 〉}_{A^{'}} 〈 H |] + \frac{T_{1} (1 - T_{2})}{4} {| vac 〉}_{B^{'}} 〈 vac | \otimes [f {| H 〉}_{A^{'}} 〈 H | + (1 - f) {| V 〉}_{A^{'}} 〈 V |] + \\ \frac{T_{1} T_{2} [f^{2} + {(1 - f)}^{2}]}{2} [f^{'} {| ϕ^{+} 〉}_{A^{'} B^{'}} 〈 ϕ^{+} + (1 - f^{'}) {| ψ^{+} 〉}_{A^{'} B^{'}} 〈 ψ^{+} | |]} \end{array}$ , where $f^{'} = \frac{f^{2}}{f^{2} + {(1 - f)}^{2}}$ .

42. N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010). [CrossRef] [PubMed]

One-step hyperentanglement purification and hyperdistillation with linear optics

Abstract

1. Introduction

2. Implementation of QND parity-check measurement and qubit amplification

3. Hyperentanglement purification in spatial-mode and polarization DOFs

4. Discussion and summary

Acknowledgments

References and links

Cited By

Figures (3)

Tables (1)

Equations (15)

Optics Express

Results of D₁D₂D₃D₄(D₅D₆D₇D₈)	Operations
\|HHHH〉, \|VVVV〉, \|HVHV〉 or \|VHVH〉	$I_{H (V)}^{S}$
\|HHVV〉, \|VVHH〉, \|HVVH〉 or \|VHHV〉	$- I_{H (V)}^{S}$
\|HHHV〉, \|VVVH〉, \|HVHH〉 or \|VHVV〉	$- σ_{Z, H (V)}^{S}$
\|HHVH〉, \|VVHV〉, \|HVVV〉 or \|VHHH〉	$σ_{Z, H (V)}^{S}$