Complete analysis of hyperentangled Bell states assisted with auxiliary hyperentanglement

Guan-Yu Wang; Bao-Cang Ren; Fu-Guo Deng; Gui-Lu Long

doi:10.1364/OE.27.008994

1. Introduction

Entanglement is a key resource for quantum communication, such as quantum teleportation [1], quantum dense coding [2], quantum key distribution [3–5], quantum secret sharing [6], quantum secure direct communication [7–14], and so on. Hyperentanglement, simultaneous entanglement in multiple degrees of freedom (DOFs) of a quantum system, has attracted much attention as all the multiple DOFs can be used to carry information in quantum information processing [15]. Some generation protocols of hyperentangled states in different types of DOFs have been proposed theoretically or implemented experimentally [16–24], such as the realization of producing a photon pair hyperentangled in polarization and two longitudinal momentum DOFs through spontaneous parametric down-conversion with β-barium borate crystal [16] and the generation of hyperentangled states of photon systems in polarization and orbital angular momentum DOFs [17]. Hyperentanglement can be used to efficiently assist the complete analysis of Bell states [25–31] and the deterministic entanglement purification [29,32–34]. Meanwhile, hyperentanglement can speed up quantum computation largely [35–40] and improve the capacity of channel for quantum communication [41–49], and it may be a good candidate to implement the high-dimensional vector in quantum machine learning [50].

Complete analysis of hyperentangled states is prerequisite for quantum communication based on hyperentanglement, such as hyperdense coding, hyperteleportation, hyperentanglement swapping, and high-capacity quantum secure direct communication [41–49]. Linear optical elements can be utilized to construct an analyzer for hyperentangled Bell states. However, there is a bottleneck to completely discriminate all the orthogonal hyperentangled Bell states with an analyzer only utilizing linear optical elements. With linear optical elements and one copy of the input state, at most 2ⁿ⁺¹ − 1 groups out of 4ⁿ hyperentangled Bell states can be distinguished by linear evolution and local projective measurement [51]. In 2007, Wei et al. implemented a complete analyzer for hyperentangled Bell states in the polarization and the momentum DOFs utilizing two copies of photon pairs with linear optical elements [48]. Meanwhile, enlarging Hilbert space is a valid way to increase the distinguishability of an analyzer for hyperentangled Bell states with linear optical elements. In 2017, Li and Ghose classified 16 hyperentangled states in two DOFs into 12 groups with the help of the third DOF using linear optical elements [49]. In addition, utilizing nonlinear optics provided by Kerr media or an artificial atom embedded in a microcavity, complete analysis protocols for entangled states or hyperentangled states can be accomplished [44–46,52–58].

Quantum hyperdense coding is one of important high-capacity quantum communication protocols, and its accomplishment is based on an analyzer for hyperentangled states. The distinguishability of an analyzer for hyperentangled states is closely associated with the transmission capacity of a quantum hyperdense coding scheme. In principle, the upper bound of transmission capacity of the quantum hyperdense coding scheme based on 16 orthogonal hyperentangled Bell states, that is log₂16 =4 bits/photon, can be achieved when the 16 orthogonal hyperentangled Bell states can be completely discriminated [41]. With an incomplete analyzer which separates the 16 hyperentangled Bell states into 7 groups, the quantum hyperdense coding can transmit up to log₂7 =2.81 bits of classical information [47,48], and with an incomplete analyzer which classifies 16 hyperentangled Bell states into 12 groups, the quantum hyperdense coding can transmit up to log₂12 =3.58 bits of classical information [49].

Due to the significance of the complete analysis of hyperentangled Bell states in high-capacity quantum communication, in this paper we present a simple protocol for complete analysis of 16 hyperentangled Bell states of two-photon system in the polarization and the first longitudinal momentum DOFs. This complete analysis protocol is accomplished with the auxiliary hyperentangled Bell state in the frequency and the second longitudinal momentum DOFs. In the establishment of our complete analysis protocol, experimentally available optical elements including linear optical elements manipulating the polarizations and the longitudinal momentums and optical elements manipulating frequencies of photons are used, and the interference between two photons is avoided. In addition, utilizing our simple complete analysis protocol, we give a practical quantum hyperdense coding scheme allowing the transmission of log₂16 = 4 bits of classical information by sending just one photon, which is the upper bound of a quantum hyperdense coding scheme based on 16 orthogonal hyperentangled Bell states. This complete analysis protocol has a potential to be experimentally realized and has an important application in high-capacity quantum communication based on hyperentanglement.

2. Complete analysis of hyperentangled Bell states

A general hyperentangled Bell state in the polarization DOF, two longitudinal momentum DOFs, and frequency DOF of two-photon system ab can be written as

| Ψ 〉 = {| Ψ 〉}^{P} \otimes {| Ψ 〉}^{F} \otimes {| ψ 〉}^{S} \otimes {| ψ 〉}^{Ω} .

Here, |Ψ〉^P is one of the four Bell states in the polarization DOF,

{| ϕ^{\pm} 〉}^{P} \equiv \frac{1}{\sqrt{2}} (| H H 〉 \pm | V V 〉), {| ψ^{\pm} 〉}^{P} \equiv \frac{1}{\sqrt{2}} (| H V 〉 \pm | V H 〉),

where H and V are the horizontal and vertical polarizations of a photon, respectively. |Ψ〉^F is one of the four Bell states in the first longitudinal momentum DOF,

{| ϕ^{\pm} 〉}^{F} \equiv \frac{1}{\sqrt{2}} (| I I 〉 \pm | E E 〉), {| ψ^{\pm} 〉}^{F} \equiv \frac{1}{\sqrt{2}} (| I E 〉 \pm | E I 〉),

where I and E are the internal and the external spatial modes of a photon, respectively.

{| ψ 〉}^{S} = \frac{1}{\sqrt{2}} (| r r 〉 + | l l 〉)

is a Bell state in the second longitudinal momentum DOF, where r and l represent the right and the left spatial modes, respectively.

{| ψ 〉}^{Ω} = \frac{1}{\sqrt{2}} (| ω_{1} ω_{2} 〉 + | ω_{2} ω_{1} 〉)

is a Bell state in the frequency DOF, where ω₁ and ω₂ are two frequency modes of a photon. Here, the hyperentangled Bell states in the polarization and two longitudinal momentum DOFs [16] and the Bell states in the frequency DOF [59] have been achieved in experiment. And the hyperentangled Bell state in the second longitudinal momentum and the frequency DOFs is used to assist the complete analysis of the 16 hyperentangled Bell states in the polarization and the first longitudinal momentum DOFs.

To completely distinguish the 16 hyprentangled Bell states in the polarization and the first longitudinal momentum DOFs, photons a and b are injected into the same setups, one of whose schematic diagram is shown in Fig. 1. First, after the spatial pathes Ir and Il are exchanged, the hyperentangled Bell states in the polarization and the frequency DOFs would not be changed, and the hyperentangled Bell states in the first longitudinal momentum and the second longitudinal momentum DOFs evolve as follows

\begin{array}{l} {| ϕ^{\pm} 〉}^{F} \otimes {| ψ 〉}^{S} \to {| ϕ^{\pm} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r r 〉 + | l l 〉), \\ {| ψ^{\pm} 〉}^{F} \otimes {| ψ 〉}^{S} \to {| ψ^{\pm} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r l 〉 + | l r 〉) . \end{array}

Second, after photons a and b pass through the FBSs, FSs, PBSs, and HWPs sequently, the hyperentangled Bell states in the two longitudinal momentum DOFs would not be changed, and the hyperentangled Bell states in the polarization and the frequency DOFs evolve as follows

\begin{array}{l} {| ϕ^{+} 〉}^{P} \otimes {| ψ 〉}^{Ω} \to {| ϕ^{+} 〉}^{P} \otimes \frac{1}{\sqrt{2}} (| a_{1} b_{2} 〉 + | a_{2} b_{1} 〉), \\ {| ϕ^{-} 〉}^{P} \otimes {| ψ 〉}^{Ω} \to {| ψ^{+} 〉}^{P} \otimes \frac{1}{\sqrt{2}} (| a_{1} b_{2} 〉 + | a_{2} b_{1} 〉), \\ {| ψ^{+} 〉}^{P} \otimes {| ψ 〉}^{Ω} \to {| ϕ^{-} 〉}^{P} \otimes \frac{1}{\sqrt{2}} (| a_{1} b_{1} 〉 + | a_{2} b_{2} 〉), \\ {| ψ^{-} 〉}^{P} \otimes {| ψ 〉}^{Ω} \to {- | ψ^{-} 〉}^{P} \otimes \frac{1}{\sqrt{2}} (| a_{1} b_{1} 〉 + | a_{2} b_{2} 〉) . \end{array}

Subsequently, after photons a and b pass through the BSs, the hyperentangled Bell states in the polarization and the frequency DOFs would not be changed, and the hyperentangled Bell states in the first and the second longitudinal momentum DOFs evolve as follows

\begin{array}{l} {| ϕ^{+} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r r 〉 + | l l 〉) \to {| ϕ^{+} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r r 〉 + | l l 〉), \\ {| ϕ^{-} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r r 〉 + | l l 〉) \to {| ψ^{+} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r r 〉 + | l l 〉), \\ {| ψ^{+} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r l 〉 + | l r 〉) \to {| ϕ^{-} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r l 〉 + | l r 〉), \\ {| ψ^{-} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r l 〉 + | l r 〉) \to - {| ψ^{-} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| r l 〉 + | l r 〉) . \end{array}

Finally, after passing through the PBSs, photons a and b in different hyperentangled Bell states would emit from different exits of the setups and trigger different single-photon detectors. We take the initial hyperentangled Bell state |ϕ⁻〉^P ⊗ |ψ⁺〉^F ⊗ |ψ〉^S ⊗ |ψ〉^Ω as an example. After the operations with the setups as shown in Fig. 1, |ϕ⁻〉^P ⊗ |ψ⁺〉^F ⊗ |ψ〉^S ⊗ |ψ〉^Ω would evolve into ${| ψ^{+} 〉}^{P} \otimes {| ϕ^{-} 〉}^{F} \otimes \frac{1}{\sqrt{2}} (| a_{1} b_{2} 〉 + | a_{2} b_{1} 〉) \otimes \frac{1}{\sqrt{2}} (| r l 〉 + | l r 〉)$ , which will lead the detection results $B_{a}^{1 H} C_{b}^{2 V}$ , $B_{a}^{2 H} C_{b}^{1 V}$ , $B_{a}^{1 V} C_{b}^{2 H}$ , $B_{a}^{2 V} C_{b}^{1 H}$ , $C_{a}^{1 H} B_{b}^{2 V}$ , $C_{a}^{2 H} B_{b}^{1 V}$ , $C_{a}^{1 V} B_{b}^{2 H}$ , $C_{a}^{2 V} B_{b}^{1 H}$ , $A_{a}^{1 H} D_{b}^{2 V}$ , $A_{a}^{2 H} D_{b}^{1 V}$ , $A_{a}^{1 V} D_{b}^{2 H}$ , $A_{a}^{2 V} D_{b}^{1 H}$ , $D_{a}^{1 H} A_{b}^{2 V}$ , $D_{a}^{2 H} A_{b}^{1 V}$ , $D_{a}^{1 V} A_{b}^{2 H}$ , or $D_{a}^{2 V} A_{b}^{1 H}$ , as the tenth group in Table 1. The complete relationship between the detection results and the initial hyperentangled Bell states of the two-photon system ab is in Table 1, according to which one can make a judgement on the hyperentangled Bell state that two-photon system ab is in.

Fig. 1 Schematic diagram of the setup for the complete analysis of the 16 hyperentangled Bell states of two-photon system ab. FBS is a frequency beam splitter which divides photon with frequency ω₁ or ω₂ into spatial mode i₁ or i₂ (i = a, b), respectively, and FS is a frequency shifter which eliminates the frequency distinguishability. PBS is a polarization beam splitter which transmits the photon in the horizontal polarization state |H〉 and reflects the photon in the vertical polarization state |V〉. HWP is a half-wave plate which performs the Hadamard operation $[| H 〉 \to \frac{1}{\sqrt{2}} (| H 〉 + | V 〉), | V 〉 \to \frac{1}{\sqrt{2}} (| H 〉 - | V 〉)]$ on the polarization DOF of a photon. BS is a 50:50 beam splitter which performs Hadamard operation $[| I 〉 \to \frac{1}{\sqrt{2}} (| I 〉 + | E 〉), | E 〉 \to \frac{1}{\sqrt{2}} (| I 〉 - | E 〉)]$ on the first longitudinal momentum DOF of a photon.

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Table 1. The relationship between the 16 hyperentangled Bell states in the polarization and the first longitudinal momentum DOFs and the detection results.

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From the above process, one can see that the complete analysis of 16 hyperentangled Bell states can be accomplished with our complete analysis protocol. In our complete analysis protocol for hyperentangled Bell states, the auxiliary Bell states in the frequency DOF and the second longitudinal momentum DOF increase the distinguishability of the hyperentangled Bell states in the polarization DOF and the first longitudinal momentum DOF, respectively. Specifically speaking, after the evolutions, the phase difference between the Bell states in the polarization DOF can be distinguished with the parity difference between the Bell states in the polarization DOF, and the parity difference between the Bell states in the polarization DOFs can be distinguished with the parity difference between the Bell states in the frequency DOF, which have been transformed into the Bell states in the spatial modes represented by a₁,₂ and b₁,₂, as shown in Eq. (5). Similarly, after the evolutions, the phase information of the Bell states in the first longitudinal momentum DOF can be reflected by the parity difference between the Bell states in the first longitudinal momentum DOF, and the parity information of the Bell states in the first longitudinal momentum DOF can be reflected by the parity difference between the Bell states in the second longitudinal momentum DOF, as shown in Eq. (6). The parity differences between the Bell states in every DOF can be read by the clicks of the single-photon detectors in different exits. Therefore, the 16 hyperentangled Bell states in the polarization and the first longitudinal momentum DOFs can be discriminated completely.

3. Discussion and summary

Our complete analysis protocol for hyperentangled Bell states of two-photon system can assist in implementing a quantum hyperdense coding scheme. Next, we implement a quantum hyperdense coding scheme based on 16 hyperentangled Bell states of a two-photon system utilizing our complete analysis protocol. The quantum hyperdense coding can be implemented with three steps as follows:

(1) Initially the receiver, say Bob, prepares a two-photon system ab in the hyperentangled Bell state (e. g., |ϕ⁺〉^P ⊗ |ϕ⁺〉^F ⊗ |ψ〉^S ⊗ |ψ〉^Ω), and sends photon a to the sender, say Alice; (2) After receiving photon a from Bob, Alice encodes the classical information through performing one of the 16 local unitary operations ${U_{m}^{P}} \otimes {U_{n}^{F}} (m, n = 0, 1, 2, 3)$ on photon a, where local unitary operations $U_{m}^{P}$ can respectively represent the classical information 00, 01, 10, 11 encoded in the polarization DOF, and local unitary operations $U_{n}^{F}$ can respectively correspond to the classical information 00, 01, 10, 11 encoded in the first longitudinal momentum DOF. $U_{0}^{P} = | H 〉〈 H | + | V 〉〈 V |$ is unit operation in the polarization DOF, which means no operation needs to be done on photon a. $U_{1}^{P} = | H 〉〈 V | + | V 〉〈 H |$ and $U_{2}^{P} = | H 〉〈 H | - | V 〉〈 V |$ can be achieved by half-wave plate set at 45° and quarter-wave plate set at 90°, respectively. $U_{3}^{P} = | V 〉〈 H | - | H 〉〈 V |$ can be achieved by the combination of half-wave plate set at 45° and quarter-wave plate set at 90°. $U_{0}^{F} = | I 〉〈 I | + | E 〉〈 E |$ is unit operation in the first longitudinal momentum DOF, which means no operation needs to be done on photon a. $U_{1}^{F} = | I 〉〈 E | + | E 〉〈 I |$ can be achieved by exchanging the two spatial modes I and E and $U_{2}^{F} = | I 〉〈 I | - | E 〉〈 E |$ can be achieved by tilting a thin glass plate in the appropriate spatial mode. The operation $U_{3}^{F} = | E 〉〈 I | - | I 〉〈 E |$ is the combination of $U_{1}^{F}$ and $U_{2}^{F}$ . After Alice encodes the classical information on photon a, the initial hyperentangled Bell state |ϕ⁺〉^P ⊗ |ϕ⁺〉^F would be changed into one of the 16 hyperentangled Bell states in the polarization and the first longitudinal momentum DOFs, and Alice returns photon a to Bob; (3) After receiving photon a from Alice, Bob decodes information by performing analysis protocol on two-photon system ab. The relationship between the classical information, the local unitary operations which Alice performs, and the hyperentangled Bell states which Alice changes the initial hyperentangled Bell state |ϕ⁺〉^P ⊗ |ϕ⁺〉^F into after performing encoding operations is shown in Table 2.

Table 2. The relationship between the classical information, the local unitary operations, and the 16 hyperentangled Bell states in the polarization and the first longitudinal momentum DOFs.

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From Table 2, one can see that there is a one-to-one correspondence between the classical information, the unitary operation, and the hyperentangled Bell state. Together with the relationship between the hyperentangled Bell states and the detecting results, the classical information is decoded. We take the classical information 1001 encoded by Alice as an example to demonstrate the process of our quantum hyperdense coding scheme. After Alice performs local unitary operation $U_{2}^{P} \otimes U_{1}^{F}$ on photon a, the initial hyperentangled Bell state |ϕ⁺〉^P ⊗ |ϕ⁺〉^F ⊗|ψ〉^S ⊗ |ψ〉^Ω of the two-photon system would be changed into |ϕ−〉^P ⊗ |ψ⁺〉^F ⊗ |ψ〉^S ⊗ |ψ〉^Ω, which would be known by Bob with our complete analysis protocol. Thus Bob knows the unitary operation performed on photon a by Alice is $U_{2}^{P} \otimes U_{1}^{F}$ and successfully decodes the classical information encoded by Alice is 1001. In this quantum hyperdense coding scheme, our complete analysis protocol for 16 hyperentangled Bell states of two-photon system plays an indispensable role, and the feature of complete discrimination of our analysis protocol allows the quantum hyperdense coding based on 16 hyperentangled Bell states to achieve its upper bound of transmission capacity, that is log₂16 = 4 bits/photon.

In our protocol for complete analysis of hyperentangled Bell states, the optical elements including linear optical elements (PBS, BS, and HWP) performing operations on the polarizations and longitudinal momentums and optical devices (FBS and FS) performing operations on the frequencies of photons are experimentally available. The FBS, which is used to divide photons with different frequencies into different spatial modes, can be realized with devices such as optical cavity [60,61], asymmetric Mach-Zehnder on the frequency encoding [62,63], and Fiber Bragg Grating [64, 65]. The FS, which is used to eliminate the frequency distinguishability, can be implemented by means of frequency up-conversion process or down-conversion process [66–70]. For instance, in 2011, Iuta et al. demonstrated the frequency down-coversion process of half a polarization-entangled photon pair and observed the entanglement between the down-converted photon and the other half being retained [70]. The detection results of our complete analysis protocol are reflected by the different coincident clicks of two single-photon detectors. Therefore, the finite efficiency of single-photon detectors has an effect on the efficiency of our complete analysis protocol. When single-photon detectors of a finite efficiency η_d are used, the efficiency of our complete analysis protocol would be decreased by a scale of $η_{d}^{2}$ . Meanwhile, during the process of hyperentangled Bell states analysis, the interference between two photons a and b is avoided, which relaxes the requirement for realization in experiment.

In summary, we present a simple protocol for complete analysis of hyperentangled Bell states of two-photon system in the polarization and the first longitudinal momentum DOFs assisted with the auxiliary hyperentangled Bell state in the frequency and the second longitudinal momentum DOFs. This simple analysis protocol can completely discriminate 16 hyperentangled Bell states in the polarization and the first longitudinal momentum DOFs of two-photon system. Meanwhile, utilizing our complete analysis protocol, we propose a quantum hyperdense coding scheme allowing the transmission of log₂16 = 4 bits of classical information, which is the upper bound of a quantum hyperdense coding scheme based on 16 orthogonal hyperentangled Bell states. In addition, the techniques required to realize our complete analysis protocol are within the current bounds, as the hyperentangled states in the four DOFs could be realized in experiment, manipulating the photons with the optical elements which are used in our protocol is skillful, and the interference between two photons is avoided. Our simple protocol for complete analysis of hyperentangled Bell states is useful for high-capacity quantum communication.

Funding

China Postdoctoral Science Foundation (2018M641318); National Natural Science Foundation of China (61727801, 11774197, 20171311628, 11604226, 11674033 ); National Key Research and Development Program of China (2017YFA0303700).

Acknowledgments

Support from Beijing Advanced Innovation Center for Future Chip (ICFC) is gratefully acknowledged).

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50. Y. B. Sheng and L. Zhou, “Distributed secure quantum machine learning,” Sience Bull. 62, 1025–1029 (2017).

51. N. Pisenti, C. P. E. Gaebler, and T. W. Lynn, “Distinguishability of hyperentangled Bell states by linear evolution and local projective measurement,” Phys. Rev. A 84, 022340 (2011). [CrossRef]

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	states	detection results
1	\|ϕ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$
2	\|ϕ⁺〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$
3	\|ϕ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$
4	\|ϕ⁺〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$
5	\|ψ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$
6	\|ψ⁺〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$
7	\|ψ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$
8	\|ψ⁺〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$
9	\|ϕ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$
10	\|ϕ⁻〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$
11	\|ϕ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$
12	\|ϕ⁻〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$
13	\|ψ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$
14	\|ψ⁻〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$
15	\|ψ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$
16	\|ψ⁻〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$

classical information	operations	states	classical information	operations	states
0000	$U_{0}^{P} \otimes U_{0}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	0001	$U_{0}^{P} \otimes U_{1}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ψ⁺〉 ^F
0010	$U_{0}^{P} \otimes U_{2}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	0011	$U_{0}^{P} \otimes U_{3}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ψ⁻〉 ^F
0100	$U_{1}^{P} \otimes U_{0}^{F}$	\|ψ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	0101	$U_{1}^{P} \otimes U_{1}^{F}$	\|ψ⁺〉 ^P ⊗ \|ψ⁺〉 ^F
0110	$U_{1}^{P} \otimes U_{2}^{F}$	\|ψ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	0111	$U_{1}^{P} \otimes U_{3}^{F}$	\|ψ⁺〉 ^P ⊗ \|ψ⁻〉 ^F
1000	$U_{2}^{P} \otimes U_{0}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	1001	$U_{2}^{P} \otimes U_{1}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ψ⁺〉 ^F
1010	$U_{2}^{P} \otimes U_{2}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	1011	$U_{2}^{P} \otimes U_{3}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ψ⁻〉 ^F
1100	$U_{3}^{P} \otimes U_{0}^{F}$	\|ψ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	1101	$U_{3}^{P} \otimes U_{1}^{F}$	\|ψ⁻〉 ^P ⊗ \|ψ⁺〉 ^F
1110	$U_{3}^{P} \otimes U_{2}^{F}$	\|ψ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	1111	$U_{3}^{P} \otimes U_{3}^{F}$	\|ψ⁻〉 ^P ⊗ \|ψ⁻〉 ^F

	states	detection results
1	\|ϕ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$
2	\|ϕ⁺〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$
3	\|ϕ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$
4	\|ϕ⁺〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{2}}^{H (V)}$ , $A_{a_{2}}^{H (V)} C_{b_{1}}^{H (V)}$ , $C_{a_{1}}^{H (V)} A_{b_{2}}^{H (V)}$ , $C_{a_{2}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{1}}^{H (V)} D_{b_{2}}^{H (V)}$ , $B_{a_{2}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{1}}^{H (V)} B_{b_{2}}^{H (V)}$ , $D_{a_{2}}^{H (V)} B_{b_{1}}^{H (V)}$
5	\|ψ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$
6	\|ψ⁺〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$
7	\|ψ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$
8	\|ψ⁺〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{1}}^{H (V)}$ , $A_{a_{2}}^{H (V)} C_{b_{2}}^{H (V)}$ , $C_{a_{1}}^{H (V)} A_{b_{1}}^{H (V)}$ , $C_{a_{2}}^{H (V)} A_{b_{2}}^{H (V)}$ , $B_{a_{1}}^{H (V)} D_{b_{1}}^{H (V)}$ , $B_{a_{2}}^{H (V)} D_{b_{2}}^{H (V)}$ , $D_{a_{1}}^{H (V)} B_{b_{1}}^{H (V)}$ , $D_{a_{2}}^{H (V)} B_{b_{2}}^{H (V)}$
9	\|ϕ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$
10	\|ϕ⁻〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$
11	\|ϕ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$
12	\|ϕ⁻〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{2}}^{V (H)}$ , $A_{a_{2}}^{H (V)} C_{b_{1}}^{V (H)}$ , $C_{a_{1}}^{H (V)} A_{b_{2}}^{V (H)}$ , $C_{a_{2}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{1}}^{H (V)} D_{b_{2}}^{V (H)}$ , $B_{a_{2}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{1}}^{H (V)} B_{b_{2}}^{V (H)}$ , $D_{a_{2}}^{H (V)} B_{b_{1}}^{V (H)}$
13	\|ψ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	$A_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$
14	\|ψ⁻〉 ^P ⊗ \|ψ⁺〉 ^F	$A_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$
15	\|ψ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	$A_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$
16	\|ψ⁻〉 ^P ⊗ \|ψ⁻〉 ^F	$A_{a_{1}}^{H (V)} C_{b_{1}}^{V (H)}$ , $A_{a_{2}}^{H (V)} C_{b_{2}}^{V (H)}$ , $C_{a_{1}}^{H (V)} A_{b_{1}}^{V (H)}$ , $C_{a_{2}}^{H (V)} A_{b_{2}}^{V (H)}$ , $B_{a_{1}}^{H (V)} D_{b_{1}}^{V (H)}$ , $B_{a_{2}}^{H (V)} D_{b_{2}}^{V (H)}$ , $D_{a_{1}}^{H (V)} B_{b_{1}}^{V (H)}$ , $D_{a_{2}}^{H (V)} B_{b_{2}}^{V (H)}$

classical information	operations	states	classical information	operations	states
0000	$U_{0}^{P} \otimes U_{0}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	0001	$U_{0}^{P} \otimes U_{1}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ψ⁺〉 ^F
0010	$U_{0}^{P} \otimes U_{2}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	0011	$U_{0}^{P} \otimes U_{3}^{F}$	\|ϕ⁺〉 ^P ⊗ \|ψ⁻〉 ^F
0100	$U_{1}^{P} \otimes U_{0}^{F}$	\|ψ⁺〉 ^P ⊗ \|ϕ⁺〉 ^F	0101	$U_{1}^{P} \otimes U_{1}^{F}$	\|ψ⁺〉 ^P ⊗ \|ψ⁺〉 ^F
0110	$U_{1}^{P} \otimes U_{2}^{F}$	\|ψ⁺〉 ^P ⊗ \|ϕ⁻〉 ^F	0111	$U_{1}^{P} \otimes U_{3}^{F}$	\|ψ⁺〉 ^P ⊗ \|ψ⁻〉 ^F
1000	$U_{2}^{P} \otimes U_{0}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	1001	$U_{2}^{P} \otimes U_{1}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ψ⁺〉 ^F
1010	$U_{2}^{P} \otimes U_{2}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	1011	$U_{2}^{P} \otimes U_{3}^{F}$	\|ϕ⁻〉 ^P ⊗ \|ψ⁻〉 ^F
1100	$U_{3}^{P} \otimes U_{0}^{F}$	\|ψ⁻〉 ^P ⊗ \|ϕ⁺〉 ^F	1101	$U_{3}^{P} \otimes U_{1}^{F}$	\|ψ⁻〉 ^P ⊗ \|ψ⁺〉 ^F
1110	$U_{3}^{P} \otimes U_{2}^{F}$	\|ψ⁻〉 ^P ⊗ \|ϕ⁻〉 ^F	1111	$U_{3}^{P} \otimes U_{3}^{F}$	\|ψ⁻〉 ^P ⊗ \|ψ⁻〉 ^F

Complete analysis of hyperentangled Bell states assisted with auxiliary hyperentanglement

Abstract

1. Introduction

2. Complete analysis of hyperentangled Bell states

3. Discussion and summary

Funding

Acknowledgments

References

Cited By

Figures (1)

Tables (2)

Equations (6)

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