Hyperentanglement concentration for time-bin and polarization hyperentangled photons

Xi-Han Li and Shohini Ghose

Phys. Rev. A 91, 062302 – Published 2 June 2015

Abstract

We present two hyperentanglement concentration schemes for two-photon states that are partially entangled in the polarization and time-bin degrees of freedom. The first scheme distills a maximally hyperentangled state from two identical less-entangled states with unknown parameters via the Schmidt projection method. The other scheme can be used to concentrate an initial state with known parameters and requires only one copy of the initial state for the concentration process. Both these two protocols can be generalized to concentrate $N$ -photon hyperentangled Greenberger-Horne-Zeilinger states that are simultaneously entangled in the polarization and time-bin degrees of freedom. Our schemes require only linear optics and are feasible with current technology. Using the time-bin degree of freedom rather than the spatial-mode degree of freedom can provide savings in quantum resources, which makes our schemes practical and useful for long-distance quantum communication.

Received 16 January 2015

DOI:https://doi.org/10.1103/PhysRevA.91.062302

Authors & Affiliations

Xi-Han Li^1,2,* and Shohini Ghose^2,3

¹Department of Physics, Chongqing University, Chongqing 401331, China
²Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
³Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

^*xihanlicqu@gmail.com

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Vol. 91, Iss. 6 — June 2015

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Figure 1
Schematic diagram of our scheme for concentration of a hyperentangled state with unknown parameters. Two identical less-entangled states, ${| ϕ 〉}_{A_{1} B_{1}}$ and ${| ϕ 〉}_{A_{2} B_{2}}$ , that are originally prepared by the source are shared by two remote parties, Alice and Bob. The two parties change the state of $A_{2}$ and $B_{2}$ to $| ϕ^{'} 〉_{A_{2} B_{2}}$ before the concentration. The operations are omitted in this figure. The ${PBS}_{i}$ $(i = a, b)$ represents a polarizing beam splitter which transmits the horizontal polarization state $| H 〉$ and reflects the vertical polarization state $| V 〉$ . ${PC}_{L}$ $({PC}_{S})$ is a Pockel cell which effects a bit flip operation when the $L (S)$ component is present. SPM denotes a single-photon measurement which is performed on the second photon of each party. With this device, Alice and Bob implement a parity check of the polarization and the time-bin DOFs, respectively.
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Figure 2
Schematic diagram of the single-photon measurement setup which consists of only passive linear optics. Here $x_{2}$ can be $a_{2} (b_{2})$ for Alice (Bob). BS denotes the 50:50 beam splitter. The PBS oriented at $45^{\circ}$ transmits the $| + 〉$ polarization states and reflects the $| - 〉$ ones. It is used to measure the polarization state in the diagonal basis.
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Figure 3
Schematic diagram of the improved single-photon measurement setup. Here $x_{2}$ can be $a_{2} (b_{2})$ for Alice (Bob). The length difference between the $L$ and $S$ paths in the UI is set to cancel the time interval between the two time-bins. After the effect of PCs and the unbalanced interferometer, the particle is measured in the diagonal basis in both the polarization and spatial DOF by a 50:50 BS, two PBSs oriented at $45^{\circ}$ , and four single-photon detectors which are omitted in this figure.
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Figure 4
The schematic of our hyperentanglement concentration scheme for an initial state with known parameters. (a) This part is used to concentrate the polarization entanglement. $R$ represents a wave plate which rotates the horizontal polarization by the angle $θ = arccos (β / α)$ . $a_{i} (i = 1, 2, 3)$ is the spatial mode of particle $A$ . The state is postselected on the condition that the photon detector placed in path $a_{3}$ (omitted in the figure) does not click and then is guided to the input port of the following device. (b) This part is used to concentrate the time-bin entanglement. UBS represents an unbalanced beam splitter with the reflection coefficient $δ$ . The desired hyperentangled state can be obtained by postselecting the situations in which particle $A$ does not arrive in the middle time slot in these two potential spatial modes.
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Figure 5
The success probabilities of our two hyperentanglement concentration schemes. Here we choose a kind of special state where $| α | = | δ |$ and $| β | = | η |$ . The solid line and dashed line correspond to the hyperentanglement concentration with unknown and known parameters, respectively.
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