Nonstandard protocols for joint remote preparation of a general quantum state and hybrid entanglement of any dimension

Nguyen Ba An, Le Thanh Dat, and Jaewan Kim

Phys. Rev. A 98, 042329 – Published 22 October 2018

Abstract

Joint remote state preparation (JRSP) is a useful way to securely transfer quantum information encoded in quantum states between distant places without physically sending the states themselves. In this paper we study JRSP of the most general $D$ -dimensional quantum state called quDit state, with arbitrary integer $D .$ We first show that, by standard procedures, i.e., by exploiting projective measurements on the Hilbert space of the systems of concern, this task can be completed for the dimensions $2, 4$ , or 8 only. We then propose a nonstandard protocol by means of a positive operator-valued measurement (POVM), which we suitably design so that our protocol works deterministically for any dimension. We also propose a nonstandard protocol for deterministic JRSP of arbitrary hybrid quDit-quNit entanglement with another POVM. Moreover, we construct quantum circuits to realize the two POVMs in the two nonstandard JRSP protocols mentioned above. Our results may be of interest not only from a theoretical point of view but also from an experimental one.

Received 28 July 2018

DOI:https://doi.org/10.1103/PhysRevA.98.042329

Physics Subject Headings (PhySH)

Quantum entanglement

Quantum Information, Science & Technology

Authors & Affiliations

Nguyen Ba An^1,2,3,*, Le Thanh Dat¹, and Jaewan Kim³

¹Thang Long Institute of Mathematics and Applied Sciences (TIMAS), Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi 10000, Vietnam
²Center for Theoretical Physics, Institute of Physics, Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi 10072, Vietnam
³School of Computational Sciences, Korea Institute for Advanced Study (KIAS), Hoegiro 85, Dongdaemun-gu, Seoul 02455, Republic of Korea

^*nban@iop.vast.ac.vn

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Vol. 98, Iss. 4 — October 2018

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Figure 1
Quantum circuit for JRSP of the general state $|Φ〉$ of dimension $D$ in Eq. (3). A solid line represents a quDit, while a dashed line represents a classical dit. $A, B$ , and $C$ are quDits of the quantum channel $|Q〉 \equiv {|Q〉}_{A B C}$ of Eq. (4). $|Π_{k l}〉 \equiv {|Π_{k l}〉}_{B C}$ is given by Eq. (13), $V^{(k, l)}$ by Eq. (14), and $R^{(k, m)}$ by Eq. (18). POVM is the positive operator-valued measurement defined by Eq. (9), while PM is the projective measurement in the basis $\{{|Λ_{m}〉}_{B}\}$ defined by Eq. (15). The wavy line embraces quDits that are entangled with each other.
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Figure 2
Quantum circuit for the POVM defined by Eq. (9): (a) the formal performance and (b) the actual realization. A solid line represents a quDit, while a dashed line represents a classical dit. $A, B$ , and $C$ are quDits of the quantum channel $|Q〉 \equiv {|Q〉}_{A B C}$ of Eq. (4), while $a$ is an ancillary quDit in state $|χ〉 \equiv {|χ〉}_{a}$ of Eq. (19). $|Θ〉 \equiv {|Θ〉}_{A B C a}$ and $|Π_{k l}〉 \equiv {|Π_{k l}〉}_{B C}$ are given by Eqs. (25) and (13), respectively. BS is short for backward-shift gate, while PM1 and PM2 are projective measurements in the bases ${{|k〉}_{a}}$ and ${{|Ω_{l}〉}_{A}}$ of Eq. (12). The wavy line embraces quDits that are entangled with each other.
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Figure 3
Quantum circuit for JRSP of an arbitrary hybrid quDit-quNit entanglement $|Ψ〉$ of any dimensions $D$ and $N$ in Eq. (2). A solid line represents a quDit or a quNit, while a dashed line represents a classical dit or nit. QuDits $A_{1}, B_{1}, C_{1}$ and quNits $A_{2}, B_{2}, C_{2}$ belong to the quantum channel $|\bar{Q}〉 \equiv {|\bar{Q}〉}_{A_{1} B_{1} C_{1} A_{2} B_{2} C_{2}}$ of Eq. (26). $|Π_{k_{1} k_{2} l_{1} l_{2}}〉 \equiv {|Π_{k_{1} k_{2} l_{1} l_{2}}〉}_{B_{1} B_{2} C_{1} C_{2}}$ is given by Eq. (29), $V^{(k_{1}, l_{1})} = X^{k_{1}} Z^{l_{1} †}, V^{(k_{2}, l_{2})} = X^{k_{2}} Z^{l_{2} †}, R^{(k_{1}, m_{1})} = Z^{m_{1} †} X^{k_{1}}$ , and $R^{(k_{2}, m_{2})} = Z^{m_{2} †} X^{k_{2}}$ . POVM is the positive operator-valued measurement defined by Eq. (27), while PM is the projective measurement in the basis $\{{|Λ_{m_{1} m_{2}}〉}_{B_{1} B_{2}}\}$ defined by Eq. (30). The wavy line embraces quDits and quNits that are entangled with each other.
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Figure 4
Quantum circuit for the POVM defined by Eq. (27): (a) the formal performance and (b) the actual realization. A solid line represents a quDit or quNit, while a dashed line represents a classical dit or nit. QuDits $A_{1}, B_{1}, C_{1}$ and quNits $A_{2}, B_{2}, C_{2}$ belong to the quantum channel $|\bar{Q}〉 \equiv {|\bar{Q}〉}_{A_{1} B_{1} C_{1} A_{2} B_{2} C_{2}}$ of Eq. (26), while ancillary quDit $a_{1}$ and quNit $a_{2}$ belong to the state $|K〉 \equiv {|K〉}_{a_{1} a_{2}}$ of Eq. (32). $|Ξ〉 \equiv {|Ξ〉}_{a_{1} a_{2} A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}}$ and $|Π_{k_{1} k_{2} l_{1} l_{2}}〉 \equiv {|Π_{k_{1} k_{2} l_{1} l_{2}}〉}_{B_{1} B_{2} C_{1} C_{2}}$ are given by Eqs. (33) and (29), respectively. BS is short for backward-shift gate, while PM1, PM2, PM3, and PM4 are projective measurements of $a_{1}$ in the basis ${{|k_{1}〉}_{a_{1}}}, a_{2}$ in the basis ${{|k_{2}〉}_{a_{2}}}, A_{1}$ in the basis ${{|Ω_{l_{1}}^{(D)}〉}_{A_{1}}}$ of Eq. (34), and $A_{2}$ in the basis ${{|Ω_{l_{2}}^{(N)}〉}_{A_{2}}}$ of Eq. (35). The wavy line embraces quDits and quNits that are entangled with each other.
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