D-DIMENSIONAL CHAIN TELEPORTATION METHOD FOR RANDOM TRANSMISSION BASED ON MEASUREMENT RESULTS OF RELAY NODES

Abstract

The present invention discloses a d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes. The method includes: two communicating parties are an information sender Alice and an information receiver Bob, a particle t carries an unknown quantum state and is held by the information sender Alice, Alice holds the particle t and a particle A.sub.1, a first intermediate node Charlie 1 holds a particle B.sub.1 and a particle A.sub.2, a second intermediate node Charlie 2 holds a particle B.sub.2 and a particle A.sub.3, . . . , and a k.sup.th(k=1, 2, 3, . . . , P) intermediate node Charlie k holds a particle B.sub.k and a particle A.sub.k+1. The beneficial effect of the present invention is as follows: any relay node can randomly transmit its generalized Bell measurement result to the information sender Alice or the information receiver Bob, thereby greatly reducing connection restrictions of a classical channel.

Claims

1. A d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes, wherein two communicating parties are an information sender Alice and an information receiver Bob, a particle t carries an unknown quantum state and is held by the information sender Alice, Alice holds the particle t and a particle A.sub.1, a first intermediate node Charlie 1 holds a particle B.sub.1 and a particle A.sub.2, a second intermediate node Charlie 2 holds a particle B.sub.2 and a particle A.sub.3, . . . , and a k.sup.th(k=1, 2, 3, . . . , P) intermediate node Charlie k holds a particle B.sub.k and a particle A.sub.k+1, wherein P is a positive integer, the information receiver Bob at a target node is a (P+2).sup.th node in a multihop quantum teleportation system and holds a particle B.sub.P+1, every two adjacent nodes share a two-bit Bell state quantum channel, to form a chain communication channel, and entangled channels have the same form and are: ${.Math. φ .Math.}_{A_{k} B_{k}} = \frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{k} B_{k}} (k = 1, 2, .Math., P + 1);$ P intermediate nodes make generalized Bell measurement on the two particles held by them respectively, to establish an entangled channel between the information sender Alice and the information receiver Bob; the P intermediate nodes respectively consider the connection condition of classical channels with the information sender Alice and the information receiver Bob, and transmit respective generalized Bell measurement results to Alice or Bob according to actual condition, Alice or Bob determines a matrix transformation operation to be performed according to the respective measurement results they receive and adjusts the entangled channel, so that at this time, a direct entangled channel between the information sender Alice and the information receiver Bob can be established; and the multihop quantum teleportation system is simplified into a single-hop teleportation system form, to perform a single-hop quantum teleportation process, the information sender Alice performs a joint Bell measurement on the held particle t and particle A.sub.1 and transmits her measurement result to the information receiver Bob, and Bob performs a corresponding unitary operation on his particle B.sub.P+1 according to the received result to recover information of the transferred unknown quantum state.

2. The d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes according to claim 1, wherein when “the P intermediate nodes respectively consider the connection condition of classical channels with the information sender Alice and the information receiver Bob, and transmit respective generalized Bell measurement results to Alice or Bob according to actual condition, Alice or Bob determines the matrix transformation operation to be performed according to the respective measurement results they receive and adjusts the entangled channel”, an adjusted quantum channel system has the following form: $.Math. φ .Math. = {(\frac{1}{\sqrt{d}})}^{P + 1} {(\frac{1}{\sqrt{d}})}^{P - 1} \overset{d - 1}{\underset{m_{1}, n_{1} = 0}{.Math.}} {.Math.}_{m_{2}, n_{2} = 0}^{d - 1} .Math. {.Math.}_{m_{P}, n_{P} = 0}^{d - 1} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} {.Math. Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} .Math. {.Math. Φ_{m_{P} n_{P}} .Math.}_{B_{P} A_{P + 1}} U_{m_{1} n_{1}}^{A_{1} / B_{P + 1}} U_{m_{1} n_{1}}^{A_{1} / B_{P + 1}} .Math. U_{m_{P} n_{P}}^{A_{1} / B_{P + 1}} {.Math. Φ)}_{A_{1} B_{P + 1}} = {(\frac{1}{\sqrt{d}})}^{P + 1} {(\frac{1}{\sqrt{d}})}^{P - 1} \overset{d - 1}{\underset{m_{1}, n_{1} = 0}{.Math.}} {.Math.}_{m_{2}, n_{2} = 0}^{d - 1} .Math. {.Math.}_{m_{P}, n_{P} = 0}^{d - 1} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} {.Math. Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} .Math. {.Math. Φ_{m_{P} n_{P}} .Math.}_{B_{P} A_{P + 1}} (\underset{{a}}{.Math.} U_{m_{a} n_{a}}^{A_{1}}) (\underset{{b}}{.Math.} U_{m_{b} n_{b}}^{B_{P + 1}}) (\frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{1} B_{P + 1}})$ in the foregoing formula, ${.Math. Φ_{m_{k} n_{k}} .Math.}_{B_{k} A_{k + 1}} (k = 1, 2, .Math., P)$ represents the generalized Bell measurement result of the k.sup.th intermediate node Charlie k, and ${.Math. Φ_{m_{k} n_{k}} .Math.}_{B_{k} A_{k + 1}} = \frac{1}{\sqrt{(1}} {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j m_{k}} .Math. j .Math. .Math. j \oplus n_{k} .Math. (k = 1, 2, .Math., P);$ $U_{m_{k} n_{k}}^{A_{1} / B_{P + 1}} (k = 1, 2, .Math., P)$ represents that after measuring the held particles B.sub.k and A.sub.k+1 to obtain |ϕ.sub.m.sub.k.sub.n.sub.k custom-character .sub.B.sub.k.sub.A.sub.k+1, Charlie k transmits the measurement result to Alice or Bob according to actual condition; it is assumed that (U.sub.A.sub.1).sup.−1=Π.sub.{a}(U.sub.m.sub.a.sub.n.sub.a.sup.A.sup.1).sup.−1, which corresponds to a unitary operation that Alice needs to perform on the particle A.sub.1 after Alice summarizing the received measurement results, wherein {a} is a set of sequence numbers of all intermediate nodes that have transmitted the measurement results to Alice; similarly, it is assumed that (U.sub.B.sub.P+1).sup.−1=Π.sub.{b}(U.sub.m.sub.b.sub.n.sub.b.sup.B.sup.P+1).sup.−1, which corresponds to a unitary operation that Bob needs to perform on the particle B.sub.P+1 after Bob summarizing the received measurement results, wherein {b} is a set of sequence numbers of all intermediate nodes that have transmitted the measurement results to Bob; and after performing the corresponding unitary operations, Alice and Bob can transform an entangled state of the particles A.sub.1 and B.sub.P+1 into the form of ${.Math. φ .Math.}_{A_{1} B_{P + 1}} = \frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{1} B_{P + 1}},$ wherein U.sub.m.sub.a.sub.n.sub.a.sup.A.sup.1(U.sub.m.sub.a.sub.n.sub.a.sup.A.sup.1).sup.−1=1; U.sub.m.sub.b.sub.n.sub.b.sup.B.sup.P+1(U.sub.m.sub.b.sub.n.sub.b.sup.B.sup.P+1).sup.−1=1.

3. The d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes according to claim 1, wherein “the P intermediate nodes respectively consider the connection condition of classical channels with the information sender Alice and the information receiver Bob” comprises whether classical information can be transmitted.

4. The d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes according to claim 1, wherein “the P intermediate nodes respectively consider the connection condition of classical channels with the information sender Alice and the information receiver Bob” comprises whether information communication is smooth.

5. The d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes according to claim 1, wherein “the P intermediate nodes respectively consider the connection condition of classical channels with the information sender Alice and the information receiver Bob” comprises transmission efficiency.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0042] FIG. 1 is a flowchart of a d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes according to the present invention.

[0043] FIG. 2 is a particle distribution diagram of a d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes according to the present invention.

[0044] FIG. 3 is a schematic diagram in which the information sender Alice, the information receiver Bob, and the P intermediate nodes perform entanglement swapping to establish quantum channels according to the present invention.

[0045] FIG. 4 is a schematic diagram of particle distribution of a 2-level 4-hop chain teleportation method according to Embodiment 1 of the present invention.

[0046] FIG. 5 is a schematic diagram of particle distribution of a 3-level 3-hop chain teleportation method according to Embodiment 2 of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0047] The present invention is further described below with reference to the accompanying drawings and specific embodiments, to enable a person skilled in the art to better understand and implement the present invention. However, the embodiments are not used to limit the present invention.

[0048] The technical terms of the present invention are described:

[0049] 1. Generalized Bell Basis

[0050] The generalized Bell basis is a maximally entangled state formed by two multi-level particles, and forms a group of complete orthogonal bases in a d (the quantity of levels)-dimensional Hilbert space, the specific form is as follows:

[00007] $| ψ_{r s} .Math. = \frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j r} .Math. j .Math. .Math. j \oplus s .Math. (r, s = 0, 1, 2, . . ., d - 1) .$

[0051] 2. Operation of Unifying Forms of Entangled Channels

[0052] In the present invention, an entangled channel system of the information sender Alice and the information receiver Bob has the following form:

[00008] $.Math. φ .Math. = {(\frac{1}{\sqrt{d}})}^{P + 1} {(\frac{1}{\sqrt{d}})}^{P - 1} \overset{d - 1}{\underset{m_{1}, n_{1} = 0}{.Math.}} {.Math.}_{m_{2}, n_{2} = 0}^{d - 1} .Math. {.Math.}_{m_{P}, n_{P} = 0}^{d - 1} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} {.Math. Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} .Math. {.Math. Φ_{m_{P} n_{P}} .Math.}_{B_{P} A_{P + 1}} U_{m_{1} n_{1}}^{A_{1} / B_{P + 1}} U_{^{m_{2} n_{2}}}^{A_{1} / B_{P + 1}} .Math. U_{m_{P} n_{P}}^{A_{1} / B_{P + 1}} {.Math. Φ .Math.}_{A_{1} B_{P + 1}} = {(\frac{1}{\sqrt{d}})}^{P + 1} {(\frac{1}{\sqrt{d}})}^{P - 1} \overset{d - 1}{\underset{m_{1}, n_{1} = 0}{.Math.}} {.Math.}_{m_{2}, n_{2} = 0}^{d - 1} .Math. {.Math.}_{m_{P}, n_{P} = 0}^{d - 1} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} {.Math. Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} .Math. {.Math. Φ_{m_{P} n_{P}} .Math.}_{B_{P} A_{P + 1}} (\underset{{a}}{.Math.} U_{m_{a} n_{a}}^{A_{1}}) (\underset{{b}}{.Math.} U_{m_{b} n_{b}}^{B_{P + 1}}) (\frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{1} B_{P + 1}})$

[0053] After a k.sup.th intermediate node Charlie k measures the held particles B.sub.k and A.sub.k+1 to obtain |ϕ.sub.m.sub.k.sub.n.sub.k custom-character .sub.B.sub.k.sub.A.sub.k+1 (k=1, 2, . . . , P), if Charlie transmits his measurement result to Alice, Alice needs to perform an operation (U.sub.m.sub.a.sub.n.sub.a.sup.A.sup.1).sup.−1 on the particle A.sub.1. If Charlie transmits his measurement result to Bob, Bob needs to perform an operation (U.sub.m.sub.k.sub.n.sub.k.sup.B.sup.P+1).sup.−1 on the particle B.sub.P+1. After summing up all received measurement results, Alice and Bob respectively perform matrix transformations (U.sub.A.sub.1).sup.−1=Π.sub.{a}(U.sub.m.sub.a.sub.n.sub.a.sup.A.sup.1).sup.−1 and (U.sub.B.sub.P+1).sup.−1=Π.sub.{b}(U.sub.m.sub.b.sub.n.sub.b.sup.B.sup.P+1).sup.−1, and respectively transform entangled states of the particles A.sub.1 and B.sub.P+1 into a unified form:

[00009] ${| φ .Math.}_{A_{1} B_{P + 1}} = \frac{1}{\sqrt{d}} Σ_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{1} B_{P + 1}} .$

The related matrix expressions are as follows:

[00010] $U_{m_{a} n_{a}}^{A_{1}} = {.Math.}_{j = 0}^{d - 1} e^{- \frac{2 π i}{d} j m_{a}} .Math. j \oplus n_{a} .Math. .Math. j .Math.$ $U_{m_{b} n_{b}}^{B_{P + 1}} = {.Math.}_{j = 0}^{d - 1} e^{- \frac{2 π i}{d} j m_{b}} .Math. j .Math. .Math. j \oplus n_{b} .Math.$ ${(U_{m_{a} n_{a}}^{A_{1}})}^{- 1} = {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j m_{a}} .Math. j .Math. .Math. j \oplus n_{a} .Math.$ $(U_{m_{b} n_{b}}^{B_{P + 1}}) = {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j m_{b}} .Math. j \oplus n_{b} .Math. .Math. j | .$

[0054] In a d-level chain teleportation communication system, the information sender Alice and the information receiver Bob that do not directly share an entangled pair can generate a direct entangled state with the help of P intermediate nodes, to establish a quantum entangled channel, to complete the process that the information sender Alice transfers a single-particle multi-level unknown quantum state to the information receiver Bob. In this multihop teleportation system, the relay node can randomly transmit its generalized Bell measurement result to the information sender Alice or the information receiver Bob. This includes the following steps:

[0055] Step 1: Construct a chain channel. Two communicating parties are the information sender Alice and the information receiver Bob, and the particle t carries an unknown quantum state and is held by the information sender Alice, Alice holds the particle t and a particle A.sub.1, the first intermediate node Charlie 1 holds a particle B.sub.1 and a particle A.sub.2, the second intermediate node Charlie 2 holds a particle B.sub.2 and the particle A.sub.3, . . . , and a k.sup.th(k=1, 2, 3, . . . , P) intermediate node Charlie k holds a particle B.sub.k and a particle A.sub.k+1, where P is a positive integer, the information receiver Bob at the target node is a (P+2).sup.th node in a multihop quantum teleportation system and holds the particle B.sub.P+1, every two adjacent nodes share the two-bit Bell state quantum channel, to form the chain communication channel, and entangled channels have the same form and are:

[00011] ${| φ .Math.}_{A_{k} B_{k}} = \frac{1}{\sqrt{d}} Σ_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{k} B_{k}} (k = 1, 2, . . ., P + 1);$

[0056] Step 2: Construct a direct channel. P intermediate nodes respectively perform generalized Bell measurements on respective held two particles. After the measurements are completed, in consideration of the connection condition of classical channels with the information sender Alice and the information receiver Bob (for example, whether classical information can be transmitted, whether information communication is smooth, and transmission efficiency), the P intermediate nodes select, according to actual condition, to transmit respective generalized Bell measurement results to Alice or Bob.

[0057] Step 3: Adjust the channel. Alice or Bob determines the matrix transformation operation to be performed according to the respective measurement results they receive and adjusts the entangled channel. An adjusted quantum channel system has the following form:

[00012] $.Math. φ .Math. = {(\frac{1}{\sqrt{d}})}^{P + 1} {(\frac{1}{\sqrt{d}})}^{P - 1} \overset{d - 1}{\underset{m_{1}, n_{1} = 0}{.Math.}} {.Math.}_{m_{2}, n_{2} = 0}^{d - 1} .Math. {.Math.}_{m_{P,} n_{P} = 0}^{d - 1} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} {.Math. Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} .Math. {.Math. Φ_{m_{P} n_{P}} .Math.}_{B_{P} A_{P + 1}} U_{m_{1} n_{1}}^{A_{1} / B_{P + 1}} U_{m_{2} n_{2}}^{A_{1} / B_{P + 1}} .Math. U_{m_{P} n_{P}}^{A_{1} / B_{P + 1}} {.Math. Φ)}_{A_{1} B_{P + 1}} = {(\frac{1}{\sqrt{d}})}^{P + 1} {(\frac{1}{\sqrt{d}})}^{P - 1} \overset{d - 1}{\underset{m_{1}, n_{1} = 0}{.Math.}} {.Math.}_{m_{2}, n_{2} = 0}^{d - 1} .Math. {.Math.}_{m_{P}, n_{P} = 0}^{d - 1} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} {.Math. Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} .Math. {.Math. Φ_{m_{P} n_{P}} .Math.}_{B_{P} A_{P + 1}} (\underset{{a}}{.Math.} U_{m_{a} n_{a}}^{A_{1}}) (\underset{{b}}{.Math.} U_{m_{b} n_{b}}^{B_{P + 1}}) (\frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{1} B_{P + 1}})$

in the foregoing formula,

[00013] ${| Φ_{m_{k} n_{k}} .Math.}_{B_{k} A_{k + 1}} (k = 1, 2, . . ., P)$

represents the generalized Bell measurement result of the k.sup.th intermediate node Charlie k, and

[00014] ${| Φ_{m_{k} n_{k}} .Math.}_{B_{k} A_{k + 1}} = \frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j m_{k}} .Math. j .Math. .Math. j \oplus n_{k} .Math. (k = 1, 2, . . ., P);$

U.sub.m.sub.k.sub.n.sub.k.sup.A.sup.1.sup./B.sup.P+1(k=1, 2, . . . , P) represents that after measuring the held particles k and B.sub.k to obtain A.sub.k+1, Charlie

[00015] ${| Φ_{m_{k} n_{k}} .Math.}_{B_{k} A_{k + 1}}$

transmits the measurement result to Alice or Bob according to actual condition (for example, whether classical information can be transmitted, whether information communication is smooth, and transmission efficiency); it is assumed that (U.sub.A.sub.1).sup.−1=Π.sub.{a}(U.sub.m.sub.a.sub.n.sub.a.sup.A.sup.1).sup.−1, which corresponds to a unitary operation that Alice needs to perform on the particle A.sub.1 after Alice sums up received measurement results, where {a} is a set of sequence numbers of all intermediate nodes that have transmitted a measurement result to Alice; similarly, it is assumed that (U.sub.B.sub.P+1).sup.−1=Π.sub.{b}(U.sub.m.sub.b.sub.n.sub.b.sup.B.sup.P+1).sup.−1, which corresponds to a unitary operation that Bob needs to perform on the particle B.sub.P+1 after Bob sums up received measurement results, where {b} is a set of sequence numbers of all intermediate nodes that have transmitted a measurement result to Bob.

[0058] After performing the corresponding unitary operations, Alice and Bob can transform an entangled state of the particles A.sub.1 and B.sub.P+1 into a form of

[00016] ${| ϕ .Math.}_{A_{1} B_{P + 1}} = \frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} {.Math. jj .Math.}_{A_{1} B_{P + 1}}, where {U_{m_{a} n_{a}}^{A_{1}} (U_{m_{a} n_{a}}^{A_{1}})}^{- 1} = I; {U_{m_{b} n_{b}}^{B_{P + 1}} (U_{m_{b} n_{b}}^{B_{P + 1}})}^{- 1} = I, and U_{m_{a} n_{a}}^{A_{1}} = {.Math.}_{j = 0}^{d - 1} e^{- \frac{2 π i}{d} j m_{a}} .Math. j \oplus n_{a} .Math. .Math. j .Math. U_{m_{b} n_{b}}^{B_{P + 1}} = {.Math.}_{j = 0}^{d - 1} e^{- \frac{2 π i}{d} j m_{b}} .Math. j .Math. .Math. j \oplus n_{b} .Math. {(U_{m_{a} n_{a}}^{A_{1}})}^{- 1} = {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j m_{a}} .Math. j .Math. .Math. j \oplus n_{a} .Math. {(U_{m_{b} n_{b}}^{B_{P + 1}})}^{- 1} = {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j m_{b}} .Math. j \oplus n_{b} .Math. .Math. j | .$

[0059] Here, a direct entangled channel in a d-dimensional multihop lossless quantum teleportation system can be obtained. The channel is formed by three parts: The first part is generalized Bell measurement results of the P intermediate nodes. The second part is a matrix transformation operation that should be performed to adjust the form of direct entangled states between a source node and the intermediate nodes corresponding to the measurement results. The third part is an entangled state of the particle A.sub.1 of the information sender Alice and the particle B.sub.P+1 of the information receiver Bob.

[0060] Step 4: Transfer information. The multihop teleportation system is simplified into a single-hop teleportation system form, to perform a single-hop lossless quantum teleportation process, the information sender Alice performs a joint Bell measurement on the held particle t and particle A.sub.1, and Alice can obtain d.sup.2 different measurement results:

[00017] ${| Φ_{r s} .Math.}_{t A_{1}} = \frac{1}{\sqrt{d}} {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j r} .Math. j .Math. .Math. j \oplus s .Math. (r, s = 0, 1, . . . d - 1) .$

Alice transmits her measurement result to the information receiver Bob, and Bob performs a corresponding unitary operation

[00018] ${(U_{rs}^{B_{P + 1}})}^{- 1} = Σ_{j = 0}^{d - 1} e^{\frac{2 π i}{d} j r} .Math. j \oplus s .Math. .Math. j |$

on his particle B.sub.P+1 according to the received result to restore information of the transferred unknown quantum state.

[0061] More specifically:

[0062] Embodiment 1: A d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes is provided, and a 2-level 4-hop example is used to implement the transfer of an unknown single particle state |χ custom-character .sub.t from the information sender Alice to the information receiver Bob. Specific steps are as follows:

[0063] Step 1: Construct a 2-level 4-hop quantum teleportation chain channel. Two communicating parties are Alice and Bob. The particle t carries an unknown quantum state |χ custom-character .sub.t=c.sub.0|, 0+c.sub.1|1 and is held by the information sender Alice. Alice wants to transmit the unknown single particle quantum state to the information receiver Bob by using three intermediate nodes. In a quantum path, entangled channels have the same form and are:

[00019] ${| φ .Math.}_{A_{k} B_{k}} = \frac{1}{\sqrt{2}} {(.Math. 00 .Math. + .Math. 11 .Math.)}_{A_{k} B_{k}} (k = 1, 2, 3, 4) .$

[0064] Step 2: Construct a direct channel. Three intermediate nodes Charlie 1, Charlie 2, and Charlie 3 respectively perform generalized Bell measurements on respective held two particles. After the measurements are completed, in consideration of the connection condition of classical channels with the information sender Alice and the information receiver Bob (for example, whether classical information can be transmitted, and transmission efficiency), the P intermediate nodes select, according to actual condition, to transmit respective generalized Bell measurement results to Alice or Bob. A direct channel is constructed. Entanglement swapping is performed on every two of the information sender Alice used as the source node, three intermediate nodes, and the information receiver Bob used as a target node, to establish direct entanglement between Alice and Bob. A tensor product operation is performed on entangled channels |φ custom-character .sub.A.sub.1.sub.B.sub.1, |φ.sub.A.sub.2.sub.B.sub.2,|φ.sub.A.sub.3.sub.B.sub.3, and |φ.sub.A.sub.4.sub.B.sub.4, and quantum states of eight particles after the operation are represented as:

[00020] $\begin{matrix} {| φ .Math. = | φ .Math.}_{A_{1} B_{1}} .Math. {.Math. φ .Math.}_{A_{2} B_{2}} .Math. {.Math. φ .Math.}_{A_{3} B_{3}} .Math. {.Math. φ .Math.}_{A_{4} B_{4}} = {.Math.}_{k = 1}^{4} \frac{1}{\sqrt{2}} {(.Math. 00 .Math. + .Math. 11 .Math.)}_{A_{4} B_{4}} = {(\frac{1}{\sqrt{2}})}^{4} {(\frac{1}{\sqrt{2}})}^{2} {.Math.}_{m_{1}, n_{1} = 0}^{1} {.Math.}_{m_{2}, n_{2} = 0}^{1} {.Math.}_{m_{3}, n_{3} = 0}^{1} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} {.Math. Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} {.Math. Φ_{m_{3} n_{3}} .Math.}_{B_{3}, A_{4}} U_{m_{1} n_{1}}^{A_{1} / B_{4}} U_{m_{2} n_{2}}^{A_{1} / B_{4}} U_{m_{3} n_{3}}^{A_{1} / B_{4}} {.Math. Φ .Math.}_{A_{1} B_{4}} \end{matrix}$

The three intermediate nodes Charlie 1, Charlie 2, and Charlie3 respectively perform Bell measurements on respective held two particles, and each intermediate nod can obtain four measurement results:

[00021] ${.Math. Φ_{00} .Math.}_{B_{k} A_{k + 1}} = \frac{1}{\sqrt{2}} {(.Math. 0 .Math. .Math. 0 .Math. + .Math. 1 .Math. .Math. 1 .Math.)}_{B_{k} A_{k + 1}}$ ${.Math. Φ_{10} .Math.}_{B_{k} A_{k + 1}} = \frac{1}{\sqrt{2}} {(.Math. 0 .Math. .Math. 0 .Math. - .Math. 1 .Math. .Math. 1 .Math.)}_{B_{k} A_{k + 1}}$ ${.Math. Φ_{01} .Math.}_{B_{k} A_{k + 1}} = \frac{1}{\sqrt{2}} {(.Math. 0 .Math. .Math. 1 .Math. + .Math. 1 .Math. .Math. 0 .Math.)}_{B_{k} A_{k + 1}}$ ${.Math. Φ_{11} .Math.}_{B_{k} A_{k + 1}} = \frac{1}{\sqrt{2}} {(.Math. 0 .Math. .Math. 1 .Math. - .Math. 1 .Math. .Math. 0 .Math.)}_{B_{k} A_{k + 1}} .$

[0065] To carry out detailed research on an operation process, the following case is used as an example: The measurement result of the intermediate node Charlie 1 is |ϕ.sub.01).sub.B.sub.1.sub.A.sub.2, and the measurement result is transmitted to Alice. Charlie 2 performs a measurement to obtain a result |ϕ.sub.11).sub.B.sub.2.sub.A.sub.3, and transmits the measurement result to Alice. Charlie 3 performs a measurement to obtain a result |ϕ.sub.10).sub.B.sub.3.sub.A.sub.4, and transmits the result to Bob. At this time, the quantum state of the system turns into:

[00022] $.Math. φ .Math. = {(\frac{1}{\sqrt{2}})}^{4} {(\frac{1}{\sqrt{2}})}^{2} {.Math. Φ_{01} .Math.}_{B_{1} A_{2}} {.Math. Φ_{11} .Math.}_{B_{2} A_{3}} {.Math. Φ_{10} .Math.}_{B_{3}, A_{4}} \frac{1}{\sqrt{2}} {(- .Math. 00 .Math. - .Math. 11 .Math.)}_{A_{1} B_{4}} = {(\frac{1}{\sqrt{2}})}^{4} {(\frac{1}{\sqrt{2}})}^{2} {.Math. Φ_{01} .Math.}_{B_{1} A_{2}} {.Math. Φ_{11} .Math.}_{B_{2} A_{3}} {.Math. Φ_{10} .Math.}_{B_{3}, A_{4}} (U_{01}^{A_{1}} U_{11}^{A_{1}}) (U_{10}^{B_{4}}) \frac{1}{\sqrt{2}} {(.Math. 00 .Math. - .Math. 11 .Math.)}_{A_{1} B_{4}} .$

[0066] Step 3: Adjust the channel. Alice performs a matrix transformation operation on the particle A.sub.1: (U.sub.A.sub.1).sup.−1=(U.sub.11.sup.A.sup.1).sup.−1(U.sub.01.sup.A.sup.1).sup.−1=(|0 custom-character <1|−|1>0|)(|01|+|10|)=|00|−|11|, and Bob performs a matrix transformation operation: (U.sub.B.sub.4)=(U.sub.10.sup.B.sup.4).sup.−1=|00|−|11|, so that the form of the directly entangled state can turn into:

[00023] ${{| φ .Math.}_{A_{1} B_{4}} = \frac{1}{\sqrt{2}} (| 00 .Math. + .Math. 11 .Math.)}_{A_{1} B_{4}} .$

[0067] Step 4: Transfer information. The multihop teleportation system is simplified into a single-hop teleportation system form, to perform a single-hop lossless quantum teleportation process, the information sender Alice transfers an unknown quantum state by using the direct entangled channel

[00024] ${{| φ .Math.}_{A_{1} B_{4}} = \frac{1}{\sqrt{2}} (| 00 .Math. + .Math. 11 .Math.)}_{A_{1} B_{4}},$

and the quantum teleportation process is simplified into the following form:

[00025] ${{{{.Math. γ .Math. = | χ .Math.}_{t} .Math. | φ .Math.}_{A_{1} B_{4}} = (c_{0} | 0 .Math. + c_{1} | 1 .Math.)}_{t} .Math. \frac{1}{\sqrt{2}} (| 0 0 .Math. + | 11 .Math.)}_{A_{1} B_{4}} = {(\frac{1}{\sqrt{2}})}^{2} [\begin{matrix} {{{{| Φ_{0 0} .Math.}_{t A_{1}} (c_{0} | 0 .Math. + c_{1} | 1 .Math.)}_{B_{4}} + | Φ_{10} .Math.}_{{tA}_{1}} (c_{0} | 0 .Math. - c_{1} | 1 .Math.)}_{B_{4}} + \\ {{{{| Φ_{0 1} .Math.}_{t A_{1}} (c_{0} | 1 .Math. + c_{1} | 0 .Math.)}_{B_{4}} + | Φ_{1 1} .Math.}_{t A_{1}} (c_{0} | 1 .Math. - c_{1} | 0 .Math.)}_{B_{4}} \end{matrix}] .$

[0068] In this case, Alice performs Bell measurements on her own particles t and A.sub.1, four measurement results |ϕ.sub.rs custom-character .sub.tA.sub.1, (r,s=0, 1) can be obtained, and Bob performs a corresponding unitary operation (U.sub.rs.sup.B.sup.4).sup.−1=Σ.sub.j=0.sup.1e.sup.πi.Math.jr|j⊕sj| according to the measurement results to restore the transferred unknown quantum state.

[0069] Embodiment 2: A d-dimensional chain teleportation method for random transmission based on measurement results of relay nodes is provided, and a 3-level 3-hop example is used to implement the transfer of an unknown single particle state |χ custom-character .sub.t from an information sender Alice to an information receiver Bob. Specific steps are as follows:

[0070] Step 1: Construct a 3-level 3-hop quantum teleportation chain channel. Two communicating parties are Alice and Bob. The particle t carries an unknown quantum state |χ custom-character .sub.t=c.sub.0|0+c.sub.1|1+c.sub.2|2 and is held by the information sender Alice. Alice wants to transmit the unknown single particle quantum state to the information receiver Bob by using three intermediate nodes. In a quantum path, entangled channels have the same form and are:

[00026] ${{| φ .Math.}_{A_{k} B_{k}} = \frac{1}{\sqrt{3}} (| 00 .Math. + | 11 .Math. + | 22 .Math.)}_{A_{k} B_{k}} (k = 1, 2, 3) .$

[0071] Step 2: Construct a direct channel. Two intermediate nodes Charlie 1 and Charlie 2 respectively perform the generalized Bell measurements on respective held two particles. After the measurements are completed, in consideration of the connection condition of classical channels with the information sender Alice and the information receiver Bob (for example, whether classical information can be transmitted, and transmission efficiency), the P intermediate nodes select, according to actual condition, to transmit respective generalized Bell measurement results to Alice or Bob. A direct channel is constructed. Entanglement swapping is performed on every two of the information sender Alice used as the source node, three intermediate nodes, and the information receiver Bob used as a target node, to establish direct entanglement between Alice and Bob. A tensor product operation is performed on entangled channels |φ custom-character .sub.A.sub.1.sub.B.sub.1, |φ.sub.A.sub.2.sub.B.sub.2, and |φ.sub.A.sub.3.sub.B.sub.3, and quantum states of six particles after the operation are represented as:

[00027] $| φ .Math. = {.Math. φ .Math.}_{A_{1} B_{1}} .Math. {.Math. φ .Math.}_{A_{2} B_{2}} .Math. {.Math. φ .Math.}_{A_{3} B_{3}} = {.Math.}_{k = 1}^{3} \frac{1}{\sqrt{3}} {(.Math. 00 .Math. + .Math. 11 .Math. + .Math. 22 .Math.)}_{A_{k} B_{k}} = {(\frac{1}{\sqrt{3}})}^{2} \frac{1}{\sqrt{3}} {.Math.}_{m_{1}, n_{1} = 0}^{2} {.Math.}_{m_{2}, n_{2} = 0}^{2} {.Math. Φ_{m_{1} n_{1}} .Math.}_{B_{1} A_{2}} .Math. {Φ_{m_{2} n_{2}} .Math.}_{B_{2} A_{3}} U_{m_{1} n_{1}}^{A_{1} / B_{3}} U_{m_{2} n_{2}}^{A_{1} / B_{3}} {.Math. Φ .Math.}_{A_{1} B_{3}} .$

[0072] The two intermediate nodes Charlie 1 and Charlie 2 respectively perform a generalized Bell measurement on respective held two particles, and each intermediate node can obtain nine measurement results

[00028] ${.Math. Φ_{m_{k} n_{k}} .Math.}_{B_{k}, A_{k + 1}} = \frac{1}{\sqrt{3}} {.Math.}_{j = 0}^{2} e^{\frac{2 π i}{3} j m_{k}} .Math. j .Math. .Math. j \oplus n_{k} .Math. (m_{k}, n_{k} = 0, 1, 2; k = 1, 2, 3) .$

[0073] To carry out detailed research on an operation process, the following case is used as an example: The measurement result of the intermediate node Charlie 1 is |ϕ.sub.11 custom-character .sub.B.sub.1.sub.A.sub.2, and the measurement result is transmitted to Alice. Charlie 2 performs a measurement to obtain a result |ϕ.sub.22.sub.B.sub.2.sub.A.sub.3, and transmits the measurement result to Bob. At this time, the quantum state of the system turns into:

[00029] $.Math. ϕ .Math. = {(\frac{1}{\sqrt{3}})}^{2} \frac{1}{\sqrt{3}} {.Math. Φ_{1 1} .Math.}_{B_{1} A_{2}} {.Math. Φ_{2 2} .Math.}_{B_{2} A_{3}} \frac{1}{\sqrt{3}} {(e^{- \frac{4 π i}{3}} .Math. 22 .Math. + e^{- \frac{4 π i}{3}} .Math. 00 .Math. + e^{- \frac{10 π i}{3}} .Math. 11 .Math.)}_{A_{1} B_{3}} = {(\frac{1}{\sqrt{3}})}^{2} \frac{1}{\sqrt{3}} {.Math. Φ_{1 1} .Math.}_{B_{1} A_{2}} {.Math. Φ_{2 2} .Math.}_{B_{2} A_{3}} U_{1 1}^{A_{1}} U_{2 2}^{B_{3}} \frac{1}{\sqrt{3}} (.Math. 00 .Math. + .Math. 11 .Math. + {.Math. 22 .Math.)}_{A_{1} B_{3}} .$

[0074] Step 3: Adjust the channel. Alice performs a matrix transformation operation on the particle A.sub.1:

[00030] ${(U_{A_{1}})}^{- 1} = {(U_{1 1}^{A_{1}})}^{- 1} = .Math. 0 .Math. .Math. 1 .Math. + e^{\frac{2 π i}{3}} .Math. 1 .Math. .Math. 2 .Math. + e^{\frac{4 π i}{3}} .Math. 2 .Math. .Math. 0 .Math.,$

and Bob performs a matrix transformation operation:

[00031] ${(U_{B_{3}})}^{- 1} = {(U_{2 2}^{B_{3}})}^{- 1} = .Math. 2 .Math. .Math. 0 .Math. + e^{\frac{4 π i}{3}} .Math. 0 .Math. .Math. 1 .Math. + e^{\frac{8 π i}{3}} .Math. 1 .Math. .Math. 2 .Math.,$

[0075] so that the form of the directly entangled state can turn into:

[00032] ${.Math. φ .Math.}_{A_{1} B_{3}} = \frac{1}{\sqrt{3}} {(.Math. 00 .Math. + .Math. 11 .Math. + .Math. 22 .Math.)}_{A_{1} B_{3}} .$

[0076] Step 4: Transfer information. The multihop teleportation system is simplified into a single-hop teleportation system form, to perform a single-hop lossless quantum teleportation process, the information sender Alice transfers an unknown quantum state by using the direct entangled channel

[00033] ${.Math. φ .Math.}_{A_{1} B_{3}} = \frac{1}{\sqrt{3}} {(.Math. 00 .Math. + .Math. 11 .Math. + .Math. 22 .Math.)}_{A_{1} B_{3}},$

and the quantum teleportation process is simplified into the following form:

[00034] $.Math. γ .Math. = {.Math. X .Math.}_{t} .Math. {.Math. φ .Math.}_{A_{1} B_{3}} = {(c_{0} .Math. 0 .Math. + c_{1} .Math. 1 .Math. + c_{2} .Math. 2 .Math.)}_{t} .Math. \frac{1}{\sqrt{3}} {(.Math. 00 .Math. + .Math. 11 .Math. + .Math. 22 .Math.)}_{A_{1} B_{3}} = {(\frac{1}{\sqrt{3}})}^{2} [\begin{matrix} {.Math. Φ_{00} .Math.}_{{tA}_{1}} {(c_{0} .Math. 0 .Math. + c_{1} .Math. 1 .Math. + c_{2} .Math. 2 .Math.)}_{B_{3}} + {.Math. Φ_{10} .Math.}_{{tA}_{1}} {(c_{0} .Math. 0 .Math. + e^{- \frac{2 π i}{3}} c_{1} .Math. 1 .Math. + e^{- \frac{4 π i}{3}} c_{2} .Math. 2 .Math.)}_{B_{3}} + {.Math. Φ_{20} .Math.}_{{tA}_{1}} {(c_{0} .Math. 0 .Math. + e^{- \frac{4 π i}{3}} c_{1} .Math. 1 .Math. + e^{- \frac{8 π i}{3}} c_{2} .Math. 2 .Math.)}_{B_{3}} \\ + {.Math. Φ_{01} .Math.}_{{tA}_{1}} {(c_{0} .Math. 1 .Math. + c_{1} .Math. 2 .Math. + c_{2} .Math. 0 .Math.)}_{B_{3}} + {.Math. Φ_{11} .Math.}_{{tA}_{1}} {(c_{0} .Math. 1 .Math. + e^{- \frac{2 π i}{3}} c_{1} .Math. 2 .Math. + e^{- \frac{4 π i}{3}} c_{2} .Math. 0 .Math.)}_{B_{3}} + {.Math. Φ_{21} .Math.}_{{tA}_{1}} {(c_{0} .Math. 1 .Math. + e^{- \frac{4 π i}{3}} c_{1} .Math. 2 .Math. + e^{- \frac{8 π i}{3}} c_{2} .Math. 0 .Math.)}_{B_{3}} \\ + {.Math. Φ_{02} .Math.}_{{tA}_{1}} {(c_{0} .Math. 2 .Math. + c_{1} .Math. 0 .Math. + c_{2} .Math. 1 .Math.)}_{B_{3}} + {.Math. Φ_{12} .Math.}_{{tA}_{1}} {(c_{0} .Math. 2 .Math. + e^{- \frac{2 π i}{3}} c_{1} .Math. 0 .Math. + e^{- \frac{4 π i}{3}} c_{2} .Math. 1 .Math.)}_{B_{3}} + {.Math. Φ_{22} .Math.}_{{tA}_{1}} {(c_{0} .Math. 2 .Math. + e^{- \frac{4 π i}{3}} c_{1} .Math. 0 .Math. + e^{- \frac{8 π i}{3}} c_{2} .Math. 1 .Math.)}_{B_{3}} \end{matrix}]$

[0077] In this case, Alice performs a Bell measurement on her own particles t and A.sub.1, nine measurement results |ϕ.sub.rs custom-character .sub.tA.sub.1(r, s=0, 1, 2) can be obtained, and Bob performs a corresponding unitary operation

[00035] ${(U_{r s}^{B_{3}})}^{- 1} = {.Math.}_{j = 0}^{1} e^{\frac{2 π i}{3} .Math. jr} .Math. j \oplus s .Math. .Math. j .Math.$

according to the measurement results to restore the transferred unknown quantum state. For example, it is assumed that Alice performs a measurement to obtain |ϕ.sub.20 custom-character .sub.tA.sub.1, and Bob performs a unitary operation on the particle B.sub.3:

[00036] ${(U_{2 0}^{B_{3}})}^{- 1} = {.Math.}_{j = 0}^{d - 1} e^{\frac{2 π i}{d} .Math. jr} .Math. j \oplus s .Math. .Math. j .Math. = .Math. 0 .Math. .Math. 0 .Math. + e^{\frac{4 π i}{3}} .Math. 1 .Math. .Math. 1 .Math. + e^{\frac{8 π i}{3}} .Math. 2 .Math. .Math. 2 .Math. .$

[0078] The information of the transferred unknown quantum state can be recovered.

[0079] The foregoing embodiments are merely preferred embodiments used to fully describe the present invention, and the protection scope of the present invention is not limited thereto. Equivalent replacements or variations made by a person skilled in the art to the present invention all fall within the protection scope of the present invention. The protection scope of the present invention is as defined in the claims.

D-DIMENSIONAL CHAIN TELEPORTATION METHOD FOR RANDOM TRANSMISSION BASED ON MEASUREMENT RESULTS OF RELAY NODES

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H04B10/0795

ELECTRICITY

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H04B10/70

ELECTRICITY

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H04B10/29

ELECTRICITY

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H04B10/70

ELECTRICITY

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H04B10/079

ELECTRICITY

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H04B10/29

ELECTRICITY

Abstract

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