CN114430299B - Indication amplifying method for three-degree-of-freedom coded single photon - Google Patents

Abstract

The present invention proposes a three-degree-of-freedom coded single-photon indication amplification method. Amplifier devices are set on four spatial modes, and each amplifier device uses an imperfect single-photon state generated by a single-photon source under current implementation conditions as an auxiliary. Passing the signal photon and auxiliary photon on each spatial mode through the amplifier is manipulated, and finally the new mixed state is extracted. This method can effectively improve the fidelity of single photons in the output mixed state, and can perfectly protect its encoded information in three degrees of freedom. This method uses the commonly used imperfect photon sources and other linear optical devices under the current experimental conditions, which can be realized under the current experimental conditions and has strong practicability.

Description

A three-degree-of-freedom coded single-photon indexing amplification method

technical field

The invention belongs to the technical field of quantum communication, and in particular relates to an indication amplification method for encoding single photons with three degrees of freedom, in particular to an indication amplification method for single photons encoded in three degrees of freedom with polarization and dual longitudinal momentum at the same time.

Background technique

In the field of quantum communication, the multi-degree-of-freedom simultaneous encoding of single photon can effectively improve the channel capacity of single photon, thereby improving the communication efficiency of quantum communication. At present, researchers have experimentally realized simultaneous encoding in three degrees of freedom, including the polarization of a single photon and the dual longitudinal momentum. The three-degree-of-freedom encoded single photon has important application prospects in the field of long-distance quantum communication. However, photon transmission loss is an important technical problem in long-distance quantum communication. When photons are transmitted in an actual quantum channel, the photon transmission loss causes the propagation of photons in the optical fiber to exhibit an exponential attenuation with the increase of the channel length. Photon loss not only seriously affects the success rate fidelity of quantum communication, but also affects its security. Quantum indicator amplification is an effective method proposed by Ralph and Lund in 2009 to solve the problem of photon transmission loss in quantum communication. In device-independent quantum key distribution (DI-QKD), quantum instruction amplification is widely used to secure single-photon qubits and entanglement.

Simultaneous encoding on multiple degrees of freedom of a single photon can effectively increase the information capacity of a single photon, thereby effectively improving the efficiency of quantum communication. At present, someone has successfully prepared a single photon encoded in three degrees of freedom, such as polarization and dual longitudinal momentum, in the laboratory. The three-degree-of-freedom encoded photon has an important application prospect in the field of long-distance quantum communication. However, when photons are transmitted in the actual noisy quantum channel, transmission loss may occur, which greatly limits the communication distance and poses a threat to the security of communication, and the existing technology has not yet achieved simultaneous polarization and dual longitudinal momentum degrees of freedom. Encoded single photons perform directed amplification.

Contents of the invention

Aiming at the deficiencies of the prior art, the object of the present invention is to provide a three-degree-of-freedom coded single photon indication amplification method. Since the single photon encoded in the polarization degree of freedom is distributed to four spatial modes with different probabilities, it is necessary to prepare the amplifier device and provide auxiliary states on the four spatial modes simultaneously. Each amplifier unit consists of two polarizing beam splitters (PBS), two 50:50 beam splitters (BS), two variable beam splitters (VBS), and four single-photon detectors, which can be adjusted according to each The single photon detector response of each amplifier unit is used to judge whether the amplification process is successful, and the success probability and fidelity of the scheme are calculated. In the case of a successful amplification scheme, the fidelity of the single photon can be effectively improved, and the encoded information of the single photon in three degrees of freedom can be perfectly preserved. The amplification scheme only needs some common optical components, and uses the imperfect single photon state generated by the imperfect single photon source commonly used under the current experimental conditions as the auxiliary state, which can be realized under the current experimental technology and has strong practicability , has important application prospects in the field of long-distance quantum communication.

The present invention provides a three-degree-of-freedom coded single photon indication amplification method, the method steps are as follows:

Step 1: User 1 prepares a single photon with encoding information on both polarization and dual longitudinal momentum degrees of freedom, forms a signal photon, and sends it to User 2;

Step 2: User 2 uses the imperfect single photon state generated by the single photon source under the current experimental conditions as the auxiliary state of each amplifier unit (amplifier);

Step 3: User 2 passes signal photons and auxiliary state photons in each spatial mode into the amplifier unit, and user 2 performs a series of operations on the signal photons and auxiliary state photons entering the amplifier. Since the amplification probability is not 100%, multiple Different output states will lead to different response effects of the detectors in each amplifier unit;

Step 4: According to the calculation, it is obtained that the output states corresponding to the response results of four types of detectors in each amplifier unit are required states and need to be retained; the output states corresponding to other detector responses are not required states and must be discarded; therefore, According to the response of the detector, you can choose to keep the required state, discard the unqualified state, and calculate the success probability of the scheme and the fidelity of the signal state according to the result.

Firstly, in the first step, a polarization modulator (POL-M) is specifically used to prepare a single photon with encoded information in both polarization and dual longitudinal momentum degrees of freedom.

A further improvement is: in the first step, user 1 prepares a single photon encoded on the polarization and dual longitudinal momentum degrees of freedom at the same time, forming a single-photon four-mode spatial entanglement state, the form of which is as follows:

where |H> and |V> are defined as horizontal and vertical polarizations, respectively, and |l>, |r>, |I> and |E> are defined as left, right internal and external modes, respectively, Coefficients α, β, δ, η, ε, ν in each degree of freedom satisfy |α| ² + |β| ² = 1, |δ| ² + |η| ² = 1, |ε| ² + |ν | ² ＝1; where the single photon α|H>+β|V> with polarization characteristics is distributed in four spatial modes |lI>, |lE>, |rI>, |rE> with different probabilities; the user 1 sends the coded single photon to user 2 through the quantum channel, and the information content is composed of the above-mentioned polarization and dual longitudinal momentum single photon state codes, but the photon transmission loss will be caused by the environmental noise in the quantum channel in reality, resulting in The entangled state degenerates into a mixed state.

The further improvement is: in the second step, in order to improve the fidelity of the single-photon spatially entangled state in the mixed state, user 2 uses the single-photon source under the current experimental conditions to prepare an imperfect auxiliary state, and the auxiliary state is in the form of Where |vac> represents an empty state.

The further improvement is: in the step 3, the signal photons and auxiliary photons in the four spatial modes are all passed into the amplifier at the same time, and the amplification scheme is run on the four spatial modes at the same time, and the main components of each amplifier unit are two polarization splitters. beam splitter (PBS), two 50:50 beam splitters (BS), two variable beam splitters (VBS), and four single-photon detectors, each photon arrives at each detector and output port with a different probability , there is a PBS before the output port, which can recombine the polarization states that were separated upon entry into a polarized single-photon state with the original form.

The further improvement is: in the step 4, according to the calculation, the output states corresponding to the four types of detector responses are required states and will be retained; while the output states corresponding to other detector response states are not required states and will be discarded ; D ₁ , D ₂ , D ₃ and D ₄ represent four detection modules in each amplifier unit, and the responses of the above four types of detectors are respectively:

The first type is that only one photon is detected by D ₁ and D ₃ , that is and

The second type is that only D ₂ and D ₄ detect one photon each, that is and

The third category is that only D ₁ and D ₄ detect one photon each, namely and

The fourth type is that only D ₂ and D ₃ detect one photon each, namely and

A further improvement is: in the fourth step, when the amplifier units on the four spatial modes |lI>, |lE>, |rI>, |rE> all obtain one of the above four successful detection results, the entire indication amplification scheme Successfully, the final preserved output state perfectly preserves the encoded information of the input signal state in three degrees of freedom, and, by adjusting the transmittance of VBS in each amplifier, user 2 can improve the fidelity of single photons in the output state.

The beneficial effects of the present invention are: the scheme sets the amplifier device on four spatial modes such as |ll>, |lE>, |rI>, |rE> and uses the imperfect single photon produced by the single photon source under the current realization conditions The auxiliary state is used as an auxiliary state, and the signal photon and auxiliary photon on each spatial mode are operated through the amplifier; if the amplification process in the four amplifiers is successful at the same time, the overall amplification scheme is successful, and a new mixed state will be output at the output port; by adjusting The transmittance of VBS in each amplifier, user 2 can effectively improve the fidelity of single photons in the output state, reduce photon transmission loss, and perfectly preserve the coding information of single photons in three degrees of freedom, such as polarization and dual longitudinal momentum. ; This amplification device uses common optical devices, especially under the existing experimental conditions, the imperfect single photon state produced by the imperfect single photon source commonly used under the current experimental conditions is used as an auxiliary instead of an ideal single photon state. The photon state is used as an auxiliary state to realize the amplification of the target single photon, so the technical solution of the present invention has stronger practicability and experimental operability.

Description of drawings

Fig. 1 is a schematic flow chart of the method of the present invention;

Fig. 2 is a schematic diagram of encoding single photons on three degrees of freedom such as polarization and double longitudinal momentum of the present invention;

Fig. 3 is a schematic diagram of a single-photon amplification scheme encoded on polarization, dual longitudinal momentum and three degrees of freedom of the present invention;

FIG. 4 is a schematic structural diagram of an amplifier unit (amplifier) of the present invention.

Detailed ways

In order to deepen the understanding of the present invention, the present invention will be further described below in conjunction with examples, which are only used to explain the present invention and do not constitute a limitation to the protection scope of the present invention.

This example provides a specific embodiment of a three-degree-of-freedom encoding single-photon bit indication amplification method, as shown in Figure 1. In this example, it is specifically assumed that the communicating party Alice, that is, user 1, the information sender uses the photon source S and the pole The polarized modulator (Pol-M) prepares a single photon with encoding information on the polarization and dual longitudinal momentum degrees of freedom. The encoding principle is shown in Figure 2, which is equivalent to a single photon four-mode space entangled state. Its form is represented by the formula:

where |H> and |V> are defined as horizontal and vertical polarizations, respectively, and |l>, |r>, |I> and |E> are defined as left, right, internal and external modes, respectively , the entanglement coefficient satisfies |α| ² + |β| ² = 1, |δ| ² + |η| ² = 1, |ε| ² + |ν| ² = 1; where the single photon α| H>+β|V> is distributed in four spatial modes |lI>, |lE>, |rI>, |rE> with different probabilities.

Alice sends the single-photon four-mode spatial entanglement state to Bob in the distance through the quantum channel, that is, user 2, the information receiver. In the process of photon transmission, channel noise will cause photons to be lost, and the probability of loss is assumed to be 1-F, which will cause the original single-photon entangled state to degenerate into a mixed state. The mixed state is expressed as:

ρ _in = F|Ψ ⁱⁿ ><Ψ ⁱⁿ |+(1-F)|vac><vac|,

Among them, |vac> represents the vacuum state.

The schematic diagram of this amplification scheme is shown in Figure 3. Bob's goal is to increase the output mixed state |Ψ ⁱⁿ > state fidelity. Within each amplifier unit in the protocol, user 2 uses the imperfect single-photon state produced by the imperfect single-photon source under the current experimental conditions as an auxiliary state of the form Bob then passes the received signal photons and auxiliary photons into each amplifier unit.

The structure of the amplifier unit used here is shown in Figure 4. The amplifier unit consists of two polarizing beam splitters (PBS), two 50:50 beam splitters (BS), two variable beam splitters (VBS), and four single-photon detectors. PBS can completely transmit horizontally polarized light and completely reflect vertically polarized light. BS transmits photons with 50% probability and reflects photons with 50% probability. VBS transmits photons with probability t and reflects photons with probability 1-t. Here D ₁ , D ₂ , D ₃ and D ₄ are four photon detection modules. We assume that the single-photon detector in the detection module can distinguish the number of incident photons, and judge whether to retain the output state according to the measurement results.

Taking the amplification of polarized photons on a spatial mode as an example, first consider the case where an auxiliary single photon source generates an ideal single photon, and assume that the input state in the a ₁ mode is not lost with probability F. Bob passes the initial photon through PBS ₁ . After passing through PBS ₁ , the state | _Ψin > changes to Next, Bob passes the auxiliary photon through the VBSs, and the state of the auxiliary photon becomes:

Thus, the entire photon state is written as:

In the above formula, Bob selects the situation that only one output port of each BS contains a photon, and the other output port does not contain a photon. In this case, the above formula will collapse to

Bob makes By BSs, /> will become:

Next, Bob detects the photons entering the D ₁ , D ₂ , D ₃ and D ₄ detection modules. All successful detection results are divided into four categories here:

The first type: when D ₁ and D ₃ each detect a photon, and D ₂ and D ₄ do not detect a photon, that is and />

The second type: when D ₂ and D ₄ each detect a photon, and D ₁ and D ₃ do not detect a photon, that is and />

The third category: when D ₁ and D ₄ each detect a photon, and D ₂ and D ₃ do not detect a photon, that is and />

The fourth type: when D ₂ and D ₃ each detect a photon, and D ₁ and D ₄ do not detect a photon, that is and />

Assuming that the detection result is that D ₁ D ₃ each detect a photon, after the detection, the entire state will collapse into:

Then, the photons on the paths a ₉ and a ₆ are output after passing through PBS ₂ , and the final output state is:

It can be seen that the output state has the same form as the original input state, and if the detection result is one of the other three successful detection results, the same output state as the above formula can be finally obtained by means of the phase flip operation.

According to the above description, the success probability can be calculated as: P ₀₀ =t(1-t).

On the other hand, if the initial photon of spatial mode a ₁ is lost in transmission with probability 1-F, then the states entering the amplified system are only auxiliary states, then the total photon state is:

Will Through the amplifier, and then select the situation that one output of each BS contains only one photon, while the other output does not contain photons, the selected successful state is:

Bob will After passing BSs, the state is obtained:

After detection, the state will collapse into an empty state. In this case, the probability of success is:

P ₀₁ =(1−t) ² .

In summary, when only the signal photon is in one spatial mode and the auxiliary state is an ideal single photon, the total probability of success of the scheme is:

P _1t ＝FP ₀₀ +(1-F)P ₀₁ ＝Ft(1-t)+(1-F)(1-t) ²

When the scheme is successful, the new mixed state can be obtained as:

ρ _out ＝ F′|Ψ ^out ><Ψ ^out |+(1-F′)|vac><vac|,

where the fidelity of the target state |Ψ ^out > is:

The resulting magnification factor is:

To achieve fidelity amplification, it is necessary to ensure that G>1. It can be seen from the calculation that when the transmittance of VBS G>1 can be achieved, so only the coefficient t of VBS needs to be adjusted to realize the amplification of the original incident state.

When a single photon is encoded in the three degrees of freedom such as polarization and dual longitudinal momentum, if the photon is not lost, the photon will appear in any one of the four spatial modes with a certain probability, then Bob must be in the four spatial modes run the scale-up scheme concurrently on . For example, if the single photon is on the |rI> spatial mode, then the probability of success of running this amplification scheme on the |rI> mode is P ₀₀ , while on the other three spatial modes (|rE>, |lE>, |lI> The success probability of running the scale-up scheme on ) is P ₀₁ . Therefore, if we consider that a single photon is encoded in three degrees of freedom simultaneously, the total probability of success of the final scheme is When the scheme is successful, the final output state obtained is:

has the same form as the original input state.

On the other hand, if the input photons are lost, then there are no photons on the four spatial modes, the probability of success of the amplification protocol is After detection, the output state eventually collapses to a vacuum state. Therefore, the overall probability of success for the entire scaling protocol is:

When the scheme is successful, the corresponding output state is:

ρ _out2 ＝ F ^* |Ψ ^out2 ><Ψ ^out2 |+(1-F ^* )|vac><vac|, |Ψ ^out2 > the fidelity is:

The magnification factor is:

It can be seen that when amplifying single photons encoded in three degrees of freedom at the same time, the amplification factor of this amplification scheme is the same as that of amplifying single photons encoded in polarization degrees of freedom.

Next consider the case where an auxiliary single photon source produces an imperfect auxiliary photon state. First consider the case where two auxiliary single-photon sources generate a two-photon and a single-photon respectively, and assume that the generated auxiliary state is The probability of this happening is /> After passing VBS, the auxiliary state becomes:

First analyze the incident photon in a spatial degree of freedom. If the input photon is not lost, the entire photon state is expressed as:

After the signal photons and auxiliary photons enter the amplifier, select the situation that one output port of each BS contains only one photon, and the other output port does not contain photons, and the selected state is:

Bob makes By BSs, /> will become:

The four types of successful detector response situations described above can result. Assuming that the detection result is that D ₁ D ₃ each has a photon, after the detection, the entire state will collapse into:

Then, let the photons on the a ₉ and a ₆ paths pass through PBS ₂ and then output, and the final output state will be obtained:

The probability of success in this case is:

P ₁₀ =α ² t ² (1-t)+2β ² t ² (1-t)=(1+β ² )t ² (1-t).

If the input photon is lost in transit, and one auxiliary single-photon source produces ditons and another auxiliary source produces single-photons, the total photon state becomes

If the above-mentioned successful detector response is obtained, the final state that can be selected is:

Bob makes by BSS, /> will become:

Then, let the photons on the paths of a _q and a ₆ output through PBS ₂ , and the final output state will be obtained

The probability of success in this case is P ₁₁ =2t(1-t) ² .

According to the above discussion, when the single photon is encoded in the three degrees of freedom such as polarization and dual longitudinal momentum, we consider that the single photon appears in four spatial modes with different probabilities. In the case where one of the auxiliary states is single-photon and the other is two-photon, the overall probability of success of the amplification scheme is:

If the auxiliary photon state generated by the spontaneous parametric down conversion (SPDC) source is Then, similar to the previous derivation, the success probability of this amplification scheme is:

Therefore, in the case where one auxiliary source produces two-photons and one auxiliary source produces single-photons, the probability of success of this amplification scheme is:

Next, we consider the case where the auxiliary photon states generated by the auxiliary photon source are all two-photon, that is, the auxiliary photon states are The probability of this case is /> Then after passing through VBSs, the auxiliary photon state becomes:

Considering that the initial photon is not lost, the entire mixed state becomes:

The output states selected based on successful detector responses are then:

Bob makes By BSs, /> will become:

Assuming that the detection result is that D ₁ D ₃ each has a photon, after the detection, the entire state will collapse into:

Then, let the photons on paths a ₉ and a ₆ output through PBS ₂ , and the final output state will be obtained:

The probability of success in this case is:

P ₂₀ =2t ³ (1-t).

If the input photon is lost during transmission, the auxiliary single-photon source both produces two-photons, then the entire state is only the auxiliary state:

When the scheme is successful, the selected states are:

Bob makes By BSs, /> will become:

Then, let the photons on paths a ₉ and a ₆ output through PBS ₂ , and the final output state will be obtained:

In this case, the probability of success is:

P ₂₁ =4t ² (1-t) ² .

In the case where the auxiliary states are all two-photon, the total probability of success is:

In summary, if the imperfect auxiliary photon source is considered, the total success probability of the polarized single-photon scheme considering only one spatial mode is:

P _tm =P _1t +P″ ₁ +P″ ₂ +P″ ₃

When considering the single photon encoded in the three degrees of freedom such as polarization and double longitudinal momentum, if the photon occurs without loss (the probability is F), it will be in any one of the four spatial modes. Bob runs the amplification scheme on four spatial modes simultaneously. If a single photon is on |rI> spatial mode, then the probability of successful amplification on |rI> is:

While the probability of success for scaling up on other spatial patterns is:

Therefore, the total probability of success in amplification of a single photon over the four spatial modes is On the other hand, if the input photon is lost (with probability (1-F)), then there are no photons on any of the four spatial modes, and the probability of success of the entire amplification scheme is /> Therefore, for the case where a single photon is simultaneously encoded in 3 degrees of freedom, the overall probability of success of the entire amplification protocol is:

The fidelity of the target output state is:

Therefore, the scaling factor of the scheme is:

It can be known by calculation that in order to make G' ^* >1, the minimum value of t obtained is still very close to 0.5, which is about 0.505. To sum up, by running this amplification scheme, communication party 2 can significantly improve the fidelity of the incident target state, and can perfectly preserve the information of the original incident state in three degrees of freedom. Because this amplification scheme uses common optical devices under the current experimental conditions, this scheme has strong practicability.

Claims (6)

1. A method for indicating amplification of a three-degree-of-freedom encoded single photon, characterized in that the steps of the method are as follows: Step 1: User 1 prepares a single photon with encoding information on both polarization and dual longitudinal momentum degrees of freedom, forms a signal photon, and sends it to User 2; Step 2: User 2 uses an imperfect single-photon source under the current experimental conditions to generate auxiliary photons as auxiliary states for each amplifier unit; Step 3: User 2 passes the signal photons and auxiliary state photons in each space mode into the amplifier unit, and user 2 performs a series of operations on the signal photons and auxiliary state photons entering the amplifier unit. Since the amplification probability is not 100%, it will produce A variety of output states, different output states will lead to different response effects of the detectors in each amplifier unit; Step 4: According to the response effect of the detector, choose to keep the required state, discard the unqualified state, and calculate the success probability of the scheme and the fidelity of the signal state; In the step three, the amplifier unit is set under the four spatial modes |lI>, |lE>, |rI>, |rE>, and the signal photons and auxiliary photons in each spatial mode are simultaneously passed into the amplifier unit, and at the same time The amplification scheme operates on four spatial modes; the main components of each amplifier unit are two polarizing beam splitters (PBS), two 50:50 beam splitters (BS), two variable beam splitters (VBS ) and four single-photon detectors, each photon will arrive at each detector and the output port with different probabilities, and there is a PBS before the output port, which can recombine the polarization states that were separated at the time of entry into a polarization with the original form single photon state. 2. A three-degree-of-freedom coded single photon indication amplification method according to claim 1, characterized in that, in said step 1, a single photon with coded information on both polarization and dual longitudinal momentum degrees of freedom is prepared , which has the following form: where |H> and |V> are defined as horizontal and vertical polarizations, respectively, and |l>, |r>, |I> and |E> are defined as left, right, internal and external modes, respectively , the coefficients α, β, δ, η, ε, v in each degree of freedom satisfy |α| ² + |β| ² = 1, |δ| ² + |η| ² = 1, |ε| ² +| v| ² =1; the above formula is equivalent to a single-photon four-mode spatial entanglement state, in which the single-photon α|H>+β|V> with polarization characteristics is distributed in four spatial modes |lI> with different probabilities , |lE>, |rI>, |rE>. 3. The instruction amplification method of a three-degree-of-freedom encoded single photon according to claim 1, wherein user 1 sends the above-mentioned single photon to user 2 through a quantum channel; due to the existence of channel noise, photon transmission may occur loss, the above-mentioned single-photon state degenerates into a mixed state. 4. A three-degree-of-freedom coded single-photon indication amplification method according to claim 1, characterized in that, in the second step, user 2 uses a single-photon source under current experimental conditions to prepare an imperfect auxiliary state. 5. A method for indicating and amplifying a three-degree-of-freedom encoded single photon according to claim 1, wherein, in said step 4, according to calculation, the output states corresponding to the response conditions of the four types of detectors in each amplifier unit are obtained is the required state and will be retained; while the output states corresponding to other detector responses are not required and will be discarded. 6. A method for indicating and amplifying a single photon with three degrees of freedom encoding according to claim 5, characterized in that, in said step 4, when the four spatial modes |lI>, |lE>, |rI>, | The amplifier units above rE> all get successful detection results, indicating that the amplification scheme is successful, and the output state preserved in the end perfectly preserves the encoding information of the input signal state in three degrees of freedom, and, by adjusting the VBS in each amplifier Transmittance, User 2 improves the fidelity of single photons in the output state.