Quantum entanglement and quantum communication technology

<Mathematical Models of Quantum Entanglement>

There are several approaches to mathematical models of quantum entanglement.

• Tensor product model: The most common mathematical model of quantum entanglement is to use tensor products in Hilbert space. In the case of multiple quantum systems, the state vectors of each quantum system can be combined using tensor products to obtain the state vector of the whole quantum system. For example, if we consider two qubits, their state vectors can be expressed as follows. \[|00⟩, |01⟩, |10⟩, |11⟩\]where |xy⟩ is the ground state of the x-th and y-th qubit. The linear combination of these ground states can represent the entangled states.
• Model by measurement of entanglement: Another way to generate quantum entanglement is to create entanglement by measurement. When considering multiple qubits with entanglement, we can consider, for example, the following state known as the Bell state: \begin{eqnarray}|Φ⁺⟩ &=& (1/√2) * (|00⟩ + |11⟩) |Φ⁻⟩ \\&=& (1/√2) * (|00⟩ – |11⟩) |Ψ⁺⟩ \\&=& (1/√2) * (|01⟩ + |10⟩) |Ψ⁻⟩ \\&=& (1/√2) * (|01⟩ – |10⟩)\end{eqnarray} These states are entangled, and when the state of one qubit is measured, the states of the other qubits are also instantly determined.

<Entanglement and its Measurement Methods>

Entanglement is a peculiar phenomenon in quantum mechanics, and refers to the state in which multiple quantum systems are strongly connected and behave as a single whole. This is a characteristic property of quantum mechanics and is not found in classical physics.

Entanglement measurement is a method to investigate the states of entangled quantum systems and to confirm their correlations, and it plays an important role in applications such as quantum information processing and quantum communications.

Measurement of entanglement requires very subtle and sophisticated techniques, and it is important to precisely control the qubits and minimize their interaction with the environment in order to measure entanglement.

With regard to methods for measuring entanglement, the main techniques are as follows

• Bell state measurement: The bell state is a special state in which two qubits are entangled. One way to measure the bell state is to measure one qubit and estimate the state of the other qubit based on the results. Since the bell state is classified into four different states (|Φ⁺⟩, |Φ-⟩, |Ψ⁺⟩, |Ψ-⟩), each measurement results in a specific state of entanglement.
• Measurement of common observables: entangled quantum systems tend to be correlated for common physical quantities. For example, if two qubits are entangled, their spins (spin angular momentum) are correlated. In this case, the presence of entanglement can be detected by measuring the spins of both qubits in the same direction or in opposite directions.
• Measurement of multiple ground states: Entangled quantum systems are correlated for multiple ground states. Entanglement can be confirmed by making measurements on these ground states.

Such measurement methods are essential for implementing quantum information processing and understanding the nature of quantum entanglement.

<Bell state>

The Bell state is one of the special two-qubit states that exhibit quantum entanglement and play an important role in quantum teleportation and quantum communication. Entanglement refers to the state of quantum entanglement, a property in which knowing the state of one qubit immediately determines the state of the other qubit.

Bell states refer to the following four states.

1. |Φ⁺⟩ = (1/√2) * (|00⟩ + |11⟩)
2. |Φ-⟩ = (1/√2) * (|00⟩ – |11⟩)
3. |Ψ⁺⟩ = (1/√2) * (|01⟩ + |10⟩)
4. |Ψ-⟩ = (1/√2) * (|01⟩ – |10⟩)

where |xy⟩ is the ground state of the x-th and y-th qubits, and these bell states are entangled states; measuring the state of one qubit instantly determines the state of the other qubits.

For example, in the case of|Φ⁺⟩, there is a high probability that two qubits are in the same state (both|0⟩ or both|1⟩), and if one is|0⟩ the other is also|0⟩ to and if one is|1⟩, the other is also|1⟩. On the other hand, in the case of|Φ-⟩, there is a high probability that the two qubits are in opposite states: if one is|0⟩, the other is|1⟩, and if one is|1⟩, the other is|0 ⟩ if one is|1⟩ and the other is|0⟩.

Next, we discuss quantum communication using these concepts.

quantum communication

Quantum communication is a communication technology that uses the principles of quantum mechanics to transmit information securely. In conventional communication methods, information is represented as classical bits (0 or 1) such as electrons or photons, but in quantum communication, information is transmitted as quantum bits or “quantum information. In quantum communication, information is transmitted as quantum bits or “quantum information.

The characteristics of quantum communication are as follows

• Quantum entanglement: Quantum communication uses quantum entanglement to transmit information, taking advantage of the fact that when two or more quantum systems are entangled, their states are highly dependent on each other, and when one state is determined, the other state is instantly determined.
• Quantum Cryptology: Quantum communication is closely related to a field called quantum cryptology. Quantum cryptography uses methods such as quantum key delivery to achieve secure communications that can detect eavesdropping and interception, thereby making information more secure than with conventional encryption methods.
• Non-locality of quantum entanglement: In quantum communication, quantum entanglement has non-local effects that are independent of distance. This property extends the range of applications of quantum communication and enables communication between remote locations.

Applications of quantum communication include the following

<Quantum Key Distribution (QKD)>

Quantum key distribution is a method of delivering cryptographic keys using a communication channel that can be eavesdropped on. This method uses the characteristics of quantum communication to detect unauthorized access and secure communication.

The procedure of quantum key delivery is as follows.

1. Prepare the qubits: First, the sender (Alice) and receiver (Bob) prepare qubits (usually photons). The state of these qubits is randomly selected as 0 or 1.
2. Sending the qubits: Alice sends the prepared qubits to Bob via a public channel. The public channel must be encrypted to guarantee the confidentiality of the information, since it is possible to eavesdrop on the communication path.
3. Receiving the quantum bits: Bob receives the quantum bits sent by Alice. This requires a means to detect modification of the qubits or eavesdropping on the communication path, typically by measuring the polarization and phase of the qubits to detect interference from outside sources.
4. Information exchange via a public channel: Alice and Bob disclose the measurement results of the received qubits to each other via a public channel. At this point, no secret key has yet been obtained, but the published information is used to generate a secret key in a later step.
5. Error correction and private key generation: Alice and Bob use the published information to perform an error correction protocol to remove the effects of errors and eavesdropping on the communication path. They then extract information for sharing a secret key. This information is randomly generated by the state of the qubits, so an outside eavesdropper cannot know the information.
6. Private Key Usage: Alice and Bob use the shared private key for secure communication and authentication, including encryption and signing.

<Quantum Teleportation>

Quantum teleportation is a method of transferring quantum information from one quantum system to another using quantum entanglement. Unlike the science fiction image that the term “teleportation” suggests, in which a physical object is instantly moved to a distant location, quantum teleportation is not a transfer of the object itself, but a means of transferring quantum information.

The steps of quantum teleportation are as follows

1. Creation of a shared entanglement: the sender (Alice) and receiver (Bob) share a pair of entangled quantum bits generated by a third party (Charlie). This entanglement is usually a pair of qubits with a special state known as a bell state.
2. Preparing the qubits: Alice entangles the qubits she wants to transmit (the quantum information she wants to teleport) with her own qubits. This operation links Alice’s qubits with Bob’s shared entanglement.
3. Bell Measurement: Alice performs a special measurement, called a bell measurement, on both her own qubit and the qubit she wants to teleport. This measurement sends information about the measurement results to Bob via the classical communication channel.
4. Reconstruction of the qubits: Bob uses the classical communication information sent by Alice to perform operations on his qubits. This results in the state of Alice’s teleported qubit being transferred to Bob’s qubit.

<Quantum Repeater>

Quantum communication is sometimes combined with quantum repeater technology to reduce information loss in long-distance communications.

The quantum repeater procedure is as follows

1. Entanglement generation: First, an entanglement is generated between the sender (Alice) and the receiver (Bob). Alice and Bob generate entangled quantum bits by a third-party relay (Charlie), and Alice and Bob retain their respective quantum bits and the entanglement generated by Charlie.
2. Communication with the relay: Alice sends a portion of the entangled quantum bits to Charlie. Charlie uses his own entangled qubits and the qubits sent by Alice to achieve entanglement at the intermediate point.
3. Repeat: The relay person (Charlie) repeats the entangled quantum bits between Alice and Bob. This allows the entanglement to be transmitted over a long distance.
4. Final entanglement generation: Once entanglement is established between the relay (Charlie) and the receiver (Bob), Bob can receive Alice’s qubit information via the relay, without communicating directly with Alice. Similarly, Alice can receive Bob’s qubit of information.