**Japanese abstract**

前のポストにおいて、２量子ビットシステムを３台のモバイルフォンで探究するためのアプリ（量子回路シミュレータ）を開発した。今回これを利用して、Superdense codingを実装してみた。Superdense codingは、少ない量子ビットを送ることでたくさんの古典ビット情報を送ることができるアルゴリズムである。このアルゴリズムは、量子重ね合わせと量子もつれを巧みに利用しており、これによって、私の開発したモバイルフォンアプリが正しく動作することを確認できた。

**English abstract**

In the previous post, I developed an app (quantum circuit simulator) to explore a two-qubit system using three mobile phones. This time I used this to implement Superdense coding. Superdense coding is an algorithm that can send many classical bits of information by sending fewer qubits. This algorithm makes clever use of quantum superposition and entanglement, and I was able to confirm that the mobile phone app I developed worked correctly.

**●Superdense coding that sends classical 2-bit information with 1 qubit**

Here, we will deal with Superdense coding, which allows you to send classical 2-bit information (any of 00, 01, 10, 11) by just sending one qubit. The details of the theory are carefully described in reference [1]. I implemented this algorithm with reference to that description. This will be explained specifically using Fig.1 and Fig.2.

Fig.1 shows the preparation phase. In the left column of Fig.1, Alice and Bob each have one qubit. Each is assigned to one mobile phone. The various quantum gates shown in the figure can be applied to freely change the qubit state. Here, (1) Alice applies Hadamard gate H to the initial state |0> to create a superposition state. On the other hand, (2) Bob applied the Pauli X gate to the initial state |1> and changed the state to |0>. These quantum states are transmitted to the control mobile phone on the right side via the real-time database CloudBD. Then, (3) tensor product is calculated to consider these two qubits as one system. Next, (4) Applying the CNOT gate creates a quantum entangled state. In other words, notice that the ”Entangled?” label has changed from false to true.

In Fig.2, (5)Alice applies Pauli gate Y. As we will see later, applying this Y gate actually means selecting the classical 2-bit information "11". It then sends this qubit to Bob. On the other hand, at this point, (6) Bob does not take any new action. The control mobile device on the right side of the figure (7)updates the tensor product to reflect Alice's application of the Y gate, but the entangled state is maintained. At this point, Bob will have a total of two qubits: the qubit sent by Alice and the qubit he originally had. Next, Bob has the control mobile device (8)apply the reverse Bell circuit. As a result, the quantum entanglement disappears ("Entangled?" false), and the probability of the basis vector |11> becomes 100%. In other words, if Bob makes a measurement here, he will get the classical bit information "11".

The most important point here is that Bob has no idea what Alice sent him before taking the measurements. Nevertheless, no matter which classical 2-bit information Alice sends, Bob can always simply apply the reverse Bell circuit. That way, Bob can get the information that Alice sent! This is the great thing about Superdense coding.

When sending classical bit information other than "11", it is shown in Fig.3. If Alice would like to send "00", she applies the I gate (no action). If she sends "01" she applies the X gate, and if she sends "10" she applies the Z gate. In any of these cases, as mentioned above, Bob only needs to perform the measurement after applying the reverse Bell circuit.

**References**

[1] Chris Bernhardt: Quantum Computing for Everyone, The MIT Press, 2020.

https://www.chrisbernhardt.info/