Era Quantic | Quantum Computing and Information Articles

  The Cauchy-Schwartz inequality is one of the important geometric properties of Hilbert space. The proof of this inequality is as follows: First, let's examine the left-hand side of the inequality. Let us now consider the right-hand side of the inequality. We can write the unit operator, which is the sum of all orthonormal basis kets in the Hilbert space. Let's look at an example that satisfies this inequality.  

    As you know, in quantum mechanics we operate with state vectors. However, it is very convenient to use the alternative notation to solve some problems. This alternative representation is expressed by the density matrix. A density matrix does not violate quantum postulates like state vectors and even postulates have equivalents just like state vectors. It is represented by the density matrix or density operator rho (ρ). The mathematical representation is as follows:, Let's take the density matrix as an example: As you can see, the density matrix is very practical and very easy to calculate. In particular, it is a pretty good alternative for computations in quantum mechanical systems that make up multiple qubits. You may have noticed another similarity right away. Yes, the trace of the density matrix is equal to 1. This means that we know that the state vector must be normalized. It means the same with the sum of all probabilities is equal to 1.

    Entropy is one of the most important concepts in physics. In my previous articles, I focused on what entropy is and its relationship with time . As you will remember, entropy is a measure of disorder and is in the positive direction of increase. In this article, I will explain the expression of entropy on quantum computers and its relationship with quantum information in its broadest form. In the last part, we will learn what another important concept, entanglement entropy, means to us. Let's consider a system. The more information we have about this system, the lower the entropy. The opposite is also true. That is, the less information we have about this system, the higher the value of entropy. So entropy is a measure of our ignorance about a system. First, let's define this definition in a classical mathematical way. Next, let's write the equivalent in quantum mechanics. Let's consider two sets A and B. Suppose that sets A and B contain one bit of information. Le

    The Stern-Gerlach experiment is one of the experiments that best demonstrates the reality of quantum physics. In 1922, Otto Stern and Walther Gerlach devised a highly effective experiment to test Bohr's theory of discrete angular momentum. Let's take an in-depth look at the experimental setup below. Stern and Gerlach used Ag atoms with 47 atoms to perform this experiment. Ag atoms with 47 atoms have 5s 1 unpaired electrons in their final orbital orbitals. Ag atoms are heated in a furnace to give them high energy. Ag atoms with high energy spins oriented in different directions are passed through the collimators in bunches and then in an inhomogeneous magnet and hit the observation screen. There are two important points here. One of these important points is that an inhomogeneous magnet was used in the experiment. As you can see in the setup, the N pole is wide and the S pole has pointed ends. The magnetic field lines are more intense at the pointed end of the S pole. If the

 One of the most important applications of quantum mechanics is Bell's inequality. In 1965, John Stewart Bell actually thought that Einstein might be right and tried to prove it, but proved that quantum logic violated classical logic. It was a revolutionary discovery. Let's look at its mathematically simple explanation. To create the analogy, let's first write a classical inequality.   This inequality above is a classic expression. The quantum analogy of this expression represents the Bell inequality. With just one difference. Bell's inequality shows that the above equation is violated. Expressing Bell's inequality by skipping some complicated mathematical intermediates; While there is no such situation in the classical world that will violate the inequality we mentioned at the beginning, it is difficult to find a situation that does not violate this inequality in the quantum world. Reference https://www.youtube.com/watch?v=lMyWl6Pq904 https://slideplay

  The entanglement properties of qubits used in quantum computing are exploited. However, the state vector of qubits need not be entangled. A composite system is formed by the tensor product of the sub-physical systems that compose it. Let's consider two systems, A and B.   If there is a correlation between these A and B systems, we can write the tensor product of these systems as follows. However, if there is no correlation between systems A and B, these systems are entangled and the above equation cannot be written. For example, Question: Can we write as |ψ> = |A>|B>? The answer is no, we cannot write this equation. because, However, we can write the tensor product expression in its most general form, whether it is entangled or not. In this case, we can write the above tensor equation of |ψ> = |00> + |11>.

  In our previous article, we had an article about quantum teleportation . In today's article, we will talk about how quantum teleportation is mathematically possible. To briefly restate quantum teleportation; Quantum teleportation allows a state vector to be transmitted somewhere without any quantum channels, even at the speed of light distance between them. For this, the following protocol is applied. A long time ago, Alice and Bob shared two qubits in an entangled state with each other. Let Alice's state vector be: The operations on the state vector |ψ> in the above protocol are | ψ 1 >, | ψ 2 >, | ψ 3 > and | ψ 4 >. Let's examine the state vectors one by one. Let's pass the state vector | ψ 1 > through the CNOT and get | ψ 2 > .   The first two qubits belong to Alice, and the last qubits to Bob. Now let's go through Hadamard. Now Alice can measure over her two qubits. As a result of the measurements, four results can be obtained with a 25%

  Time is a strange phenomenon. We talked about entropy in our, thermodynamics article. In our classical perception, the time has a relationship with entropy. We know that the direction of the arrow of time always flows forward with the positive progression of entropy. Because common sense tells us that this is so. So is it really so? Let go of whatever work you are doing for a moment and imagine your past, present, and future. Which one do you think actually existed? Many of you have lived the past and it was real, I am living in the present moment and this was real but I did not live in the future, you may not be sure of its reality. If you are someone who accepts only the present, you are presentism. How can we agree on a time when we even doubt reality? We cannot be… In his theories, Newton defined space and time as absolute. In fact, he was intuitively right in his own reference. However, our deeper-thinking genius, Einstein, had stated in his theory of relativity that space and

    In today's article, I will write the proof that quantum cloning, which is an important feature of quantum mechanics, is not possible. The Quantum no-cloning theorem tells us that it is not possible to create any quantum state vector with the unitary operator. To prove this, let's consider a unitary operator machine that creates two state vectors from a state vector.   Clone |0>, |1>, and even the most general state vector |ψ> as follows.   We see that the 1st and 2nd equations above should be equal to each other, but they are not. This situation shows us that quantum cloning contains mathematical contradictions.       Reference https://www.youtube.com/watch?v=qUo3zjkNE24&list=PLqNc_xpYGu77u6dI_RijjCLd-nP8p_4pI&index=23 https://physicsworld.com/a/cloning-quantum-steering-is-a-no-go/

  One of the most basic applications of quantum algorithms is superdense coding. It is a simple protocol, but it is also an important protocol. Now let's take a detailed look. The setup of the protocol is as follows: Alice wants to send 2-qubit information to Bob, but she can only send a single-qubit due to her possibilities. But how does Bob get a 2-qubit of information from Alice? This is where Caren comes in. Caren entanglements 2 qubits through Hadamard and CNOT. It sends one of the entangled particles to Alice and the other to Bob. Alice and Bob have no information as they do not measure which particle has come to them. From here on, Alice will be able to send the qubits she wants to Bob. Let's examine them all in turn. Let Alice wants to send the 00 qubits. In this case, it will use the I gate. Let Alice wants to send the 01 qubits. In this case, it will use the X gate.  Let Alice wants to send the 10 qubits. In this case, it will use the Z gate.                Let Alice

 Classic NOT Gate: The NOT gate converts bit 0 to bit 1 and bit 1 to bit 0.   Sigma X Gate: The X gate, also known as the Pauli X matrix, also refers to the NOT gate. The X matrix and the expressions |0> and |1> are as follows.   If we make the X gate act on the qubits |0> and |1>;   In general, if we write; For example, let's draw a 2-qubit circuit; Sigma Z Gate:  The Pauli Z gate does not change the result when it acts on the |0> qubit. However, when it acts on the |1> qubit, it changes phase, that is, it introduces a minus sign. If we make the Z gate act on the qubits |0> and |1>; Sigma Y Gate:  If we make the Y gate act on the qubits |0> and |1>; Hadamard Gate:  The Hadamard gate [H] is used to superposition our qubit. If we make the H gate act on the qubits |0> and |1>; The gates you've read so far were single qubit gates. However, the CNOT gate we will see now is used for multiple qubits.   CNOT (Controlled NOT) Gate: Nota