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 th...

 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% ...