[MUSIC] This week we'll take a look at physical system described by so called continuous variables. Money physical system are described by this type of variable. For example, continuous variables include the quid, richer of the electromagnetic field, the collective spin of an example of atoms, the position and momentum of mirrors or membranes, of up to mechanical systems and many others. It easy to understand that for example, a laser field consisting of about a million forums is practically impossible to describe and discrete variables. We will be forced to live in the states of each for them in the field, we mainly use continuous variables for describing quantum microsystems, although it is also suitable for describing single quantumologists. We will introduce mathematical apparatus that allow us to characterize and describe such systems in general without being tied to specific physical system. Also, we will find out how such systems can be used to implement quantum computations and how such computations different from other, more familiar computations on Qbits.. Before proceeding to the description of quantum systems in continuous variables, let's talk about the canonical description of classical systems. Any classical system with one degree of freedom is given by two countries. Get quantities called the generalized coordinates Q. And a journalist momentum P. The evolution of this wearable sometime is given by the economical Hamilton equation. In these questions the time the relative of one quantity of GOP is equal to the partial derivative for the system from within an age with respect to the con drug quantity P or Q. The Hamiltonian is the energy of the system expressed in terms of canonical variables. For example, a simple harmonic oscillator has a Hamiltonian H, which is expressed as the sum of the kinetic and potential challengers of the escalator. Using this Hamiltonian, I'm looking right equations for the generalized cars night at momentum. These equations show that the general like momentum is equal to the time derivative of the generalist coordinate and the generalist coordinate is proportional to the time derivative of the generalist momentum. Different state in the generalist pulse and time, and then substituting it into another equation, which is a well known equation of the harmonic oscillator. To describe the dynamic evolution of physical systems in time, it is convenient to use the so called classical posts on brackets, which are defiant on the slide. Because in these brackets were able to write the derivative over physical quantity. Let's assume that we have a physical quantity that depends on journalists, impulses, journalist coordinates and time and time. Then the total gelato for this quantity can be written as you see on the slide. Using the canonical Hamiltonian equations, one can rewrite the situation using Poisson bracket. Here F if some physical quantity and H is the Hamiltonian of the system under study as an example if it considers Poisson bracket between the generalist carbonate and momentum, then we see that such a bracket is equal to one. This bracket is called the Fundamental Poisson bracket. After we decided on general economical description of classical physical systems, we can proceed to the description of quantum system in continuous variables. To do this, we use the quantization procedure. This whole procedure consists of several steps. First, canonical conjugate variables are declared to Hamilton operators that correspond to observables. In doing so the canonical Hamilton equations are preserved in the same form seconds the classical Poisson brackets are replaced by commutation relations divided by I and H bear. Here square brackets as a commentator between operators F and G. It follows that the operators of canonical variables are related by the commutation relations shown on the slide at the same time, the equation for the evolution of an arbitrary operator F is similar to the classical case. This equation is called the Heisenberg equation and it represents a Heisenberg version of quantum mechanics where the operators are time dependent, but the white function are not. And this equation age is the Hamilton operator, which as in the classical case, is expressed in terms of the kind of the canonical operations Q and P. Along with the conversation procedure, we need to introduce the reverse procedure namely the transition from quantum operations to the average values these values are measured an experiment. To calculate the average value of the operator A in the quantum state, we need to take the trace of the product of operator A as the density metrics role, which describes this state. In the Heisenberg representation ,a quantum state is described by its density metrics that does not change overtime. In this case the dynamics of the main values of the observed quantities are specified by the dynamics of quantum operators. Let's look at the quantization of the oscillator as an example. First we will write a Hamiltonian of the oscillator using the Hamilton operators of generalized car energy and momentum. Next we'll write the commutation relations with the Hamiltonian as the canonical operators Q and P. Since the operator and it's square compute only the commutate between the conjugator operator remain non zero. Applying economical commutation relation twice we get two equalities substitute into the evolution equation we find that the dynamics in the quantum case as the same as in the classical one. Using the present and formalism, we can define the physical quantum system in continuous variables as a system described by two observable canonical operations Q and P, which obeys the canonical commutation relation. Physical systems in continuous variables has an infinite dimensional basis in a Hilbert space. This easy to prove to do this, let's take a look at the canonical commutation relations operators that describes are considered system. Let's take a trade separation from the right and left hand side of the commutation relation. If the canonical operators Q and P acted infinite dimensional space, then due to the properties of the trace, the left hand side of the equation would be zero. In this case on the right head, stand on the equality, we obtain a non zero constant proportional to the dimension of the space. This contradiction means that we are dealing with operators and acts and infinite dimensional hill world space. Moreover since the operators Q and P turn their again values can take any real values. The two canonical operators Q and P describes the so called single mode quantum system if you have an and mode systems, and it is described by a set of two and canonical variables. If we combine all these operator and director M, then we can write down the generalized canonical commutation relations. Here Omega is the metrics that the direct some of the fixed to multiply by two non singular excuse and metric metrics.