{"id":55,"date":"2008-08-17T17:31:02","date_gmt":"2008-08-17T16:31:02","guid":{"rendered":"http:\/\/www.quantum-bits.org\/?p=55"},"modified":"2018-06-08T14:44:29","modified_gmt":"2018-06-08T13:44:29","slug":"information-and-quantum-mechanics-part-1","status":"publish","type":"post","link":"https:\/\/www.quantum-bits.org\/?p=55","title":{"rendered":"Information and quantum mechanics (part 1)"},"content":{"rendered":"

Okay, at last, a first post about information<\/a> and quantum mechanics<\/a>. Let’s begin with the very basic building blocks of all this: quantum states<\/a>.<\/p>\n

A qubit is a short for “quantum bit”. It is to quantum information what a bit<\/a> is to computer information. Bits and qubits are nevertheless quite different. A computer bit is either a 0 or a 1 (switch off\/on, true\/false, etc.) whereas a qubit is a superposition of these two orthogonal states. It is a single piece of information with a probability of being true.<\/p>\n

In more formal terms, quantum information is described by state vectors in a 2-level quantum mechanical system<\/a>. In this post, we’ll go into what a “quantum state” is.<\/p>\n

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\nThe classical world<\/b><\/p>\n

The simplest classical system is composed of a single non-relativistic particle. Its motion is easily described by the evolution of its coordinates and momentum as a function of time.<\/p>\n

This can be generalized to complex systems with the introduction of generalized coordinates and momenta .<\/p>\n

The space of all the possible positions that the physical system may attain is called the configuration space<\/a>. For a single non-relativistic particle, this space is simply . For a n-particles system, this would be .<\/p>\n

To take account of both generalized positions and momenta, one has to extend that to introduce a larger manifold<\/a> called the phase space<\/a>. Each point of the phase space correspond to a given set of coordinates and momenta .<\/p>\n

The dynamics of the physical system is given by all the possible trajectories inside the phase space. It is ruled by the existence of a relation connecting coordinates and momenta formally called an Hamiltonian: .<\/p>\n

The equation of motion is given by a variational principle<\/a> called the principle of least action<\/a>. Simply speaking, it states that Mother Nature is a big fat lazy girl: among all the possible paths inside the phase space, she will choose the one that minimizes her efforts.<\/p>\n

The equation allows one to calculate the dynamical state of the system at time given that its the initial state is known at time .<\/p>\n

Practically speaking, the initial state is usually not perfectly known. It is measured by the mean of an experimental apparatus and is actually known within a margin of error given by the precision of this apparatus. From a theoretical point of view, it means that the inital state is known to be somewhere within a defined small hypervolume inside the phase space.<\/p>\n

If, at time the system is in two radically different position inside the phase space given two initial states inside that same small hypervolume, the system is said to be chaotic<\/a>. Even so, the system is still deterministic: if the initial states is perfectly determined, the state at will be perfectly determined.<\/p>\n

To sum up, the dynamical state of a classical physical system is fully characterized by its initial state . The values of the measurable quantities (observables) and are thus functions of . One can say that and are the values of the observables<\/b> and at time for the prepared state<\/b> .<\/p>\n

Now, imagine an experiment where we conduct a series of measures of and at a given fixed time and where the initial state is random. This initial state is called a mixed state<\/b>, as opposed to pure (defined) states<\/b> like : it is a statistical mixture<\/b> of pure states. Both and are now random variables. Even though a single measurement cannot be predicted, on can measure the expectation values of the observables in the state : and .<\/p>\n

Observables and states<\/strong><\/p>\n

Now, let’s jump into the quantum world. Here, even pure states show statistical behaviours: regardless of how carefully we prepare the initial state of the system, measurement results are generally not repeatable<\/b>. This is not the weird prediction of some kind of crackpot theory from a mad scientist. This is just… Nature at work.<\/p>\n

At the quantum level, one must understand the results of the measurements of an observable A as a statistical mean. And this mean is what physical theories have to predict. Many experiments have been conducted, setting the fundations of quantum mechanics. I strongly recommend reading about Stern-Gerlach<\/a> and Bell test<\/a> experiments.<\/p>\n

What are these observations telling us ? Well, many many fundamental things. Among others:<\/p>\n