# How meaurements increase entropy?

1. May 24, 2013

### StarsRuler

¿ How meaurements increase entropy?

The decoherence before the measurement increase entropy but collapse returns the state of the system to a pure state. ¿Why then measurements increase entropy

2. May 24, 2013

### DrDu

Ideally measurements don't increase entropy.

3. May 24, 2013

### StarsRuler

Sorry, I don´t know what is the difference between ideal and not ideal measurements. What is a no ideal measurement, and how does that increase entropy anyway??

4. May 24, 2013

### DrDu

You may have to cool a detector, which generates lots of entropy, or you may not register all of the information principally available. Then the system is also left in a statistical mixture with higher entropy. But in principle, a quantum mechanical measurement can be performed in such a way that no or arbitrary little entropy is produced.

5. May 24, 2013

### Jano L.

This is very sensitive to what we mean by "entropy".

If it means "information entropy" regarding the knowledge of the state of the system, then the measurement usually decreases it, because usually we know more about the state after the measurement than we did before.

For example, in the measurement of the atom spin, before the measurement we do not know anything about its value at all, so the probability $p_1$ for obtaining spin z+ is one half and the probability $p_2$ for obtaining spin z- is also one half. The information entropy is

$$I =\sum_k -p_k\ln p_k = -\frac{1}{2}\ln \frac{1}{2} -\frac{1}{2}\ln \frac{1}{2} = \ln 2\approx 1.4~~.$$

After the measurement by the Stern-Gerlach magnet, we know the spin exactly, let it be z+. Then $p_1 = 1$, so the information entropy is

$$I' =\sum_k -p'_k\ln p'_k = -1.\ln 1 - 0.\ln 0 = 0.$$

($0.\ln 0$ is defined to be 0, probably based on the limit $\lim_{p\rightarrow 0} p\ln p = 0$.)

It follows that the measurement is followed by decrease of information entropy describing the knowledge of the system.

6. May 24, 2013

### StarsRuler

But then, ¿where does come the second principle of thermodynamics, measurements are the unique proccess that change entropy

7. May 24, 2013

### DrDu

Who sais so?

8. May 24, 2013

### Jano L.

Second principle of thermodynamics says that it is impossible that a system undergoes a cyclic process and during this process converts energy received from thermal reservoir into equivalent amount of work. It has no direct connection to measurements.

In quantum theory, one usually deals with measurements of microscopic systems. Such systems do not admit thermodynamic description and the second thermodynamic law is not valid for them (perhaps only in a statistical sense).

9. May 24, 2013

### StarsRuler

Well, this is the primitive formulation of the second principle. Really, I never understanded it. And in the career litte and simple thermodynamics we studied, I really understanded thermodynamics with the Landau Book about it. The modern formulation is , you know, that entropy do not reduce in a isolated system, and entropy is the Boltzmann entropy, that in a temporal promedium in a time long for local relaxation time, coincides with shannon entropy for microstates. Sorry, I don´t know use latex in this forums, normal prefix [tex] doesnt work. What is the prefix? Landau is certain doesn´t deduce second principle, only postulate that is a consequence of measurement proccess, but assumes it like a postulate. But obviously, the other principles are deduced, it is a better conceptual treatment ( thermodynamics deduced from statistical mechanics) than exclusive thermodynamic treatment.

10. May 25, 2013