I think it is impossible generally. Especially for an open quantum system, the information of the system of interest is lost during the evolution, so the reverse process can work as usual with less information. If we consider such a question by the quantum depolarizing channel, one can also find the reverse process is unphysical.

@C.-S. Yu

Dear Yu,

you are right, we can't protect our open system from interacting with environment.But in a closed system, we can access information from evolving Quantum system by reducing interaction from outside.Quantum non-demolishing and back action evading are techniques from which we can avoid interaction up to great extend.if our system has evolved from time t=t' to t=t''.now can we evolve somehow our system back to initial state?

Since the microscopic laws of physics obey time reversal symmetry this question comes up often. However, even though a systems can exist in principle doesn't imply that it may be constructed by any means available to us. I like to think of the analogy of factoring large integers. Just because the prime factors of an integer are assure to exist in all cases doesn't imply one could find them even given cosmological time scales.

IMHO, in quantum field theory the arrow of time is chosen arbitrarily. You can work with advanced solutions as well as with retarded. BUT THIS CHOICE MUST BE THE SAME FOR THE SYSTEM AND THE OBSERVER. You cannot evolve the system ahead and the observer in reverse. And a system interacting with an observer is ipso facto open. Thus, if you believe that you go ahead in time, you MUST solve equations for the system you observe ahead in time. Conversely, assuming that a system is closed at the fundamental level renders it unobservable.

A quick addition to my post. If BY DEFINITION the observer and the system do not interact, their arrows of time are independent. But, by the same definition, this cannot be observed.

More on the same. A closed system is an approximation neglecting at least the back action of the observation. But neglect does not mean forget. In principle it is always there synchronizing the arrows of time.

It is not possible to do so by reversing the time arrow. A state under evolution will lose information and it will not be possible to trace back from which state it evolved. 

Consider any state which is a linear superposition of many other states (linearly independent). Upon evolution, say under a magnetic field, the amplitudes of each linearly independent state will change (this happens for a spin system). And there is no way by which we can tell the amplitudes of the states before evolution.

If the evolution is unitary, this means that the evolution operator is. This statement has content that the adjoint operator exists and using the adjoint operator it's possible to compute the time reversed states. This is all standard material. The problem with evolving backwards in  time regarding cosmology is that we do not know the evolution operator, so we do not know its adjoint. The reason is that we do not know  a unitary description of gravity, beyond the classical approximation, which is general relativity-which predicts the existence of black holes, which give rise to caually disconnected patches of spacetime, behind event horizons. And we do not understand Hawking radiation beyond the semi-classical approximation. For certain special cases, only, can unitary evolution be established in the presence of black holes.  And the Standard Model, while providing a unitary description within the framework of perturbation theory, is not a complete description of matter to arbitrarily high energies. The discovery of the Higgs indicates that there must exist additional degrees of freedom, that stabilize the corrections to its vacuum expectation value, for instance.

If you are talking about generating quantum states that travel backwards in time, see Pregnell and Pegg "Retrodictive quantum state engineering" J. Mod. Opt. 51, 1613 (2004).  If you are talking about retrodicting the past (as against predicting the future) by means of quantum mechanics see Barnett et al "Master equation for retrodiction of quantum communication signals" Phys. Rev. Lett 86, 2455 (2001) and the references therein.

To elaborate a little on my previous answer, in a closed system one can use the unitary time evolution operator to find the state at a later time and its inverse to find the state at an earlier time.  In an open system to predict the state (density operator) at a later time one can use the usual (i.e. predictive) master equation. To retrodict the state at an earlier time in an open system one can use the retrodictive master equation mentioned in my previous answer.

Most of the systems with which we do experiments either evolve non-unitary, or it is very difficult to reverse their evolution.

I will begin with the latter case. Imagine a Gaussian beam (did you study such beams?). Well, the beam may be quite localized in space in the beginning. But if so, the indetermination in linear momenta is very big. The distribution of the linear momenta is the Fourier transform of the Gaussian, and has also Gaussian form, so let's call them the Gaussian of positions, and Gaussian of linear momenta. So, the Gaussian of the linear moments is wide in our case. In short, it shows that big velocity and also small velocities are probable,

In short, very soon after the initial beam starts its travel, it widens and widens because the particles with small velocities advance little, while those with big velocities advance a lot. The reverse in time can be simulated by reversing the movement. We can surround our wave-packet by mirrors and reflect all the particle back, with the hope to obtain again the localized wave-packet. Well, it won't be easy. The fast particles will reach the initial positions rapidly, and leave these positions again, the slow particles will hardly move. We won't get back the small wave-packet despite the unitary transformation.

Now, another trouble are NON-UNITARY processes: the decay, the thermo-nuclear reactions, the chemical reaction which emit energy in the space...

Most of the processes in our world evolve non-unitarily.

Good luck!

Probability is declared in QM, but not clearly confirmed. Probability was also declared in CM, as Laplace demon, but has not been confirmed yet, except a new picture of deterministic chaos.

In discussing this question, we must distinguish between two cases:                  

(1) Continuous evolution of a system under a wave equation with known Hamiltonian, and (2) Evolution including measurements. In the first case, the evolution is, as explained in some previous answers, theoretically reversible. But I agree with Sofia Wechsler that it is still difficult to realize it experimentally.

In the second case, it is generally impossible even theoretically. According to the conventional Quantum Mechanics, a measurement usually disrupts continuous evolution, and the system "jumps"  into a totally new state with some sufficiently definite value of the measured observable (the only exception is when the system already had this value before the measurement).  Generally, all information about the previous state is irretrievably lost. Instead, the measurement creates some new information embodied in the new state. We can predict only the probability of this or that outcome, but not the outcome itself. These features of measurement (emergence of a specific outcome from a range of possibilities) are not described by any equation and severely restrict classical determinism. An example may be the photon frequency measurement in a pure ensemble of photons from a pulse laser. We can do it by placing detectors tuned each to a specific frequency along the  beam's path and dimming the laser so that we are dealing only with one photon at a time. If one of detectors clicked, we know the frequency of the given photon, but even that - only in retrospect. The information was obtained only about the past state at the cost of the photon itself - it is absorbed in the detector. Another example - polarization measurement . Suppose we have a diagonally polarized photon, but we do not know its polarization state, and pass it through a polarizing filter with vertical transmission axis. In a less favorable outcome, when the photon does not pass, we learn retrospectively that its polarization had a non-zero horizontal component, without its exact value, and even this very restricted information comes at the cost of the photon itself. In a more favorable outcome when the photon passes, its polarization state is abruptly changed from diagonal to vertical. We only learn retrospectively that its polarization had a non-zero vertical component, and nothing about its horizontal component. The time reverse of this process would be the vertically polarized photon incident on the back side of the same filter and emerging from the front side as diagonally polarized. Such a process never happens in the real world, so we have no time-symmetry in this case.

There is so-called "Time-symmetric" formulation of QM, but in my view, it is highly debatable, and, to my knowledge, does not extend to the above-described cases.

What is a system state in QM? You know, not coordinates nor momenta. The question is much more complex, since the wave function characterizing evolution of states can be reversible, very simple and transparent, but its arguments are supposed probabilistic without any explanation the nature of this probability. In CM limit one can propose some realistic scenario of the probability origin, nice or no so fine, doesn't matter. But, there is no some similar viewpoint in QM while. So, how to access to reversibility without any working idea to explain this one?

"If that control were possible we could, e.g., resuscitate a dead cat"

But is it fair to say that to control all particles in the whole universe we would have to borrow some resources from that universe and those resources then would play a different role in the reversal process making the full reversal simply impossible.

Only a deity could do this while remaining unnoticed. And then when reversing time to the past moment 100%, all memories of the future would reverse to the past state so whats the difference in saying that everything irreversibly "moves forward".

I want to correct myself in my previous answer as of Nov. 29 (2014). I wrote about the photon frequency measurement:

"... We can do it by placing detectors tuned each to a specific frequency along the beam's path and dimming the laser so that we are dealing only with one photon at a time. If one of detectors clicked, we know the frequency of the given photon, but even that - only in retrospect."

The last statement would be correct only for a special case when the measured photon had already had definite frequency, equal to that of the detector, before measurement (was in an eigenstate of the energy operator - monochromatic state). Generally, even a single photon is in a superposition of such states, therefore the described frequency measurement would not tell us the exact frequency even in retrospect. It would only tell us the frequency the photon "chose" to collapse to and thus to disappear as separate entity. We still could get the retrospect info about the frequency distribution, but only in the set of trials with a pure ensemble of such photons and recording the number of clicks of different detectors. But even that will not tell us about the shape of the photon's wave packet in configuration space, since the information about the phases of its superposition amplitudes will be obliterated. The original bottom line is the same - the restrictions on the classical determinism in QM measurements make the time reverse of a quantum state generally impossible.  

I cannot discuss the concept of "weak measurements" allegedly conserving the time symmetry, which is developed by Aharonov a.o. This is a murky and controversial issue and needs thorough scrutiny.

"observing an object at a later time can allow a better guess about its earlier quantum state"

http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.114.090403