Is there a relationship between Average energy of transmission and power saving As mentioned  in many books, the average energy of transmitted symbol is given by: 

$E_s = (2/3)(M - 1) $ 

where $M$ indicated the modulation order. 
Suppose that we are using in the first scenario $SISO$ system, and we need to transmit 8bit/channel use, so the needed modulation is $256QAM$ which is equivalent to $E_s = 170$.  now suppose that we are using $MIMO$ system with two transmitters antennas and two receiver antennas, so to transmit the data rate of  8bit/channel use, we need to transmit 4bit/channel use in each antenna, we will need to use $16QAM$, so the needed $E_s = 10$ in each antennas, which is in total 20.  

can we say, the power saving for transmitting 8bit/s/Hz in case of using MIMO system is 170/20  ? 

Is that right? thank you
First of all, 8 b/s/Hz is not a bit rate, but rather a spectral efficiency.

Second, you don't transmit $\log_2(M)$ bits per second, but rather $\log_2(M)$ bits per channel use, which is roughly $T_s$ seconds, where $T_s$ is the symbol time. So, the bit rate is $\frac{\log_2(M)}{T_s}$.

If you adjust your wording to the above by saying 8 bits/channel use, then, yes, you are right, you save energy by using MIMO, but the saving is $170/20$ not the reciprocal.