Current Amplification
A recent review of the CH Precision P1 phono preamplifier ($48,000) states that Current amplification circuits produce the best signal/noise ratios [for moving coil cartridges].
Their spec is -138  dBu [.775 volt reference] with a 1 ohm source and 22 kHz bandwidth, or -140 dBV.
A current preamplifier responds to the current output of the source and has near zero input impedance*.  A voltage preamplifier has a high input impedance and responds to the voltage of the source.
In a voltage amplifier the principal noise sources are the source resistance, the input transistor, and the feedback resistor to ground.   In a current amplifier there is no resistor to ground; the source resistance is the feedback resistor.  So one noise source is eliminated and the theoretical noise is lower.  A side affect which may be beneficial is that the gain is inversely proportional to the source resistance, so the lower the source resistance the higher the gain.  Since in general the low output MC cartridges also have low resistance,  that is self compensating.

Is it low noise?
 The DB Systems DB-4B head amp, introduced in 1976,  specs -86 dB A Weighted [1 mV reference]  That is -146 dBV, or about -143 with a 22 kHz bandwidth.  So it is 3 dB better, using conventional voltage amplification.  BTW the noise of a 1 ohm resistor is -154.4.

Note: This calculation doesn't take into account the RIAA curve which would give the DB-4B a few dB more advantage.

I emailed CH Precision on their website and got an error message.

*Speaking in terms of operational amplifiers, the + input is grounded and a resistor is connected from the output to the – input which is the summing node.  The cartridge has one side grounded and the other is connected to the summing node which is essentially zero impedance and has no voltage associated with it.
A current amplifier output has a very high impedance output but that is a different story.