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How many parameters are required to specify the density matrix of a $n$-qubit system, and how many parameters are required to specify a quantum operation (completely positive maps between states) on an $n$-qubit system?

For example, when $n=1$, there are 3 parameters required to specify the density matrix and 12 parameters are required to specify a completely positive map.

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1) Density matrix

Let us admit that the most general quantum description of a system is a "not normed" density matrix (I mean a density matrix with a trace not necessarily equals to $1$).

The density matrix is a hermitian matrix, so in a $N$ dimensional space, it has $N$ real diagonal parameters and $\frac{N(N-1)}{2}$ non-diagonal complex parameters, so the total is : $N + 2 * \frac{N(N-1)}{2} = N^2$ real parameters.

To describe a n-qubit, you have $N = 2^n$.

So you have $2^{2n}$ real parameters to describe the most general n-qbit density matrix.

If you normalize the density matrix ($Tr(\rho)=1$), you need only only $2^{2n}-1$ real parameters, so for $n=1$, you will recover your $3$ parameters.

2) Unitary transformations

A unitary transformation ($U(N)$), in a $N$ dimensional space, requires $N^2$ real parameters, so in our case ($N = 2^n$), it requires $2^{2n}$ real parameters. However, mathematical states which differ only by a global phase correspond to the same physical state, so we may consider, that physically, we have only $2^{2n}-1$ real parameters, which correspond to special unitary transformations ($SU(N)$). I must say I don't understand where you get your "$12$".

**UPDATE**

3) Completely positive maps

According to this paper, the dimension of completely positive maps seems to be $N^4-N^2$. Here, we have $N=2^n$, so the dimension is $2^{4n}-2^{2n}$.

So for one qubit $(n=1; N=2)$, we would have a dimension $2^4-2^2=16-4=12$.

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  • $\begingroup$ My bad, I meant general quantum operations, not just unitary transforms. $\endgroup$
    – ruadath
    Jun 13, 2014 at 14:11
  • $\begingroup$ @ruadath : What do yo mean exactly by "general quantum operations" ? $\endgroup$
    – Trimok
    Jun 14, 2014 at 10:57
  • $\begingroup$ Completely positive maps $\endgroup$
    – ruadath
    Jun 16, 2014 at 12:55
  • $\begingroup$ @ruadath : I update the answer $\endgroup$
    – Trimok
    Jun 17, 2014 at 8:35
  • $\begingroup$ Alright cool, I will vote up once I get the necessary reputation. $\endgroup$
    – ruadath
    Jun 17, 2014 at 11:11

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