Qubit mixed state density matrix coordinates on a Bloch ball

In summary, to find the coordinates on the 3D Bloch ball of a qubit's mixed state, you must use the formula (\sum p_i x_i, \sum p_i y_i,\sum p_i z_i) after diagonalizing the density matrix. This process may seem impractical, but it is a simple calculation involving only three additions/subtractions.
  • #1
maxverywell
197
2
What are the coordinates on the 3D Bloch ball of a qubit's mixed state of the form:
##\rho=p_{00}|0\rangle \langle 0|+p_{01}|0\rangle \langle 1|+p_{10}|1\rangle \langle 0|+p_{11}|1\rangle \langle 1|##

451px-Bloch_sphere.svg.png
 
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  • #2
Do we have to diagonalize the density matrix everytime and then use the forumla for the coordinates
##(\sum p_i x_i, \sum p_i y_i,\sum p_i z_i)## ?
https://en.wikipedia.org/wiki/Bloch_sphere

Not very practical.
 
  • #3
From stackoverflow: "Convert from qubit density matrix to Bloch vector"

Code:
def toBloch(matrix):
   [[a, b], [c, d]] = matrix
   x = complex(c + b).real
   y = complex(c - b).imag
   z = complex(d - a).real
   return x, y, z

In other words, yes you have to use the formula. But I'm not sure why you think it's impractical... it's just three additions/subtractions.
 

1. What is a qubit mixed state?

A qubit mixed state is a quantum state that cannot be described by a single pure state, but rather is a combination of two or more pure states. This means that the qubit has a certain probability of being in each of these pure states, making it a probabilistic state.

2. What is a density matrix?

The density matrix is a mathematical representation of a quantum state, which takes into account the probabilities of being in different pure states. It is a square matrix that contains information about the state's quantum coherence and entanglement with other states.

3. What are the coordinates on a Bloch ball?

The coordinates on a Bloch ball represent the state of a qubit in terms of its quantum mechanical properties, such as spin and phase. The x, y, and z coordinates correspond to the probabilities of the qubit being in the 0 and 1 states, as well as the relative phase between these states.

4. How are qubit mixed state density matrix coordinates visualized on a Bloch ball?

The density matrix coordinates are visualized on a Bloch ball by plotting them as a point on the surface of the ball. The x, y, and z coordinates correspond to the point's position on the x, y, and z axes, respectively. The distance from the origin of the ball represents the magnitude of the state's coherence.

5. What is the significance of qubit mixed state density matrix coordinates on a Bloch ball?

The Bloch ball representation of qubit mixed state density matrix coordinates provides a convenient way to visualize and understand the properties of a quantum state. It allows for easy comparison between different states and helps in studying their quantum behavior and interactions.

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