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I know that a one-qubit $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, where $\alpha, \beta \in \mathbb C$, can be represented geometrically on a Bloch sphere as $|\psi\rangle = \cos\theta |0\rangle +e^{\text{i}\phi}\sin \theta |1\rangle$.

Now, for a two-qubit state $|\psi\rangle= \alpha |00\rangle + \beta |11\rangle+\gamma|01\rangle+\delta|10\rangle$, where $\alpha, \beta, \gamma, \delta \in \mathbb C$, can we find a geometrical representation analog to that of the Bloch sphere?

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  • $\begingroup$ Your question is a special case of this slightly more general one which concerns analogue to the Bloch sphere representation for mixed two-qubit states. Does it help? $\endgroup$ – Adam Zalcman Apr 6 at 23:29
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A two qubit pure state can be decomposed into 8 "geometric" real parameters:

  • (4) Latitudes and longitudes of the normalized bloch vectors for the two qubits.
  • (3) Euler angles for a 3d rotation specifying how entanglement translates from one qubit to the other.
  • (1) A "how entangled is it?" weighting value between 0 and 1.

To compute these values you perform a Schmidt decomposition of the system. The "how entangled is it?" value is the minimum of the two Schmidt coefficients.

To get the entanglement rotation's Euler angles: mutate both Schmidt coefficients to be 0.5, put the state vector back together using the mutated coefficients, then interpret it as a 2x2 unitary matrix (instead of a 1x4 vector) and map that unitary matrix from SU(2) to SO(3) in the usual way.

To get the normalized bloch vectorss: compute bloch vectors in the usual way, normalize them, then convert to latitude and longitude.

Resources: entangled states are like unitary matrices and visualizing 2 qubit entanglement.

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  • $\begingroup$ can you please give me more information on how to factorize the parameters? I mean how to calculate those parameters. $\endgroup$ – Nehad Apr 9 at 18:52
  • $\begingroup$ @Nehad Each of the steps is non-trivial amounts of work, or I'd have listed them. The bloch sphere computation you can get from e.g. cirq. The Schmidt decomposition you can do using numpy's SVD. The conversion to Euler angles you can do by decompose the matrix into the Pauli basis realizing the coefficients are a quaternion and looking up quaternion-to-euler-angle. $\endgroup$ – Craig Gidney Apr 9 at 20:16
  • $\begingroup$ Thank you very much $\endgroup$ – Nehad Apr 9 at 21:41

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