.. _sphx_glr_api_gallery_quantum_entanglement:

Entanglement and topology
--------------------------

Bell/EPR spin correlations and the CHSH inequality, the Aharonov-Bohm
effect (a purely topological consequence of the vector potential),
Bloch-sphere qubit dynamics (Larmor precession and an animated Rabi
oscillation), the dynamical generation of entanglement between two
qubits coupled by an Ising interaction, and why no fixed unitary can
clone an arbitrary qubit state (the no-cloning theorem).


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="A charged particle confined to a 1D ring of radius R threads a magnetic flux \Phi through its center; even though the field B=0 everywhere the particle can actually be, the eigenspectrum">

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  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_aharonov_bohm_ring_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_aharonov_bohm_ring`

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      <div class="sphx-glr-thumbnail-title">The Aharonov-Bohm effect</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A spin-1/2 (qubit) state evolves under H=-(\omega/2)\sigma_z (a static field along z) or H=-(\omega_R/2)\sigma_x (a resonant drive along x). Since \sigma_z^2=\sigma_x^2=I, the propagator has the closed form e^{-i\theta\sigma} = \cos\theta\, I - i\sin\theta\, \sigma, used directly here instead of a general matrix exponential. A final section drives the same kind of two-level system with Rabi&#x27;s own closed-form resonance formula (~physicskit.quantum.chapters.spin.RabiProblem), animating the Bloch vector itself as it nutates under the drive.">

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  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_bloch_sphere_spin_dynamics_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_bloch_sphere_spin_dynamics`

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      <div class="sphx-glr-thumbnail-title">Bloch-sphere spin dynamics</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A textbook demonstration that quantum mechanics has no local, realistic description. Two spin-1/2 particles prepared in the entangled singlet state">

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  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_entanglement_and_topology_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_entanglement_and_topology`

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      <div class="sphx-glr-thumbnail-title">Bell correlations and the CHSH inequality</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Two qubits, each individually prepared in the unbiased state \lvert+\rangle = (\lvert0\rangle+\lvert1\rangle)/\sqrt2 -- an unentangled product state -- are coupled by an Ising interaction \hat H = \hbar J\,\sigma_z^{(1)}\otimes\sigma_z^{(2)} and become genuinely entangled purely through that interaction&#x27;s time evolution, reaching a maximally entangled (Bell-equivalent) state at Jt=\pi/4. Unlike the static Bell/EPR correlations of /api/gallery/quantum/entanglement/plot_entanglement_and_topology (which start from an already-entangled state), this traces entanglement&#x27;s actual dynamical genesis.">

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  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_ising_entanglement_growth_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_ising_entanglement_growth`

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      <div class="sphx-glr-thumbnail-title">Dynamical entanglement generation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Wootters, Zurek, and (independently) Dieks proved that no unitary process can take an arbitrary, unknown quantum state and produce two independent copies of it. The proof is a direct consequence of linearity: a unitary that faithfully clones two particular states must, by linearity, act on their superposition in a way that is not a faithful copy of that superposition. This example makes the failure completely explicit with the simplest possible candidate &quot;cloning machine,&quot; the CNOT gate: it successfully copies the computational basis states \lvert0\rangle and \lvert1\rangle, but applied to a superposition it produces exactly a Bell state -- physicskit.quantum.chapters.entanglement.bell_state -- rather than two independent copies, entangling the two qubits instead of cloning either one.">

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  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_no_cloning_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_no_cloning_theorem`

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      <div class="sphx-glr-thumbnail-title">The no-cloning theorem: why a fixed unitary can copy a basis, not a state</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/quantum/entanglement/plot_aharonov_bohm_ring
   /api/gallery/quantum/entanglement/plot_bloch_sphere_spin_dynamics
   /api/gallery/quantum/entanglement/plot_entanglement_and_topology
   /api/gallery/quantum/entanglement/plot_ising_entanglement_growth
   /api/gallery/quantum/entanglement/plot_no_cloning_theorem