Can the Bloch sphere be generalized to two qubits?











up vote
10
down vote

favorite












The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy to visualize.



What attempts have been made to extend visualizations based on the Bloch sphere to two qubits?










share|improve this question




















  • 4




    related on physics.SE: physics.stackexchange.com/q/41223/58382
    – glS
    Nov 13 at 10:48















up vote
10
down vote

favorite












The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy to visualize.



What attempts have been made to extend visualizations based on the Bloch sphere to two qubits?










share|improve this question




















  • 4




    related on physics.SE: physics.stackexchange.com/q/41223/58382
    – glS
    Nov 13 at 10:48













up vote
10
down vote

favorite









up vote
10
down vote

favorite











The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy to visualize.



What attempts have been made to extend visualizations based on the Bloch sphere to two qubits?










share|improve this question















The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy to visualize.



What attempts have been made to extend visualizations based on the Bloch sphere to two qubits?







resource-request bloch-sphere






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited Nov 14 at 7:57









Peter Mortensen

1053




1053










asked Nov 13 at 10:05









James Wootton

5,7501941




5,7501941








  • 4




    related on physics.SE: physics.stackexchange.com/q/41223/58382
    – glS
    Nov 13 at 10:48














  • 4




    related on physics.SE: physics.stackexchange.com/q/41223/58382
    – glS
    Nov 13 at 10:48








4




4




related on physics.SE: physics.stackexchange.com/q/41223/58382
– glS
Nov 13 at 10:48




related on physics.SE: physics.stackexchange.com/q/41223/58382
– glS
Nov 13 at 10:48










5 Answers
5






active

oldest

votes

















up vote
8
down vote













For pure states, there is a reasonably simple way to make a "2 qubit bloch sphere". You basically use the Schmidt decomposition to divide your state into two cases: not entangled and fully entangled. For the not-entangled part, you just use two bloch spheres. And then the entangled part is isomorphic to the set of possible rotations in 3d space (the rotation is how you translate measurements on one qubit into predictions on the other qubit). This gives you a representation with eight real parameters:



1) A real value w between 0 and 1 indicating the weight of not-entangled vs fully-entangled.



2+3) The not-entangled unit bloch vector for qubit 1.



4+5) The not-entangled unit bloch vector for qubit 2.



6+7+8) The fully-entangled rotation.



Here's what it looks like if you show the rotation part as "where X Y and Z axes get mapped", and additionally scale the axes by w so that it gets bigger the more entangled you are:



entangled view



(The bouncing in the middle one is due to a numerical degeneracy in my code.)



For mixed states, I've had a bit of success showing the envelope of bloch vectors predicted for qubit 2 given every possible measurement of qubit 1. That looks like this:



mixed state envelope



But note that a) this 'envelope' representation is not symmetric (one of the qubits is the control and the other is the target) and b) although it looks pretty it's not algebraically compact.



This display is available in the alternate dev-entanglement-display branch of Quirk. If you're able to follow the build instructions, then you can play with it directly.






share|improve this answer




























    up vote
    7
    down vote













    Since a spin $j$ irreducible representation of $SU(2)$ has a dimension $2j+1$ ($j$ is half integer), any finite dimensional Hilbert space can be obtained as a representation space of $SU(2)$. Moreover, since all irreducible representations of $SU(2)$ are symmetric tensor products of the fundamental spinor representation, therefore every finite dimensional Hilbert space can be thought of as a symmetric tensor product of fundamental $SU(2)$ fundamental representation spaces.



    This is the basis of the Majorana stellar representation construction. A state of a qudit living in a Hilbert space of dimension $2j+1$ can be represented by $2j$ points on the Bloch sphere. The state vector can be reconstructed from the $2j$ (2-dimensional) spin vectors of the $2j$ points by a symmetrized tensor product.



    Given a state vector in a $2j+1$ dimensional Hilbert space (Please see Liu, Fu and Wang, section 2.1)
    $$|psirangle = sum_{m=-j}^{j} C_m |j, mrangle, $$
    The locations of the corresponding points (the Majorana stars) on the Bloch sphere are given by the roots of the equation:
    $$sum_{k=0}^{2j} frac{(-1)^k C_{j-k}}{(2j-k)! k!} z^{2j-k}=0.$$



    (The parametrization is by means of the stereographic projection coordinate $ z = tan theta e^{iphi}$ ($theta$, $phi$ are the spherical coordinates))



    One application of this representation to quantum computation, is in the visualization of the trajectories giving rise to geometric phases, which serve as the gates in holonomic quantum computation. These trajectories are reflected as trajectories of the Majorana stars on the Bloch spheres and the geometric phases can be computed from the solid angles enclosed by these trajectories. Please see Liu and Fu's work on Abelian geometric phases. A treatment of some non-Abelian cases is given by Liu Roy and Stone.



    Finally, let me remark that there are many geometric representations relevant to quantum computation, but they are multidimensional and may be not useful in general as visualization tools. Please see for example Bernatska and Holod treating coadjoint orbits which can serve as phase spaces of the finite dimensional Hilbert spaces used in quantum computation. The Grassmannian which parametrizes the ground state manifold of adiabatic quantum Hamiltonians is a particular example of these spaces.






    share|improve this answer























    • I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate?
      – Phil H
      Nov 15 at 8:42










    • In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7,
      – David Bar Moshe
      Nov 15 at 15:09










    • cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place.
      – David Bar Moshe
      Nov 15 at 15:09




















    up vote
    4
    down vote













    For more than 1-qubit visualization, we will need more complex visualizations than a Bloch sphere. The below answer from Physics Stack Exchange explains this concept quite authoritatively:



    Bloch sphere for 2 and more qubits



    In another article, the two qubit representation is described as a seven-dimensional sphere, S 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive.



    Geometry of entangled states, Bloch spheres and Hopf fibrations



    Having said that, a Bloch sphere based approach is quite useful even to model the behavior of qubits in a noisy environment. There has been analysis of the two-qubit system by use of the generalized Bloch vector to generate tractable analytic equations for the dynamics of the four-level Bloch vectors. This is based on the application of geometrical concepts from the well-known two-level Bloch sphere.



    We can find that in the presence of correlated or anti-correlated noise, the rate of decoherence is very sensitive to the initial two-qubit state, as well as to the symmetry of the Hamiltonian. In the absence of symmetry in the Hamiltonian, correlations only weakly impact the decoherence rate:



    Bloch-sphere approach to correlated noise in coupled qubits



    There is another interesting research article on the representation of the two-qubit pure state parameterized by three unit 2-spheres and a phase factor.For separable states, two of the three unit spheres are the Bloch spheres of each qubit with coordinates (A,A) and (B,B). The third sphere parameterises the degree and phase of concurrence, an entanglement measure.



    This sphere may be considered a ‘variable’ complex imaginary unit t where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states.



    As per this hypothesis, the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuthal angles, A and B, of the two quasi-Bloch spheres.



    Bloch sphere model for two






    share|improve this answer



















    • 3




      Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...)
      – Niel de Beaudrap
      Nov 13 at 23:44










    • Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"?
      – Peter Mortensen
      Nov 14 at 6:57




















    up vote
    3
    down vote













    We have some multiqubit visualizations within Q-CTRL's Black Opal package.



    These are all fully interactive and are designed to help build intuition about correlations in interacting two-qubit systems.



    The two Bloch spheres represent the relevant separable states of two qubits. The tetrahedra in the middle visually capture correlations between certain projections of the two qubits. When there is no entanglement, the Bloch vectors live entirely on the surfaces of the respective spheres. However, a fully entangled state lives exclusively in the space of correlations in this representation. The extrema of these spaces will always be maximally entangled states like Bell states, but maximally entangled states can also reside within multiple tetrahedra simultaneously.



    enter image description here






    share|improve this answer










    New contributor




    Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.














    • 1




      Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer.
      – Niel de Beaudrap
      Nov 13 at 23:39










    • edited to add further material.
      – Michael Biercuk
      Nov 14 at 0:39










    • Thanks @MichaelBiercuk, and good to see you here.
      – James Wootton
      Nov 14 at 6:17


















    up vote
    1
    down vote













    A paper has been published on the subject, called "Bloch sphere model for two-qubit pure states"



    https://arxiv.org/abs/1403.8069






    share|improve this answer



















    • 4




      Hi Adrien! Welcome to Quantum Computing SE. We discourage link-only answers on this site. Please add a paragraph to summarize the paper. Thank you.
      – Blue
      Nov 13 at 13:07













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "694"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    convertImagesToLinks: false,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: null,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














     

    draft saved


    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fquantumcomputing.stackexchange.com%2fquestions%2f4699%2fcan-the-bloch-sphere-be-generalized-to-two-qubits%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    5 Answers
    5






    active

    oldest

    votes








    5 Answers
    5






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    8
    down vote













    For pure states, there is a reasonably simple way to make a "2 qubit bloch sphere". You basically use the Schmidt decomposition to divide your state into two cases: not entangled and fully entangled. For the not-entangled part, you just use two bloch spheres. And then the entangled part is isomorphic to the set of possible rotations in 3d space (the rotation is how you translate measurements on one qubit into predictions on the other qubit). This gives you a representation with eight real parameters:



    1) A real value w between 0 and 1 indicating the weight of not-entangled vs fully-entangled.



    2+3) The not-entangled unit bloch vector for qubit 1.



    4+5) The not-entangled unit bloch vector for qubit 2.



    6+7+8) The fully-entangled rotation.



    Here's what it looks like if you show the rotation part as "where X Y and Z axes get mapped", and additionally scale the axes by w so that it gets bigger the more entangled you are:



    entangled view



    (The bouncing in the middle one is due to a numerical degeneracy in my code.)



    For mixed states, I've had a bit of success showing the envelope of bloch vectors predicted for qubit 2 given every possible measurement of qubit 1. That looks like this:



    mixed state envelope



    But note that a) this 'envelope' representation is not symmetric (one of the qubits is the control and the other is the target) and b) although it looks pretty it's not algebraically compact.



    This display is available in the alternate dev-entanglement-display branch of Quirk. If you're able to follow the build instructions, then you can play with it directly.






    share|improve this answer

























      up vote
      8
      down vote













      For pure states, there is a reasonably simple way to make a "2 qubit bloch sphere". You basically use the Schmidt decomposition to divide your state into two cases: not entangled and fully entangled. For the not-entangled part, you just use two bloch spheres. And then the entangled part is isomorphic to the set of possible rotations in 3d space (the rotation is how you translate measurements on one qubit into predictions on the other qubit). This gives you a representation with eight real parameters:



      1) A real value w between 0 and 1 indicating the weight of not-entangled vs fully-entangled.



      2+3) The not-entangled unit bloch vector for qubit 1.



      4+5) The not-entangled unit bloch vector for qubit 2.



      6+7+8) The fully-entangled rotation.



      Here's what it looks like if you show the rotation part as "where X Y and Z axes get mapped", and additionally scale the axes by w so that it gets bigger the more entangled you are:



      entangled view



      (The bouncing in the middle one is due to a numerical degeneracy in my code.)



      For mixed states, I've had a bit of success showing the envelope of bloch vectors predicted for qubit 2 given every possible measurement of qubit 1. That looks like this:



      mixed state envelope



      But note that a) this 'envelope' representation is not symmetric (one of the qubits is the control and the other is the target) and b) although it looks pretty it's not algebraically compact.



      This display is available in the alternate dev-entanglement-display branch of Quirk. If you're able to follow the build instructions, then you can play with it directly.






      share|improve this answer























        up vote
        8
        down vote










        up vote
        8
        down vote









        For pure states, there is a reasonably simple way to make a "2 qubit bloch sphere". You basically use the Schmidt decomposition to divide your state into two cases: not entangled and fully entangled. For the not-entangled part, you just use two bloch spheres. And then the entangled part is isomorphic to the set of possible rotations in 3d space (the rotation is how you translate measurements on one qubit into predictions on the other qubit). This gives you a representation with eight real parameters:



        1) A real value w between 0 and 1 indicating the weight of not-entangled vs fully-entangled.



        2+3) The not-entangled unit bloch vector for qubit 1.



        4+5) The not-entangled unit bloch vector for qubit 2.



        6+7+8) The fully-entangled rotation.



        Here's what it looks like if you show the rotation part as "where X Y and Z axes get mapped", and additionally scale the axes by w so that it gets bigger the more entangled you are:



        entangled view



        (The bouncing in the middle one is due to a numerical degeneracy in my code.)



        For mixed states, I've had a bit of success showing the envelope of bloch vectors predicted for qubit 2 given every possible measurement of qubit 1. That looks like this:



        mixed state envelope



        But note that a) this 'envelope' representation is not symmetric (one of the qubits is the control and the other is the target) and b) although it looks pretty it's not algebraically compact.



        This display is available in the alternate dev-entanglement-display branch of Quirk. If you're able to follow the build instructions, then you can play with it directly.






        share|improve this answer












        For pure states, there is a reasonably simple way to make a "2 qubit bloch sphere". You basically use the Schmidt decomposition to divide your state into two cases: not entangled and fully entangled. For the not-entangled part, you just use two bloch spheres. And then the entangled part is isomorphic to the set of possible rotations in 3d space (the rotation is how you translate measurements on one qubit into predictions on the other qubit). This gives you a representation with eight real parameters:



        1) A real value w between 0 and 1 indicating the weight of not-entangled vs fully-entangled.



        2+3) The not-entangled unit bloch vector for qubit 1.



        4+5) The not-entangled unit bloch vector for qubit 2.



        6+7+8) The fully-entangled rotation.



        Here's what it looks like if you show the rotation part as "where X Y and Z axes get mapped", and additionally scale the axes by w so that it gets bigger the more entangled you are:



        entangled view



        (The bouncing in the middle one is due to a numerical degeneracy in my code.)



        For mixed states, I've had a bit of success showing the envelope of bloch vectors predicted for qubit 2 given every possible measurement of qubit 1. That looks like this:



        mixed state envelope



        But note that a) this 'envelope' representation is not symmetric (one of the qubits is the control and the other is the target) and b) although it looks pretty it's not algebraically compact.



        This display is available in the alternate dev-entanglement-display branch of Quirk. If you're able to follow the build instructions, then you can play with it directly.







        share|improve this answer












        share|improve this answer



        share|improve this answer










        answered Nov 13 at 23:42









        Craig Gidney

        3,107119




        3,107119
























            up vote
            7
            down vote













            Since a spin $j$ irreducible representation of $SU(2)$ has a dimension $2j+1$ ($j$ is half integer), any finite dimensional Hilbert space can be obtained as a representation space of $SU(2)$. Moreover, since all irreducible representations of $SU(2)$ are symmetric tensor products of the fundamental spinor representation, therefore every finite dimensional Hilbert space can be thought of as a symmetric tensor product of fundamental $SU(2)$ fundamental representation spaces.



            This is the basis of the Majorana stellar representation construction. A state of a qudit living in a Hilbert space of dimension $2j+1$ can be represented by $2j$ points on the Bloch sphere. The state vector can be reconstructed from the $2j$ (2-dimensional) spin vectors of the $2j$ points by a symmetrized tensor product.



            Given a state vector in a $2j+1$ dimensional Hilbert space (Please see Liu, Fu and Wang, section 2.1)
            $$|psirangle = sum_{m=-j}^{j} C_m |j, mrangle, $$
            The locations of the corresponding points (the Majorana stars) on the Bloch sphere are given by the roots of the equation:
            $$sum_{k=0}^{2j} frac{(-1)^k C_{j-k}}{(2j-k)! k!} z^{2j-k}=0.$$



            (The parametrization is by means of the stereographic projection coordinate $ z = tan theta e^{iphi}$ ($theta$, $phi$ are the spherical coordinates))



            One application of this representation to quantum computation, is in the visualization of the trajectories giving rise to geometric phases, which serve as the gates in holonomic quantum computation. These trajectories are reflected as trajectories of the Majorana stars on the Bloch spheres and the geometric phases can be computed from the solid angles enclosed by these trajectories. Please see Liu and Fu's work on Abelian geometric phases. A treatment of some non-Abelian cases is given by Liu Roy and Stone.



            Finally, let me remark that there are many geometric representations relevant to quantum computation, but they are multidimensional and may be not useful in general as visualization tools. Please see for example Bernatska and Holod treating coadjoint orbits which can serve as phase spaces of the finite dimensional Hilbert spaces used in quantum computation. The Grassmannian which parametrizes the ground state manifold of adiabatic quantum Hamiltonians is a particular example of these spaces.






            share|improve this answer























            • I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate?
              – Phil H
              Nov 15 at 8:42










            • In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7,
              – David Bar Moshe
              Nov 15 at 15:09










            • cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place.
              – David Bar Moshe
              Nov 15 at 15:09

















            up vote
            7
            down vote













            Since a spin $j$ irreducible representation of $SU(2)$ has a dimension $2j+1$ ($j$ is half integer), any finite dimensional Hilbert space can be obtained as a representation space of $SU(2)$. Moreover, since all irreducible representations of $SU(2)$ are symmetric tensor products of the fundamental spinor representation, therefore every finite dimensional Hilbert space can be thought of as a symmetric tensor product of fundamental $SU(2)$ fundamental representation spaces.



            This is the basis of the Majorana stellar representation construction. A state of a qudit living in a Hilbert space of dimension $2j+1$ can be represented by $2j$ points on the Bloch sphere. The state vector can be reconstructed from the $2j$ (2-dimensional) spin vectors of the $2j$ points by a symmetrized tensor product.



            Given a state vector in a $2j+1$ dimensional Hilbert space (Please see Liu, Fu and Wang, section 2.1)
            $$|psirangle = sum_{m=-j}^{j} C_m |j, mrangle, $$
            The locations of the corresponding points (the Majorana stars) on the Bloch sphere are given by the roots of the equation:
            $$sum_{k=0}^{2j} frac{(-1)^k C_{j-k}}{(2j-k)! k!} z^{2j-k}=0.$$



            (The parametrization is by means of the stereographic projection coordinate $ z = tan theta e^{iphi}$ ($theta$, $phi$ are the spherical coordinates))



            One application of this representation to quantum computation, is in the visualization of the trajectories giving rise to geometric phases, which serve as the gates in holonomic quantum computation. These trajectories are reflected as trajectories of the Majorana stars on the Bloch spheres and the geometric phases can be computed from the solid angles enclosed by these trajectories. Please see Liu and Fu's work on Abelian geometric phases. A treatment of some non-Abelian cases is given by Liu Roy and Stone.



            Finally, let me remark that there are many geometric representations relevant to quantum computation, but they are multidimensional and may be not useful in general as visualization tools. Please see for example Bernatska and Holod treating coadjoint orbits which can serve as phase spaces of the finite dimensional Hilbert spaces used in quantum computation. The Grassmannian which parametrizes the ground state manifold of adiabatic quantum Hamiltonians is a particular example of these spaces.






            share|improve this answer























            • I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate?
              – Phil H
              Nov 15 at 8:42










            • In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7,
              – David Bar Moshe
              Nov 15 at 15:09










            • cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place.
              – David Bar Moshe
              Nov 15 at 15:09















            up vote
            7
            down vote










            up vote
            7
            down vote









            Since a spin $j$ irreducible representation of $SU(2)$ has a dimension $2j+1$ ($j$ is half integer), any finite dimensional Hilbert space can be obtained as a representation space of $SU(2)$. Moreover, since all irreducible representations of $SU(2)$ are symmetric tensor products of the fundamental spinor representation, therefore every finite dimensional Hilbert space can be thought of as a symmetric tensor product of fundamental $SU(2)$ fundamental representation spaces.



            This is the basis of the Majorana stellar representation construction. A state of a qudit living in a Hilbert space of dimension $2j+1$ can be represented by $2j$ points on the Bloch sphere. The state vector can be reconstructed from the $2j$ (2-dimensional) spin vectors of the $2j$ points by a symmetrized tensor product.



            Given a state vector in a $2j+1$ dimensional Hilbert space (Please see Liu, Fu and Wang, section 2.1)
            $$|psirangle = sum_{m=-j}^{j} C_m |j, mrangle, $$
            The locations of the corresponding points (the Majorana stars) on the Bloch sphere are given by the roots of the equation:
            $$sum_{k=0}^{2j} frac{(-1)^k C_{j-k}}{(2j-k)! k!} z^{2j-k}=0.$$



            (The parametrization is by means of the stereographic projection coordinate $ z = tan theta e^{iphi}$ ($theta$, $phi$ are the spherical coordinates))



            One application of this representation to quantum computation, is in the visualization of the trajectories giving rise to geometric phases, which serve as the gates in holonomic quantum computation. These trajectories are reflected as trajectories of the Majorana stars on the Bloch spheres and the geometric phases can be computed from the solid angles enclosed by these trajectories. Please see Liu and Fu's work on Abelian geometric phases. A treatment of some non-Abelian cases is given by Liu Roy and Stone.



            Finally, let me remark that there are many geometric representations relevant to quantum computation, but they are multidimensional and may be not useful in general as visualization tools. Please see for example Bernatska and Holod treating coadjoint orbits which can serve as phase spaces of the finite dimensional Hilbert spaces used in quantum computation. The Grassmannian which parametrizes the ground state manifold of adiabatic quantum Hamiltonians is a particular example of these spaces.






            share|improve this answer














            Since a spin $j$ irreducible representation of $SU(2)$ has a dimension $2j+1$ ($j$ is half integer), any finite dimensional Hilbert space can be obtained as a representation space of $SU(2)$. Moreover, since all irreducible representations of $SU(2)$ are symmetric tensor products of the fundamental spinor representation, therefore every finite dimensional Hilbert space can be thought of as a symmetric tensor product of fundamental $SU(2)$ fundamental representation spaces.



            This is the basis of the Majorana stellar representation construction. A state of a qudit living in a Hilbert space of dimension $2j+1$ can be represented by $2j$ points on the Bloch sphere. The state vector can be reconstructed from the $2j$ (2-dimensional) spin vectors of the $2j$ points by a symmetrized tensor product.



            Given a state vector in a $2j+1$ dimensional Hilbert space (Please see Liu, Fu and Wang, section 2.1)
            $$|psirangle = sum_{m=-j}^{j} C_m |j, mrangle, $$
            The locations of the corresponding points (the Majorana stars) on the Bloch sphere are given by the roots of the equation:
            $$sum_{k=0}^{2j} frac{(-1)^k C_{j-k}}{(2j-k)! k!} z^{2j-k}=0.$$



            (The parametrization is by means of the stereographic projection coordinate $ z = tan theta e^{iphi}$ ($theta$, $phi$ are the spherical coordinates))



            One application of this representation to quantum computation, is in the visualization of the trajectories giving rise to geometric phases, which serve as the gates in holonomic quantum computation. These trajectories are reflected as trajectories of the Majorana stars on the Bloch spheres and the geometric phases can be computed from the solid angles enclosed by these trajectories. Please see Liu and Fu's work on Abelian geometric phases. A treatment of some non-Abelian cases is given by Liu Roy and Stone.



            Finally, let me remark that there are many geometric representations relevant to quantum computation, but they are multidimensional and may be not useful in general as visualization tools. Please see for example Bernatska and Holod treating coadjoint orbits which can serve as phase spaces of the finite dimensional Hilbert spaces used in quantum computation. The Grassmannian which parametrizes the ground state manifold of adiabatic quantum Hamiltonians is a particular example of these spaces.







            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited Nov 14 at 9:44

























            answered Nov 13 at 12:42









            David Bar Moshe

            6445




            6445












            • I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate?
              – Phil H
              Nov 15 at 8:42










            • In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7,
              – David Bar Moshe
              Nov 15 at 15:09










            • cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place.
              – David Bar Moshe
              Nov 15 at 15:09




















            • I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate?
              – Phil H
              Nov 15 at 8:42










            • In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7,
              – David Bar Moshe
              Nov 15 at 15:09










            • cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place.
              – David Bar Moshe
              Nov 15 at 15:09


















            I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate?
            – Phil H
            Nov 15 at 8:42




            I know they are time consuming to find or make, but is there any chance you could illustrate this answer with such visualisations? Perhaps an example of a CNOT gate?
            – Phil H
            Nov 15 at 8:42












            In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7,
            – David Bar Moshe
            Nov 15 at 15:09




            In general, a unitary transformation of a state will move its constellation to new locations such that the coordinate of a star in the final state depends algebraically on all the coordinates of all stars in the initial state. However, in simple cases, we can perform the computation by a simple inspection. Please see for example Bengtsson and Życzkowski: researchgate.net/profile/Karol_Zyczkowski/publication/… page 103, figure 4.7,
            – David Bar Moshe
            Nov 15 at 15:09












            cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place.
            – David Bar Moshe
            Nov 15 at 15:09






            cont. where for example, the CNOT gate action on a state with three stars at the north pole shifts one of the stars to the south pole while keeping the other two stars in place.
            – David Bar Moshe
            Nov 15 at 15:09












            up vote
            4
            down vote













            For more than 1-qubit visualization, we will need more complex visualizations than a Bloch sphere. The below answer from Physics Stack Exchange explains this concept quite authoritatively:



            Bloch sphere for 2 and more qubits



            In another article, the two qubit representation is described as a seven-dimensional sphere, S 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive.



            Geometry of entangled states, Bloch spheres and Hopf fibrations



            Having said that, a Bloch sphere based approach is quite useful even to model the behavior of qubits in a noisy environment. There has been analysis of the two-qubit system by use of the generalized Bloch vector to generate tractable analytic equations for the dynamics of the four-level Bloch vectors. This is based on the application of geometrical concepts from the well-known two-level Bloch sphere.



            We can find that in the presence of correlated or anti-correlated noise, the rate of decoherence is very sensitive to the initial two-qubit state, as well as to the symmetry of the Hamiltonian. In the absence of symmetry in the Hamiltonian, correlations only weakly impact the decoherence rate:



            Bloch-sphere approach to correlated noise in coupled qubits



            There is another interesting research article on the representation of the two-qubit pure state parameterized by three unit 2-spheres and a phase factor.For separable states, two of the three unit spheres are the Bloch spheres of each qubit with coordinates (A,A) and (B,B). The third sphere parameterises the degree and phase of concurrence, an entanglement measure.



            This sphere may be considered a ‘variable’ complex imaginary unit t where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states.



            As per this hypothesis, the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuthal angles, A and B, of the two quasi-Bloch spheres.



            Bloch sphere model for two






            share|improve this answer



















            • 3




              Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...)
              – Niel de Beaudrap
              Nov 13 at 23:44










            • Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"?
              – Peter Mortensen
              Nov 14 at 6:57

















            up vote
            4
            down vote













            For more than 1-qubit visualization, we will need more complex visualizations than a Bloch sphere. The below answer from Physics Stack Exchange explains this concept quite authoritatively:



            Bloch sphere for 2 and more qubits



            In another article, the two qubit representation is described as a seven-dimensional sphere, S 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive.



            Geometry of entangled states, Bloch spheres and Hopf fibrations



            Having said that, a Bloch sphere based approach is quite useful even to model the behavior of qubits in a noisy environment. There has been analysis of the two-qubit system by use of the generalized Bloch vector to generate tractable analytic equations for the dynamics of the four-level Bloch vectors. This is based on the application of geometrical concepts from the well-known two-level Bloch sphere.



            We can find that in the presence of correlated or anti-correlated noise, the rate of decoherence is very sensitive to the initial two-qubit state, as well as to the symmetry of the Hamiltonian. In the absence of symmetry in the Hamiltonian, correlations only weakly impact the decoherence rate:



            Bloch-sphere approach to correlated noise in coupled qubits



            There is another interesting research article on the representation of the two-qubit pure state parameterized by three unit 2-spheres and a phase factor.For separable states, two of the three unit spheres are the Bloch spheres of each qubit with coordinates (A,A) and (B,B). The third sphere parameterises the degree and phase of concurrence, an entanglement measure.



            This sphere may be considered a ‘variable’ complex imaginary unit t where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states.



            As per this hypothesis, the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuthal angles, A and B, of the two quasi-Bloch spheres.



            Bloch sphere model for two






            share|improve this answer



















            • 3




              Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...)
              – Niel de Beaudrap
              Nov 13 at 23:44










            • Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"?
              – Peter Mortensen
              Nov 14 at 6:57















            up vote
            4
            down vote










            up vote
            4
            down vote









            For more than 1-qubit visualization, we will need more complex visualizations than a Bloch sphere. The below answer from Physics Stack Exchange explains this concept quite authoritatively:



            Bloch sphere for 2 and more qubits



            In another article, the two qubit representation is described as a seven-dimensional sphere, S 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive.



            Geometry of entangled states, Bloch spheres and Hopf fibrations



            Having said that, a Bloch sphere based approach is quite useful even to model the behavior of qubits in a noisy environment. There has been analysis of the two-qubit system by use of the generalized Bloch vector to generate tractable analytic equations for the dynamics of the four-level Bloch vectors. This is based on the application of geometrical concepts from the well-known two-level Bloch sphere.



            We can find that in the presence of correlated or anti-correlated noise, the rate of decoherence is very sensitive to the initial two-qubit state, as well as to the symmetry of the Hamiltonian. In the absence of symmetry in the Hamiltonian, correlations only weakly impact the decoherence rate:



            Bloch-sphere approach to correlated noise in coupled qubits



            There is another interesting research article on the representation of the two-qubit pure state parameterized by three unit 2-spheres and a phase factor.For separable states, two of the three unit spheres are the Bloch spheres of each qubit with coordinates (A,A) and (B,B). The third sphere parameterises the degree and phase of concurrence, an entanglement measure.



            This sphere may be considered a ‘variable’ complex imaginary unit t where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states.



            As per this hypothesis, the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuthal angles, A and B, of the two quasi-Bloch spheres.



            Bloch sphere model for two






            share|improve this answer














            For more than 1-qubit visualization, we will need more complex visualizations than a Bloch sphere. The below answer from Physics Stack Exchange explains this concept quite authoritatively:



            Bloch sphere for 2 and more qubits



            In another article, the two qubit representation is described as a seven-dimensional sphere, S 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive.



            Geometry of entangled states, Bloch spheres and Hopf fibrations



            Having said that, a Bloch sphere based approach is quite useful even to model the behavior of qubits in a noisy environment. There has been analysis of the two-qubit system by use of the generalized Bloch vector to generate tractable analytic equations for the dynamics of the four-level Bloch vectors. This is based on the application of geometrical concepts from the well-known two-level Bloch sphere.



            We can find that in the presence of correlated or anti-correlated noise, the rate of decoherence is very sensitive to the initial two-qubit state, as well as to the symmetry of the Hamiltonian. In the absence of symmetry in the Hamiltonian, correlations only weakly impact the decoherence rate:



            Bloch-sphere approach to correlated noise in coupled qubits



            There is another interesting research article on the representation of the two-qubit pure state parameterized by three unit 2-spheres and a phase factor.For separable states, two of the three unit spheres are the Bloch spheres of each qubit with coordinates (A,A) and (B,B). The third sphere parameterises the degree and phase of concurrence, an entanglement measure.



            This sphere may be considered a ‘variable’ complex imaginary unit t where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states.



            As per this hypothesis, the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuthal angles, A and B, of the two quasi-Bloch spheres.



            Bloch sphere model for two







            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited Nov 14 at 7:44









            Peter Mortensen

            1053




            1053










            answered Nov 13 at 10:51









            Gokul B. Alex

            1093




            1093








            • 3




              Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...)
              – Niel de Beaudrap
              Nov 13 at 23:44










            • Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"?
              – Peter Mortensen
              Nov 14 at 6:57
















            • 3




              Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...)
              – Niel de Beaudrap
              Nov 13 at 23:44










            • Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"?
              – Peter Mortensen
              Nov 14 at 6:57










            3




            3




            Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...)
            – Niel de Beaudrap
            Nov 13 at 23:44




            Would it be possible to expand a bit on these remarks? Rather than linking to these articles, it would be good to describe the relevant ideas in some detail to keep the answer self-contained. (Also, in your third answer in this post, the symbols are not rendering properly...)
            – Niel de Beaudrap
            Nov 13 at 23:44












            Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"?
            – Peter Mortensen
            Nov 14 at 6:57






            Near "the azimuthal angles": What is it before "A" and "B"? Firefox shows it as "F066". Also near "qubit with coordinates", before A and B (four in total), two of them "F071"?
            – Peter Mortensen
            Nov 14 at 6:57












            up vote
            3
            down vote













            We have some multiqubit visualizations within Q-CTRL's Black Opal package.



            These are all fully interactive and are designed to help build intuition about correlations in interacting two-qubit systems.



            The two Bloch spheres represent the relevant separable states of two qubits. The tetrahedra in the middle visually capture correlations between certain projections of the two qubits. When there is no entanglement, the Bloch vectors live entirely on the surfaces of the respective spheres. However, a fully entangled state lives exclusively in the space of correlations in this representation. The extrema of these spaces will always be maximally entangled states like Bell states, but maximally entangled states can also reside within multiple tetrahedra simultaneously.



            enter image description here






            share|improve this answer










            New contributor




            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.














            • 1




              Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer.
              – Niel de Beaudrap
              Nov 13 at 23:39










            • edited to add further material.
              – Michael Biercuk
              Nov 14 at 0:39










            • Thanks @MichaelBiercuk, and good to see you here.
              – James Wootton
              Nov 14 at 6:17















            up vote
            3
            down vote













            We have some multiqubit visualizations within Q-CTRL's Black Opal package.



            These are all fully interactive and are designed to help build intuition about correlations in interacting two-qubit systems.



            The two Bloch spheres represent the relevant separable states of two qubits. The tetrahedra in the middle visually capture correlations between certain projections of the two qubits. When there is no entanglement, the Bloch vectors live entirely on the surfaces of the respective spheres. However, a fully entangled state lives exclusively in the space of correlations in this representation. The extrema of these spaces will always be maximally entangled states like Bell states, but maximally entangled states can also reside within multiple tetrahedra simultaneously.



            enter image description here






            share|improve this answer










            New contributor




            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.














            • 1




              Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer.
              – Niel de Beaudrap
              Nov 13 at 23:39










            • edited to add further material.
              – Michael Biercuk
              Nov 14 at 0:39










            • Thanks @MichaelBiercuk, and good to see you here.
              – James Wootton
              Nov 14 at 6:17













            up vote
            3
            down vote










            up vote
            3
            down vote









            We have some multiqubit visualizations within Q-CTRL's Black Opal package.



            These are all fully interactive and are designed to help build intuition about correlations in interacting two-qubit systems.



            The two Bloch spheres represent the relevant separable states of two qubits. The tetrahedra in the middle visually capture correlations between certain projections of the two qubits. When there is no entanglement, the Bloch vectors live entirely on the surfaces of the respective spheres. However, a fully entangled state lives exclusively in the space of correlations in this representation. The extrema of these spaces will always be maximally entangled states like Bell states, but maximally entangled states can also reside within multiple tetrahedra simultaneously.



            enter image description here






            share|improve this answer










            New contributor




            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            We have some multiqubit visualizations within Q-CTRL's Black Opal package.



            These are all fully interactive and are designed to help build intuition about correlations in interacting two-qubit systems.



            The two Bloch spheres represent the relevant separable states of two qubits. The tetrahedra in the middle visually capture correlations between certain projections of the two qubits. When there is no entanglement, the Bloch vectors live entirely on the surfaces of the respective spheres. However, a fully entangled state lives exclusively in the space of correlations in this representation. The extrema of these spaces will always be maximally entangled states like Bell states, but maximally entangled states can also reside within multiple tetrahedra simultaneously.



            enter image description here







            share|improve this answer










            New contributor




            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            share|improve this answer



            share|improve this answer








            edited Nov 14 at 3:54









            Blue

            5,60011249




            5,60011249






            New contributor




            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            answered Nov 13 at 22:53









            Michael Biercuk

            412




            412




            New contributor




            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.





            New contributor





            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            Michael Biercuk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.








            • 1




              Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer.
              – Niel de Beaudrap
              Nov 13 at 23:39










            • edited to add further material.
              – Michael Biercuk
              Nov 14 at 0:39










            • Thanks @MichaelBiercuk, and good to see you here.
              – James Wootton
              Nov 14 at 6:17














            • 1




              Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer.
              – Niel de Beaudrap
              Nov 13 at 23:39










            • edited to add further material.
              – Michael Biercuk
              Nov 14 at 0:39










            • Thanks @MichaelBiercuk, and good to see you here.
              – James Wootton
              Nov 14 at 6:17








            1




            1




            Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer.
            – Niel de Beaudrap
            Nov 13 at 23:39




            Would you be able to describe these representations? It would be nice if you could expand this into a self-contained answer.
            – Niel de Beaudrap
            Nov 13 at 23:39












            edited to add further material.
            – Michael Biercuk
            Nov 14 at 0:39




            edited to add further material.
            – Michael Biercuk
            Nov 14 at 0:39












            Thanks @MichaelBiercuk, and good to see you here.
            – James Wootton
            Nov 14 at 6:17




            Thanks @MichaelBiercuk, and good to see you here.
            – James Wootton
            Nov 14 at 6:17










            up vote
            1
            down vote













            A paper has been published on the subject, called "Bloch sphere model for two-qubit pure states"



            https://arxiv.org/abs/1403.8069






            share|improve this answer



















            • 4




              Hi Adrien! Welcome to Quantum Computing SE. We discourage link-only answers on this site. Please add a paragraph to summarize the paper. Thank you.
              – Blue
              Nov 13 at 13:07

















            up vote
            1
            down vote













            A paper has been published on the subject, called "Bloch sphere model for two-qubit pure states"



            https://arxiv.org/abs/1403.8069






            share|improve this answer



















            • 4




              Hi Adrien! Welcome to Quantum Computing SE. We discourage link-only answers on this site. Please add a paragraph to summarize the paper. Thank you.
              – Blue
              Nov 13 at 13:07















            up vote
            1
            down vote










            up vote
            1
            down vote









            A paper has been published on the subject, called "Bloch sphere model for two-qubit pure states"



            https://arxiv.org/abs/1403.8069






            share|improve this answer














            A paper has been published on the subject, called "Bloch sphere model for two-qubit pure states"



            https://arxiv.org/abs/1403.8069







            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited Nov 13 at 13:06









            Blue

            5,60011249




            5,60011249










            answered Nov 13 at 10:44









            Adrien Nivaggioli

            516




            516








            • 4




              Hi Adrien! Welcome to Quantum Computing SE. We discourage link-only answers on this site. Please add a paragraph to summarize the paper. Thank you.
              – Blue
              Nov 13 at 13:07
















            • 4




              Hi Adrien! Welcome to Quantum Computing SE. We discourage link-only answers on this site. Please add a paragraph to summarize the paper. Thank you.
              – Blue
              Nov 13 at 13:07










            4




            4




            Hi Adrien! Welcome to Quantum Computing SE. We discourage link-only answers on this site. Please add a paragraph to summarize the paper. Thank you.
            – Blue
            Nov 13 at 13:07






            Hi Adrien! Welcome to Quantum Computing SE. We discourage link-only answers on this site. Please add a paragraph to summarize the paper. Thank you.
            – Blue
            Nov 13 at 13:07




















             

            draft saved


            draft discarded



















































             


            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fquantumcomputing.stackexchange.com%2fquestions%2f4699%2fcan-the-bloch-sphere-be-generalized-to-two-qubits%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Probability when a professor distributes a quiz and homework assignment to a class of n students.

            Aardman Animations

            Are they similar matrix