Meaning of Bloch representationCan the Bloch sphere be generalized to two qubits?What's the difference...

Why does academia still use scientific journals and not peer-reviewed government funded alternatives?

Why do games have consumables?

Does holding a wand and speaking its command word count as V/S/M spell components?

Why do Computer Science majors learn Calculus?

Why was Germany not as successful as other Europeans in establishing overseas colonies?

How to stop co-workers from teasing me because I know Russian?

Is there an official tutorial for installing Ubuntu 18.04+ on a device with an SSD and an additional internal hard drive?

To say I met a person for the first time

Please, smoke with good manners

What language was spoken in East Asia before Proto-Turkic?

What route did the Hindenburg take when traveling from Germany to the U.S.?

How do I deal with a coworker that keeps asking to make small superficial changes to a report, and it is seriously triggering my anxiety?

Apply MapThread to all but one variable

Are Boeing 737-800’s grounded?

Is there any limitation with Arduino Nano serial communication distance?

How come there are so many candidates for the 2020 Democratic party presidential nomination?

The Defining Moment

Use tikz commands in caption

what is the sudo password for a --disabled-password user

Error message with tabularx

Meaning of Bloch representation

Rivers without rain

Does a semiconductor follow Ohm's law?

What does KSP mean?



Meaning of Bloch representation


Can the Bloch sphere be generalized to two qubits?What's the difference between a pure and mixed quantum state?Why do Bloch sphere wavefunctions have half angles?Density matrices for pure states and mixed statesWhy is an entangled qubit shown at the origin of a Bloch sphere?Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch spherePurity of mixed states as a function of radial distance from origin of Bloch ballCompute average value of two-qubit systemWhy do Bloch sphere wavefunctions have half angles?Why is an entangled qubit shown at the origin of a Bloch sphere?Is there a place online where I can catch up with all the notational syntax associated with quantum computing?What is the meaning of the state $|1rangle-|1rangle$?What's a vector in the format of the Bloch Sphere?What utility is provided by the Bloch sphere visualization?Purity of mixed states as a function of radial distance from origin of Bloch ballNotation for two qubit composite product stateWhat does it mean to express a gate in Dirac notation?






.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0;
}







1












$begingroup$


What is the meaning of writing a state $|psirangle$ in its Bloch representation. Would I be correct in saying it's just writing out its Bloch vector?










share|improve this question











$endgroup$








  • 1




    $begingroup$
    The short answer is "yes." Read Sanchayan's answer for a complete understanding :)
    $endgroup$
    – Will
    15 mins ago


















1












$begingroup$


What is the meaning of writing a state $|psirangle$ in its Bloch representation. Would I be correct in saying it's just writing out its Bloch vector?










share|improve this question











$endgroup$








  • 1




    $begingroup$
    The short answer is "yes." Read Sanchayan's answer for a complete understanding :)
    $endgroup$
    – Will
    15 mins ago














1












1








1





$begingroup$


What is the meaning of writing a state $|psirangle$ in its Bloch representation. Would I be correct in saying it's just writing out its Bloch vector?










share|improve this question











$endgroup$




What is the meaning of writing a state $|psirangle$ in its Bloch representation. Would I be correct in saying it's just writing out its Bloch vector?







notation bloch-sphere






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 50 mins ago









Sanchayan Dutta

6,75841556




6,75841556










asked 2 hours ago









can'tcauchycan'tcauchy

2015




2015








  • 1




    $begingroup$
    The short answer is "yes." Read Sanchayan's answer for a complete understanding :)
    $endgroup$
    – Will
    15 mins ago














  • 1




    $begingroup$
    The short answer is "yes." Read Sanchayan's answer for a complete understanding :)
    $endgroup$
    – Will
    15 mins ago








1




1




$begingroup$
The short answer is "yes." Read Sanchayan's answer for a complete understanding :)
$endgroup$
– Will
15 mins ago




$begingroup$
The short answer is "yes." Read Sanchayan's answer for a complete understanding :)
$endgroup$
– Will
15 mins ago










1 Answer
1






active

oldest

votes


















2












$begingroup$

The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|Psirangle$ of a qubit can be written in the form:



$$|Psirangle = cosfrac{theta}{2}|0rangle + e^{iphi}sinfrac{theta}{2}|1rangle$$ where $0leq thetaleq pi$ and $0leq phileq 2pi$. This $|Psirangle$ can be represented on the Bloch sphere as:



Bloch Sphere



The Bloch vector $vec{a}in Bbb R^3$ is basically $(sintheta cosphi, sinthetasinphi, cos theta) = (a_1,a_2,a_3)$.



To represent mixed states you need to consider the corresponding density operator $rho$. the set of states of a single qubit can be described in terms of $2times 2$ density matrices and as ${I,X,Y,Z}$ forms a basis for the vector space of $2times 2$ Hermitian matrices, you can write the density operator as $$rho = a_0I+a_1X+a_2Y+a_3Z = frac{1}{2}begin{pmatrix}1+a_3 & a_1-ia_2 \ a_1+ia_2 & 1-a_3end{pmatrix}.$$ As density matrices always have trace $1$, and here $mathrm{tr}(rho)=2a_0$, so $a_0$ is necessarily $frac{1}{2}$. So from here you can find out the Bloch coordinates of the any mixed state i.e. $(a_1,a_2,a_3)$ after performing the Pauli decompostion of the density matrix. If you're wondering what ensures that $|vec{a}|leq 1$, it's the positive semidefiniteness! The two eigenvalues of $rho$ are $frac{1}{2}(1+|vec{a}|)$ and $frac{1}{2}(1-|vec{a}|)$. Thus, to ensure that the second eigenvalue is non-negative, $|vec{a}|leq 1$. The three properties of density matrices which you should drill into your brain are: self-adjointness, positive-semidefiniteness and unit trace.



Once you determine the values $a_1,a_2$ and $a_3$ from the density operator, you can easily find the location of the qubit state $(sintheta cosphi, sinthetasinphi, cos theta)$ inside the Bloch sphere. Let me emphasize on this point: pure states lie on the Bloch sphere (i.e. $|vec{a}|=1$) whereas mixed states lie inside the Bloch sphere (i.e. $|vec{a}|<1$); prove this as an exercise. If you're mathematically inclined, you'll also love to think about the Bloch sphere in terms of stereographic projections; it's excellently summarized in this Physics SE answer. I'll attach the image from there, which is originally from this blogpost (the article is in French, sorry :).



enter image description here



Here are a few "further readings" for you:




  • Density matrices for pure states and mixed states

  • Why do Bloch sphere wavefunctions have half angles?

  • Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere


  • Purity of mixed states as a function of radial distance from origin of Bloch ball


  • Can the Bloch sphere be generalized to two qubits?


  • Why is an entangled qubit shown at the origin of a Bloch sphere?



Essentially, go through the bloch-sphere tag; you'll find several interesting questions and answers, which should clarify most of your beginner confusions about the Bloch sphere formalism.






share|improve this answer











$endgroup$














    Your Answer








    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',
    autoActivateHeartbeat: false,
    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%2f5993%2fmeaning-of-bloch-representation%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|Psirangle$ of a qubit can be written in the form:



    $$|Psirangle = cosfrac{theta}{2}|0rangle + e^{iphi}sinfrac{theta}{2}|1rangle$$ where $0leq thetaleq pi$ and $0leq phileq 2pi$. This $|Psirangle$ can be represented on the Bloch sphere as:



    Bloch Sphere



    The Bloch vector $vec{a}in Bbb R^3$ is basically $(sintheta cosphi, sinthetasinphi, cos theta) = (a_1,a_2,a_3)$.



    To represent mixed states you need to consider the corresponding density operator $rho$. the set of states of a single qubit can be described in terms of $2times 2$ density matrices and as ${I,X,Y,Z}$ forms a basis for the vector space of $2times 2$ Hermitian matrices, you can write the density operator as $$rho = a_0I+a_1X+a_2Y+a_3Z = frac{1}{2}begin{pmatrix}1+a_3 & a_1-ia_2 \ a_1+ia_2 & 1-a_3end{pmatrix}.$$ As density matrices always have trace $1$, and here $mathrm{tr}(rho)=2a_0$, so $a_0$ is necessarily $frac{1}{2}$. So from here you can find out the Bloch coordinates of the any mixed state i.e. $(a_1,a_2,a_3)$ after performing the Pauli decompostion of the density matrix. If you're wondering what ensures that $|vec{a}|leq 1$, it's the positive semidefiniteness! The two eigenvalues of $rho$ are $frac{1}{2}(1+|vec{a}|)$ and $frac{1}{2}(1-|vec{a}|)$. Thus, to ensure that the second eigenvalue is non-negative, $|vec{a}|leq 1$. The three properties of density matrices which you should drill into your brain are: self-adjointness, positive-semidefiniteness and unit trace.



    Once you determine the values $a_1,a_2$ and $a_3$ from the density operator, you can easily find the location of the qubit state $(sintheta cosphi, sinthetasinphi, cos theta)$ inside the Bloch sphere. Let me emphasize on this point: pure states lie on the Bloch sphere (i.e. $|vec{a}|=1$) whereas mixed states lie inside the Bloch sphere (i.e. $|vec{a}|<1$); prove this as an exercise. If you're mathematically inclined, you'll also love to think about the Bloch sphere in terms of stereographic projections; it's excellently summarized in this Physics SE answer. I'll attach the image from there, which is originally from this blogpost (the article is in French, sorry :).



    enter image description here



    Here are a few "further readings" for you:




    • Density matrices for pure states and mixed states

    • Why do Bloch sphere wavefunctions have half angles?

    • Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere


    • Purity of mixed states as a function of radial distance from origin of Bloch ball


    • Can the Bloch sphere be generalized to two qubits?


    • Why is an entangled qubit shown at the origin of a Bloch sphere?



    Essentially, go through the bloch-sphere tag; you'll find several interesting questions and answers, which should clarify most of your beginner confusions about the Bloch sphere formalism.






    share|improve this answer











    $endgroup$


















      2












      $begingroup$

      The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|Psirangle$ of a qubit can be written in the form:



      $$|Psirangle = cosfrac{theta}{2}|0rangle + e^{iphi}sinfrac{theta}{2}|1rangle$$ where $0leq thetaleq pi$ and $0leq phileq 2pi$. This $|Psirangle$ can be represented on the Bloch sphere as:



      Bloch Sphere



      The Bloch vector $vec{a}in Bbb R^3$ is basically $(sintheta cosphi, sinthetasinphi, cos theta) = (a_1,a_2,a_3)$.



      To represent mixed states you need to consider the corresponding density operator $rho$. the set of states of a single qubit can be described in terms of $2times 2$ density matrices and as ${I,X,Y,Z}$ forms a basis for the vector space of $2times 2$ Hermitian matrices, you can write the density operator as $$rho = a_0I+a_1X+a_2Y+a_3Z = frac{1}{2}begin{pmatrix}1+a_3 & a_1-ia_2 \ a_1+ia_2 & 1-a_3end{pmatrix}.$$ As density matrices always have trace $1$, and here $mathrm{tr}(rho)=2a_0$, so $a_0$ is necessarily $frac{1}{2}$. So from here you can find out the Bloch coordinates of the any mixed state i.e. $(a_1,a_2,a_3)$ after performing the Pauli decompostion of the density matrix. If you're wondering what ensures that $|vec{a}|leq 1$, it's the positive semidefiniteness! The two eigenvalues of $rho$ are $frac{1}{2}(1+|vec{a}|)$ and $frac{1}{2}(1-|vec{a}|)$. Thus, to ensure that the second eigenvalue is non-negative, $|vec{a}|leq 1$. The three properties of density matrices which you should drill into your brain are: self-adjointness, positive-semidefiniteness and unit trace.



      Once you determine the values $a_1,a_2$ and $a_3$ from the density operator, you can easily find the location of the qubit state $(sintheta cosphi, sinthetasinphi, cos theta)$ inside the Bloch sphere. Let me emphasize on this point: pure states lie on the Bloch sphere (i.e. $|vec{a}|=1$) whereas mixed states lie inside the Bloch sphere (i.e. $|vec{a}|<1$); prove this as an exercise. If you're mathematically inclined, you'll also love to think about the Bloch sphere in terms of stereographic projections; it's excellently summarized in this Physics SE answer. I'll attach the image from there, which is originally from this blogpost (the article is in French, sorry :).



      enter image description here



      Here are a few "further readings" for you:




      • Density matrices for pure states and mixed states

      • Why do Bloch sphere wavefunctions have half angles?

      • Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere


      • Purity of mixed states as a function of radial distance from origin of Bloch ball


      • Can the Bloch sphere be generalized to two qubits?


      • Why is an entangled qubit shown at the origin of a Bloch sphere?



      Essentially, go through the bloch-sphere tag; you'll find several interesting questions and answers, which should clarify most of your beginner confusions about the Bloch sphere formalism.






      share|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|Psirangle$ of a qubit can be written in the form:



        $$|Psirangle = cosfrac{theta}{2}|0rangle + e^{iphi}sinfrac{theta}{2}|1rangle$$ where $0leq thetaleq pi$ and $0leq phileq 2pi$. This $|Psirangle$ can be represented on the Bloch sphere as:



        Bloch Sphere



        The Bloch vector $vec{a}in Bbb R^3$ is basically $(sintheta cosphi, sinthetasinphi, cos theta) = (a_1,a_2,a_3)$.



        To represent mixed states you need to consider the corresponding density operator $rho$. the set of states of a single qubit can be described in terms of $2times 2$ density matrices and as ${I,X,Y,Z}$ forms a basis for the vector space of $2times 2$ Hermitian matrices, you can write the density operator as $$rho = a_0I+a_1X+a_2Y+a_3Z = frac{1}{2}begin{pmatrix}1+a_3 & a_1-ia_2 \ a_1+ia_2 & 1-a_3end{pmatrix}.$$ As density matrices always have trace $1$, and here $mathrm{tr}(rho)=2a_0$, so $a_0$ is necessarily $frac{1}{2}$. So from here you can find out the Bloch coordinates of the any mixed state i.e. $(a_1,a_2,a_3)$ after performing the Pauli decompostion of the density matrix. If you're wondering what ensures that $|vec{a}|leq 1$, it's the positive semidefiniteness! The two eigenvalues of $rho$ are $frac{1}{2}(1+|vec{a}|)$ and $frac{1}{2}(1-|vec{a}|)$. Thus, to ensure that the second eigenvalue is non-negative, $|vec{a}|leq 1$. The three properties of density matrices which you should drill into your brain are: self-adjointness, positive-semidefiniteness and unit trace.



        Once you determine the values $a_1,a_2$ and $a_3$ from the density operator, you can easily find the location of the qubit state $(sintheta cosphi, sinthetasinphi, cos theta)$ inside the Bloch sphere. Let me emphasize on this point: pure states lie on the Bloch sphere (i.e. $|vec{a}|=1$) whereas mixed states lie inside the Bloch sphere (i.e. $|vec{a}|<1$); prove this as an exercise. If you're mathematically inclined, you'll also love to think about the Bloch sphere in terms of stereographic projections; it's excellently summarized in this Physics SE answer. I'll attach the image from there, which is originally from this blogpost (the article is in French, sorry :).



        enter image description here



        Here are a few "further readings" for you:




        • Density matrices for pure states and mixed states

        • Why do Bloch sphere wavefunctions have half angles?

        • Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere


        • Purity of mixed states as a function of radial distance from origin of Bloch ball


        • Can the Bloch sphere be generalized to two qubits?


        • Why is an entangled qubit shown at the origin of a Bloch sphere?



        Essentially, go through the bloch-sphere tag; you'll find several interesting questions and answers, which should clarify most of your beginner confusions about the Bloch sphere formalism.






        share|improve this answer











        $endgroup$



        The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|Psirangle$ of a qubit can be written in the form:



        $$|Psirangle = cosfrac{theta}{2}|0rangle + e^{iphi}sinfrac{theta}{2}|1rangle$$ where $0leq thetaleq pi$ and $0leq phileq 2pi$. This $|Psirangle$ can be represented on the Bloch sphere as:



        Bloch Sphere



        The Bloch vector $vec{a}in Bbb R^3$ is basically $(sintheta cosphi, sinthetasinphi, cos theta) = (a_1,a_2,a_3)$.



        To represent mixed states you need to consider the corresponding density operator $rho$. the set of states of a single qubit can be described in terms of $2times 2$ density matrices and as ${I,X,Y,Z}$ forms a basis for the vector space of $2times 2$ Hermitian matrices, you can write the density operator as $$rho = a_0I+a_1X+a_2Y+a_3Z = frac{1}{2}begin{pmatrix}1+a_3 & a_1-ia_2 \ a_1+ia_2 & 1-a_3end{pmatrix}.$$ As density matrices always have trace $1$, and here $mathrm{tr}(rho)=2a_0$, so $a_0$ is necessarily $frac{1}{2}$. So from here you can find out the Bloch coordinates of the any mixed state i.e. $(a_1,a_2,a_3)$ after performing the Pauli decompostion of the density matrix. If you're wondering what ensures that $|vec{a}|leq 1$, it's the positive semidefiniteness! The two eigenvalues of $rho$ are $frac{1}{2}(1+|vec{a}|)$ and $frac{1}{2}(1-|vec{a}|)$. Thus, to ensure that the second eigenvalue is non-negative, $|vec{a}|leq 1$. The three properties of density matrices which you should drill into your brain are: self-adjointness, positive-semidefiniteness and unit trace.



        Once you determine the values $a_1,a_2$ and $a_3$ from the density operator, you can easily find the location of the qubit state $(sintheta cosphi, sinthetasinphi, cos theta)$ inside the Bloch sphere. Let me emphasize on this point: pure states lie on the Bloch sphere (i.e. $|vec{a}|=1$) whereas mixed states lie inside the Bloch sphere (i.e. $|vec{a}|<1$); prove this as an exercise. If you're mathematically inclined, you'll also love to think about the Bloch sphere in terms of stereographic projections; it's excellently summarized in this Physics SE answer. I'll attach the image from there, which is originally from this blogpost (the article is in French, sorry :).



        enter image description here



        Here are a few "further readings" for you:




        • Density matrices for pure states and mixed states

        • Why do Bloch sphere wavefunctions have half angles?

        • Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere


        • Purity of mixed states as a function of radial distance from origin of Bloch ball


        • Can the Bloch sphere be generalized to two qubits?


        • Why is an entangled qubit shown at the origin of a Bloch sphere?



        Essentially, go through the bloch-sphere tag; you'll find several interesting questions and answers, which should clarify most of your beginner confusions about the Bloch sphere formalism.







        share|improve this answer














        share|improve this answer



        share|improve this answer








        edited 42 mins ago

























        answered 55 mins ago









        Sanchayan DuttaSanchayan Dutta

        6,75841556




        6,75841556






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Quantum Computing Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fquantumcomputing.stackexchange.com%2fquestions%2f5993%2fmeaning-of-bloch-representation%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

            El tren de la libertad Índice Antecedentes "Porque yo decido" Desarrollo de la...

            Puerta de Hutt Referencias Enlaces externos Menú de navegación15°58′00″S 5°42′00″O /...

            Castillo d'Acher Características Menú de navegación