How to show a density matrix is in a pure/mixed state? Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Density matrices for pure states and mixed statesHow do density matrices act on $mathcalH_A$?How is measurement modelled when using the density operator?How do we derive the density operator of a subsystem?Density matrix after measurement on density matrixPartial Trace over a complicated looking stateNon-uniqueness of pure states ensemble decompositionWhy is a density operator defined the way it's defined?Computing von Neumann entropy of pure state in density matrixQuantum teleportation with “noisy” entangled state

One-one communication

New Order #6: Easter Egg

Should man-made satellites feature an intelligent inverted "cow catcher"?

draw a pulley system

Can gravitational waves pass through a black hole?

Why are current probes so expensive?

Shimano 105 brifters (5800) and Avid BB5 compatibility

The test team as an enemy of development? And how can this be avoided?

Combining list in a Cartesian product format with addition operation?

why doesn't university give past final exams' answers

Why is there so little support for joining EFTA in the British parliament?

Magento 2 - Add additional attributes in register

Centre cell vertically in tabularx across multiple multiline rows

Dinosaur Word Search, Letter Solve, and Unscramble

Did pre-Columbian Americans know the spherical shape of the Earth?

Marquee sign letters

How does Billy Russo acquire his 'Jigsaw' mask?

Is there a canonical “inverse” of Abelianization?

Was the pager message from Nick Fury to Captain Marvel unnecessary?

By what mechanism was the 2017 UK General Election called?

Can I take recommendation from someone I met at a conference?

How to make an animal which can only breed for a certain number of generations?

Where did Ptolemy compare the Earth to the distance of fixed stars?

How many time has Arya actually used Needle?



How to show a density matrix is in a pure/mixed state?



Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Density matrices for pure states and mixed statesHow do density matrices act on $mathcalH_A$?How is measurement modelled when using the density operator?How do we derive the density operator of a subsystem?Density matrix after measurement on density matrixPartial Trace over a complicated looking stateNon-uniqueness of pure states ensemble decompositionWhy is a density operator defined the way it's defined?Computing von Neumann entropy of pure state in density matrixQuantum teleportation with “noisy” entangled state



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








1












$begingroup$


Say we have a single qubit with some density matrix, for example lets say we have the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/2endpmatrix$. I would like to know what is the procedure for checking whether this state is pure or mixed.



I know that a pure state is one which can be written $rho=|psiranglelanglepsi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_kranglelanglepsi_k|$. But I have two problems. Firstly I'm not sure what the $p_k$ is, I know it's called the weight function but I don't understand it significance, how to find it, or its role mathematically. Secondly I don't see how it is possible to determine from these definition whether the sate I mentioned above is pure or mixed. How can we? Or perhaps there's some method that doesn't use the definitions directly?



Could anyone please clear up this problem for me ?






quantum-state quantum-information mathematics density-matrix matrix-representation







share|improve this question











$endgroup$











  • $begingroup$
    Is it possible that you instead mean the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/4endpmatrix$?
    $endgroup$
    – Mithrandir24601♦
    29 mins ago

















1












$begingroup$


Say we have a single qubit with some density matrix, for example lets say we have the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/2endpmatrix$. I would like to know what is the procedure for checking whether this state is pure or mixed.



I know that a pure state is one which can be written $rho=|psiranglelanglepsi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_kranglelanglepsi_k|$. But I have two problems. Firstly I'm not sure what the $p_k$ is, I know it's called the weight function but I don't understand it significance, how to find it, or its role mathematically. Secondly I don't see how it is possible to determine from these definition whether the sate I mentioned above is pure or mixed. How can we? Or perhaps there's some method that doesn't use the definitions directly?



Could anyone please clear up this problem for me ?






quantum-state quantum-information mathematics density-matrix matrix-representation







share|improve this question











$endgroup$











  • $begingroup$
    Is it possible that you instead mean the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/4endpmatrix$?
    $endgroup$
    – Mithrandir24601♦
    29 mins ago













1












1








1





$begingroup$


Say we have a single qubit with some density matrix, for example lets say we have the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/2endpmatrix$. I would like to know what is the procedure for checking whether this state is pure or mixed.



I know that a pure state is one which can be written $rho=|psiranglelanglepsi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_kranglelanglepsi_k|$. But I have two problems. Firstly I'm not sure what the $p_k$ is, I know it's called the weight function but I don't understand it significance, how to find it, or its role mathematically. Secondly I don't see how it is possible to determine from these definition whether the sate I mentioned above is pure or mixed. How can we? Or perhaps there's some method that doesn't use the definitions directly?



Could anyone please clear up this problem for me ?






quantum-state quantum-information mathematics density-matrix matrix-representation







share|improve this question











$endgroup$




Say we have a single qubit with some density matrix, for example lets say we have the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/2endpmatrix$. I would like to know what is the procedure for checking whether this state is pure or mixed.



I know that a pure state is one which can be written $rho=|psiranglelanglepsi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_kranglelanglepsi_k|$. But I have two problems. Firstly I'm not sure what the $p_k$ is, I know it's called the weight function but I don't understand it significance, how to find it, or its role mathematically. Secondly I don't see how it is possible to determine from these definition whether the sate I mentioned above is pure or mixed. How can we? Or perhaps there's some method that doesn't use the definitions directly?



Could anyone please clear up this problem for me ?






quantum-state quantum-information mathematics density-matrix matrix-representation




quantum-state quantum-information mathematics density-matrix matrix-representation






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 30 mins ago









Mithrandir24601♦

2,4352932




2,4352932










asked 4 hours ago









can'tcauchycan'tcauchy

1455




1455











  • $begingroup$
    Is it possible that you instead mean the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/4endpmatrix$?
    $endgroup$
    – Mithrandir24601♦
    29 mins ago
















  • $begingroup$
    Is it possible that you instead mean the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/4endpmatrix$?
    $endgroup$
    – Mithrandir24601♦
    29 mins ago















$begingroup$
Is it possible that you instead mean the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/4endpmatrix$?
$endgroup$
– Mithrandir24601♦
29 mins ago




$begingroup$
Is it possible that you instead mean the density matrix $rho=beginpmatrix3/4&1/2\1/2&1/4endpmatrix$?
$endgroup$
– Mithrandir24601♦
29 mins ago










2 Answers
2






active

oldest

votes


















4












$begingroup$

First, the example that you give is not a density matrix (they must have trace 1).



Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are many ways of doing this. However, a particularly natural way of decomposing it is using the spectral decomposition. The weights are the eigenvalues and the states are the corresponding eigenvectors.



However, if all you want to do is determine if the state is mixed, there’s a simpler way: calculate the trace of the square of the density matrix. If it’s 1, the state is pure. If it’s less than 1, the state is mixed. If it’s more than 1, you’ve messed up.






share|improve this answer









$endgroup$




















    1












    $begingroup$

    By spectral theorem density matrices are diagonizable, since they are hermitian (also they are positive semi-definite and have trace 1). That means that there is a set of $n$ non-negative eigenvalues $lambda_i$ with $n$ corresponding mutually orthogonal eigenvectors $|v_irangle$ such that
    $$
    rho = sum_i=1^nv_irangle langle v_i
    $$
    This matrix represents pure state only if it has exactly one non-zero eigenvalue (it must be equal to 1 since $tr(rho)=sum_i=1^nlambda_i=1$).



    So, to analyze density matrix you just need to find eigenvalues and eigenvectors.

    Though, to check if $rho$ is pure it is enough to verify the equality
    $$
    tr(rho^2)=1
    $$






    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%2f5952%2fhow-to-show-a-density-matrix-is-in-a-pure-mixed-state%23new-answer', 'question_page');

      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      First, the example that you give is not a density matrix (they must have trace 1).



      Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are many ways of doing this. However, a particularly natural way of decomposing it is using the spectral decomposition. The weights are the eigenvalues and the states are the corresponding eigenvectors.



      However, if all you want to do is determine if the state is mixed, there’s a simpler way: calculate the trace of the square of the density matrix. If it’s 1, the state is pure. If it’s less than 1, the state is mixed. If it’s more than 1, you’ve messed up.






      share|improve this answer









      $endgroup$

















        4












        $begingroup$

        First, the example that you give is not a density matrix (they must have trace 1).



        Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are many ways of doing this. However, a particularly natural way of decomposing it is using the spectral decomposition. The weights are the eigenvalues and the states are the corresponding eigenvectors.



        However, if all you want to do is determine if the state is mixed, there’s a simpler way: calculate the trace of the square of the density matrix. If it’s 1, the state is pure. If it’s less than 1, the state is mixed. If it’s more than 1, you’ve messed up.






        share|improve this answer









        $endgroup$















          4












          4








          4





          $begingroup$

          First, the example that you give is not a density matrix (they must have trace 1).



          Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are many ways of doing this. However, a particularly natural way of decomposing it is using the spectral decomposition. The weights are the eigenvalues and the states are the corresponding eigenvectors.



          However, if all you want to do is determine if the state is mixed, there’s a simpler way: calculate the trace of the square of the density matrix. If it’s 1, the state is pure. If it’s less than 1, the state is mixed. If it’s more than 1, you’ve messed up.






          share|improve this answer









          $endgroup$



          First, the example that you give is not a density matrix (they must have trace 1).



          Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are many ways of doing this. However, a particularly natural way of decomposing it is using the spectral decomposition. The weights are the eigenvalues and the states are the corresponding eigenvectors.



          However, if all you want to do is determine if the state is mixed, there’s a simpler way: calculate the trace of the square of the density matrix. If it’s 1, the state is pure. If it’s less than 1, the state is mixed. If it’s more than 1, you’ve messed up.







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered 4 hours ago









          DaftWullieDaftWullie

          15.5k1642




          15.5k1642























              1












              $begingroup$

              By spectral theorem density matrices are diagonizable, since they are hermitian (also they are positive semi-definite and have trace 1). That means that there is a set of $n$ non-negative eigenvalues $lambda_i$ with $n$ corresponding mutually orthogonal eigenvectors $|v_irangle$ such that
              $$
              rho = sum_i=1^nv_irangle langle v_i
              $$
              This matrix represents pure state only if it has exactly one non-zero eigenvalue (it must be equal to 1 since $tr(rho)=sum_i=1^nlambda_i=1$).



              So, to analyze density matrix you just need to find eigenvalues and eigenvectors.

              Though, to check if $rho$ is pure it is enough to verify the equality
              $$
              tr(rho^2)=1
              $$






              share|improve this answer











              $endgroup$

















                1












                $begingroup$

                By spectral theorem density matrices are diagonizable, since they are hermitian (also they are positive semi-definite and have trace 1). That means that there is a set of $n$ non-negative eigenvalues $lambda_i$ with $n$ corresponding mutually orthogonal eigenvectors $|v_irangle$ such that
                $$
                rho = sum_i=1^nv_irangle langle v_i
                $$
                This matrix represents pure state only if it has exactly one non-zero eigenvalue (it must be equal to 1 since $tr(rho)=sum_i=1^nlambda_i=1$).



                So, to analyze density matrix you just need to find eigenvalues and eigenvectors.

                Though, to check if $rho$ is pure it is enough to verify the equality
                $$
                tr(rho^2)=1
                $$






                share|improve this answer











                $endgroup$















                  1












                  1








                  1





                  $begingroup$

                  By spectral theorem density matrices are diagonizable, since they are hermitian (also they are positive semi-definite and have trace 1). That means that there is a set of $n$ non-negative eigenvalues $lambda_i$ with $n$ corresponding mutually orthogonal eigenvectors $|v_irangle$ such that
                  $$
                  rho = sum_i=1^nv_irangle langle v_i
                  $$
                  This matrix represents pure state only if it has exactly one non-zero eigenvalue (it must be equal to 1 since $tr(rho)=sum_i=1^nlambda_i=1$).



                  So, to analyze density matrix you just need to find eigenvalues and eigenvectors.

                  Though, to check if $rho$ is pure it is enough to verify the equality
                  $$
                  tr(rho^2)=1
                  $$






                  share|improve this answer











                  $endgroup$



                  By spectral theorem density matrices are diagonizable, since they are hermitian (also they are positive semi-definite and have trace 1). That means that there is a set of $n$ non-negative eigenvalues $lambda_i$ with $n$ corresponding mutually orthogonal eigenvectors $|v_irangle$ such that
                  $$
                  rho = sum_i=1^nv_irangle langle v_i
                  $$
                  This matrix represents pure state only if it has exactly one non-zero eigenvalue (it must be equal to 1 since $tr(rho)=sum_i=1^nlambda_i=1$).



                  So, to analyze density matrix you just need to find eigenvalues and eigenvectors.

                  Though, to check if $rho$ is pure it is enough to verify the equality
                  $$
                  tr(rho^2)=1
                  $$







                  share|improve this answer














                  share|improve this answer



                  share|improve this answer








                  edited 3 hours ago

























                  answered 3 hours ago









                  Danylo YDanylo Y

                  67016




                  67016



























                      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%2f5952%2fhow-to-show-a-density-matrix-is-in-a-pure-mixed-state%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

                      Are there any comparative studies done between Ashtavakra Gita and Buddhim?How is it wrong to believe that a self exists, or that it doesn't?Can you criticise or improve Ven. Bodhi's description of MahayanaWas the doctrine of 'Anatta', accepted as doctrine by modern Buddhism, actually taught by the Buddha?Relationship between Buddhism, Hinduism and Yoga?Comparison of Nirvana, Tao and Brahman/AtmaIs there a distinction between “ego identity” and “craving/hating”?Are there many differences between Taoism and Buddhism?Loss of “faith” in buddhismSimilarity between creation in Abrahamic religions and beginning of life in Earth mentioned Agganna Sutta?Are there studies about the difference between meditating in the morning versus in the evening?Can one follow Hinduism and Buddhism at the same time?Are there any prohibitions on participating in other religion's practices?Psychology of 'flow'

                      Image
                      Why "be dealt cards" rather than "be dealing cards"? Have I saved too much for retirement so far? The plural of 'stomach" How do I rename a LINUX host without needing to reboot for the rename to take effect? What defines a dissertation? Why are on-board computers allowed to change controls without notifying the pilots? Bash method for viewing beginning and end of file Student evaluations of teaching assistants Opposite of a diet Go Pregnant or Go Home Valid Badminton Score? What's the purpose of "true" in bash "if sudo true; then" What is the intuitive meaning of having a linear relationship between the logs of two variables? The baby cries all morning Is there a good way to store credentials outside of a password manager? Can I use my Chinese passport to enter China after I acquired another citizenship? Why Were Madagascar and New Zealand Discovered So Late? What are the ramifications of creating a ...
                      Read more

                      Where else does the Shulchan Aruch quote an authority by name?Parashat Metzora+HagadolPesach/PassoverShulchan Aruch UTF-8Anonymous glosses in the Shulchan AruchWhy is the Shulchan Aruch definitive?Siman 32, Kitzur Shulchan Aruch: UntranslatedLitvaks/Yeshivish and Shulchan AruchBuying a Shulchan AruchEnglish version of SHULCHAN ARUCHIs there any place where Shulchan Aruch rules with the Rosh against the Rif and Rambam?Are there practices where Sepharadim do not hold by Shulchan Aruch?5th part of the shulchan aruch

                      Image
                      Landlord wants to switch my lease to a "Land contract" to "get back at the city" Was there ever an axiom rendered a theorem? Finding files for which a command fails Is Social Media Science Fiction? What does "enim et" mean? Are objects structures and/or vice versa? Is every set a filtered colimit of finite sets? Doomsday-clock for my fantasy planet "My colleague's body is amazing" Why do UK politicians seemingly ignore opinion polls on Brexit? Does bootstrapped regression allow for inference? Is it legal to have the "// (c) 2019 John Smith" header in all files when there are hundreds of contributors? Is there a familial term for apples and pears? How did the USSR manage to innovate in an environment characterized by government censorship and high bureaucracy? I see my dog run How to make payment on the internet without leaving a money trail? Could a US political party gain complete control over the go...
                      Read more

                      fallocate: fallocate failed: Text file busy in Ubuntu 17.04? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)defragmenting and increasing performance of old lubuntu system with swap partitionIssue with increasing the root partition from the swapthis /usr/bin/dpkg returned error || ubuntu-16.04, 64bitDefault 17.04 swap file locationHow to Resize Ubuntu 17.04 Zesty Swap file size?Ubuntu freezes from online formsMy Laptop is not starting after upgrade ubuntu 16.04 (Kernel 4.8.0-38 to 04.10.0-36)hcp: ERROR: FALLOCATE FAILED!Not sure my swap is being usedWine 3.0 asking for more virtual free swap

                      Image
                      Why does Python start at index 1 when iterating an array backwards? Letter Boxed validator What's the purpose of writing one's academic bio in 3rd person? What is the longest distance a 13th-level monk can jump while attacking on the same turn? What are 'alternative tunings' of a guitar and why would you use them? Doesn't it make it more difficult to play? Can Pao de Queijo, and similar foods, be kosher for Passover? When -s is used with third person singular. What's its use in this context? Right-skewed distribution with mean equals to mode? Should gear shift center itself while in neutral? Are my PIs rude or am I just being too sensitive? When is phishing education going too far? If 'B is more likely given A', then 'A is more likely given B' List *all* the tuples! Is high blood pressure ever a symptom attributable solely to dehydration? Is there a service that would inform me whenever a new direct route is scheduled ...
                      Read more