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?What is the difference between superpositions and mixed states?Density matrices for pure states and mixed statesHow do density matrices act on $mathcalH_A$?Modeling energy relaxation effects with density matrix formalismHow is measurement modelled when using the density operator?How do we derive the density operator of a subsystem?Partial Trace over a complicated looking stateWhy 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
Why are two-digit numbers in Jonathan Swift's "Gulliver's Travels" (1726) written in "German style"?
Why did Bronn offer to be Tyrion Lannister's champion in trial by combat?
JImage - Set generated image quality
Weaponising the Grasp-at-a-Distance spell
Random body shuffle every night—can we still function?
calculator's angle answer for trig ratios that can work in more than 1 quadrant on the unit circle
Did pre-Columbian Americans know the spherical shape of the Earth?
How do I find my Spellcasting Ability for my D&D character?
Does the universe have a fixed centre of mass?
Is there a verb for listening stealthily?
Improvising over quartal voicings
How to name indistinguishable henchmen in a screenplay?
Shimano 105 brifters (5800) and Avid BB5 compatibility
How to achieve cat-like agility?
3D Masyu - A Die
Why can't fire hurt Daenerys but it did to Jon Snow in season 1?
How do you cope with tons of web fonts when copying and pasting from web pages?
What helicopter has the most rotor blades?
geoserver.catalog.FailedRequestError: Tried to make a GET request to http://localhost:8080/geoserver/workspaces.xml but got a 404 status code
why doesn't university give past final exams' answers
Hide attachment record without code
Restricting the Object Type for the get method in java HashMap
Marquee sign letters
Any stored/leased 737s that could substitute for grounded MAXs?
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?What is the difference between superpositions and mixed states?Density matrices for pure states and mixed statesHow do density matrices act on $mathcalH_A$?Modeling energy relaxation effects with density matrix formalismHow is measurement modelled when using the density operator?How do we derive the density operator of a subsystem?Partial Trace over a complicated looking stateWhy 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;
$begingroup$
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 $rho=|psi><psi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_k><psi_k|$. 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
$endgroup$
add a comment |
$begingroup$
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 $rho=|psi><psi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_k><psi_k|$. 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
$endgroup$
add a comment |
$begingroup$
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 $rho=|psi><psi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_k><psi_k|$. 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
$endgroup$
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 $rho=|psi><psi|$, and that a mixed state is one which can be written $rho=sum_k=1^Np_k|psi_k><psi_k|$. 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
asked 3 hours ago
can'tcauchycan'tcauchy
1455
1455
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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
$$
rho = sum_i=1^n
$$
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
$$
tr(rho^2)=1
$$
$endgroup$
add a comment |
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
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered 3 hours ago
DaftWullieDaftWullie
15.5k1642
15.5k1642
add a comment |
add a comment |
$begingroup$
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
$$
rho = sum_i=1^n
$$
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
$$
tr(rho^2)=1
$$
$endgroup$
add a comment |
$begingroup$
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
$$
rho = sum_i=1^n
$$
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
$$
tr(rho^2)=1
$$
$endgroup$
add a comment |
$begingroup$
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
$$
rho = sum_i=1^n
$$
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
$$
tr(rho^2)=1
$$
$endgroup$
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
$$
rho = sum_i=1^n
$$
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
$$
tr(rho^2)=1
$$
edited 2 hours ago
answered 3 hours ago
Danylo YDanylo Y
67016
67016
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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