Otdfbt

Pull-up sequence accumulator counter

Pronunciation of "œuf" in "deux œufs kinder" and "bœuf "in "deux bœufs bourguignons" as an exception to silent /f/ in the plural

Are there any vegetarian astronauts?

Should I report a leak of confidential HR information?

Is adding a new player (or players) a DM decision, or a group decision?

Stepcounter after paragraph

Wilcoxon signed rank test – critical value for n>50

Analog is Obtuse!

Bash echo $-1 prints hb1. Why?

How often can a PC check with passive perception during a combat turn?

Why cruise at 7000' in an A319?

Magnitude of vector quantities

Forgot chonantanu after already making havdalah over wine

What happens when your group is victim of a surprise attack but you can't be surprised?

Why is Madam Hooch not a professor?

Why is a blank required between "[[" and "-e xxx" in ksh?

Procedurally generate regions on island

Mount a folder with a space on Linux

Do scales have a relation to certain genres or themes?

How hard is it to sell a home which is currently mortgaged?

How are monocytes larger than capillaries?

Was touching your nose a greeting in second millenium Mesopotamia?

Why does the numerical solution of an ODE move away from an unstable equilibrium?

Layout of complex table

How can I write the maximally mixed state on m qubits as a linear combination of basis vectors?

Density matrices for pure states and mixed statesHow to show a density matrix is in a pure/mixed state?What is the difference between superpositions and mixed states?Embedding classical information into norm of a quantum stateHow is measurement modelled when using the density operator?How do we derive the density operator of a subsystem?Suggest, partly based upon limited numerical results, possible “elegant” exact formulas for Bures two-qubit separability probabilityMaximally mixed states for more than 1 qubitIs the set of classical-quantum states convex?Quantum proof for the group non-membership problemBetter Way Of Separating Two CQ-StatesMaking a maximally mixed 2-qubit state in the IBM Q

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

2

$begingroup$

The maximally mixed state on m qubits is defined to be the quantum state with associated density operator $rho_m = frac12^m I$. Examples are

• On one qubit this is $rho_1 = frac12(|0ranglelangle0|+|1ranglelangle1|) = frac12I$
  


• On two qubits we have $rho_2 = frac14 (|00ranglelangle00| +|01ranglelangle01|+|10ranglelangle10|+|11ranglelangle11|$.

My question is the following: how can the corresponding state vector $|phi_mrangle$ be expressed in terms of the standard basis elements, eg. $|phi_2rangle = sum_i,j a_i,j|ijrangle$? What are the values of $a_i,j$?

quantum-state density-matrix

share|improve this question

edited Jun 10 at 17:19

Sanchayan Dutta

7,87744 gold badges1616 silver badges6262 bronze badges

asked Jun 9 at 14:09

gengen

11333 bronze badges

$endgroup$

add a comment | 

2

$begingroup$

The maximally mixed state on m qubits is defined to be the quantum state with associated density operator $rho_m = frac12^m I$. Examples are

• On one qubit this is $rho_1 = frac12(|0ranglelangle0|+|1ranglelangle1|) = frac12I$
  


• On two qubits we have $rho_2 = frac14 (|00ranglelangle00| +|01ranglelangle01|+|10ranglelangle10|+|11ranglelangle11|$.

My question is the following: how can the corresponding state vector $|phi_mrangle$ be expressed in terms of the standard basis elements, eg. $|phi_2rangle = sum_i,j a_i,j|ijrangle$? What are the values of $a_i,j$?

quantum-state density-matrix

share|improve this question

edited Jun 10 at 17:19

Sanchayan Dutta

7,87744 gold badges1616 silver badges6262 bronze badges

asked Jun 9 at 14:09

gengen

11333 bronze badges

$endgroup$

add a comment | 

$begingroup$

The maximally mixed state on m qubits is defined to be the quantum state with associated density operator $rho_m = frac12^m I$. Examples are

• On one qubit this is $rho_1 = frac12(|0ranglelangle0|+|1ranglelangle1|) = frac12I$
  


• On two qubits we have $rho_2 = frac14 (|00ranglelangle00| +|01ranglelangle01|+|10ranglelangle10|+|11ranglelangle11|$.

My question is the following: how can the corresponding state vector $|phi_mrangle$ be expressed in terms of the standard basis elements, eg. $|phi_2rangle = sum_i,j a_i,j|ijrangle$? What are the values of $a_i,j$?

quantum-state density-matrix

share|improve this question

edited Jun 10 at 17:19

Sanchayan Dutta

7,87744 gold badges1616 silver badges6262 bronze badges

asked Jun 9 at 14:09

gengen

11333 bronze badges

$endgroup$

share|improve this question

edited Jun 10 at 17:19

Sanchayan Dutta

7,87744 gold badges1616 silver badges6262 bronze badges

asked Jun 9 at 14:09

gengen

11333 bronze badges

share|improve this question

edited Jun 10 at 17:19

Sanchayan Dutta

7,87744 gold badges1616 silver badges6262 bronze badges

asked Jun 9 at 14:09

gengen

11333 bronze badges

edited Jun 10 at 17:19

Sanchayan Dutta

7,87744 gold badges1616 silver badges6262 bronze badges

edited Jun 10 at 17:19

Sanchayan Dutta

7,87744 gold badges1616 silver badges6262 bronze badges

asked Jun 9 at 14:09

gengen

11333 bronze badges

asked Jun 9 at 14:09

gengen

11333 bronze badges

1 Answer
1

active

oldest

votes

2

$begingroup$

If by "corresponding state vector" you mean a pure state $lvertpsirangle$ such that $lvertpsirangle!langlepsirvert$ is maximally mixed, then the answer is that there isn't one.

A density matrix $rho$ can be written as $rho=lvertpsirangle!langlepsirvert$ for some ket state $lvertpsirangle$ if and only if it is pure. One easy way to check for this is to compute $mathrmTr(rho^2)$, which is the so-called purity of the state, and equals $1$ if and only if the state is pure.

A maximally mixed state is as far away as possible from this situation, in the sense that it is the state (or one of the states) which corresponds to the minimal possible value of the purity.

You might also want to have a look at these related questions:

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


• Density matrices for pure states and mixed states

share|improve this answer

answered Jun 9 at 14:46

glSglS

5,24011 gold badge99 silver badges4545 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well what I meant by corresponding is this: a one qubit system can be described either via it's density matrix or just as an element of the Hilbert space in question ie as $|phirangle = sum_i |phi_irangle$ for basis vectors $|phi_irangle$. Hmm, on a second look I think I might be confusing mixed states with superposition... So I am guessing when I say $|phirangle$ can be written as linear combination of basis vectors, that's only true for pure states, right? That would explain my confusion.
  $endgroup$
  – gen
  Jun 9 at 14:53
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @gen yes, when you say that a state can be described either via its density matrix $rho$ or its ket state $lvertpsirangle$, the correspondence between the two is that $rho=lvertpsirangle!langlepsirvert$. Not all density matrices correspond to a ket (pure) state. They only do when the state is pure. Otherwise, the density matrices are mixtures of DMs corresponding to pure states, that is, objects of the form $rho=sum_k p_klvertpsi_krangle!langlepsi_krvert$ for some ensemble of pure states $lvertpsi_krangle$
  $endgroup$
  – glS
  Jun 9 at 14:55
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  if the question is what mixture gives a maximally mixed state, then the answer is that you can get via any mixture with equal probabilities of an orthonormal set for the space
  $endgroup$
  – glS
  Jun 9 at 14:58
  

  
  
  
  

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
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fquantumcomputing.stackexchange.com%2fquestions%2f6388%2fhow-can-i-write-the-maximally-mixed-state-on-m-qubits-as-a-linear-combination-of%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

2

$begingroup$

If by "corresponding state vector" you mean a pure state $lvertpsirangle$ such that $lvertpsirangle!langlepsirvert$ is maximally mixed, then the answer is that there isn't one.

A density matrix $rho$ can be written as $rho=lvertpsirangle!langlepsirvert$ for some ket state $lvertpsirangle$ if and only if it is pure. One easy way to check for this is to compute $mathrmTr(rho^2)$, which is the so-called purity of the state, and equals $1$ if and only if the state is pure.

A maximally mixed state is as far away as possible from this situation, in the sense that it is the state (or one of the states) which corresponds to the minimal possible value of the purity.

You might also want to have a look at these related questions:

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


• Density matrices for pure states and mixed states

share|improve this answer

answered Jun 9 at 14:46

glSglS

5,24011 gold badge99 silver badges4545 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well what I meant by corresponding is this: a one qubit system can be described either via it's density matrix or just as an element of the Hilbert space in question ie as $|phirangle = sum_i |phi_irangle$ for basis vectors $|phi_irangle$. Hmm, on a second look I think I might be confusing mixed states with superposition... So I am guessing when I say $|phirangle$ can be written as linear combination of basis vectors, that's only true for pure states, right? That would explain my confusion.
  $endgroup$
  – gen
  Jun 9 at 14:53
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @gen yes, when you say that a state can be described either via its density matrix $rho$ or its ket state $lvertpsirangle$, the correspondence between the two is that $rho=lvertpsirangle!langlepsirvert$. Not all density matrices correspond to a ket (pure) state. They only do when the state is pure. Otherwise, the density matrices are mixtures of DMs corresponding to pure states, that is, objects of the form $rho=sum_k p_klvertpsi_krangle!langlepsi_krvert$ for some ensemble of pure states $lvertpsi_krangle$
  $endgroup$
  – glS
  Jun 9 at 14:55
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  if the question is what mixture gives a maximally mixed state, then the answer is that you can get via any mixture with equal probabilities of an orthonormal set for the space
  $endgroup$
  – glS
  Jun 9 at 14:58
  

  
  
  
  

add a comment | 

2

$begingroup$

If by "corresponding state vector" you mean a pure state $lvertpsirangle$ such that $lvertpsirangle!langlepsirvert$ is maximally mixed, then the answer is that there isn't one.

A density matrix $rho$ can be written as $rho=lvertpsirangle!langlepsirvert$ for some ket state $lvertpsirangle$ if and only if it is pure. One easy way to check for this is to compute $mathrmTr(rho^2)$, which is the so-called purity of the state, and equals $1$ if and only if the state is pure.

A maximally mixed state is as far away as possible from this situation, in the sense that it is the state (or one of the states) which corresponds to the minimal possible value of the purity.

You might also want to have a look at these related questions:

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


• Density matrices for pure states and mixed states

share|improve this answer

answered Jun 9 at 14:46

glSglS

5,24011 gold badge99 silver badges4545 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well what I meant by corresponding is this: a one qubit system can be described either via it's density matrix or just as an element of the Hilbert space in question ie as $|phirangle = sum_i |phi_irangle$ for basis vectors $|phi_irangle$. Hmm, on a second look I think I might be confusing mixed states with superposition... So I am guessing when I say $|phirangle$ can be written as linear combination of basis vectors, that's only true for pure states, right? That would explain my confusion.
  $endgroup$
  – gen
  Jun 9 at 14:53
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @gen yes, when you say that a state can be described either via its density matrix $rho$ or its ket state $lvertpsirangle$, the correspondence between the two is that $rho=lvertpsirangle!langlepsirvert$. Not all density matrices correspond to a ket (pure) state. They only do when the state is pure. Otherwise, the density matrices are mixtures of DMs corresponding to pure states, that is, objects of the form $rho=sum_k p_klvertpsi_krangle!langlepsi_krvert$ for some ensemble of pure states $lvertpsi_krangle$
  $endgroup$
  – glS
  Jun 9 at 14:55
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  if the question is what mixture gives a maximally mixed state, then the answer is that you can get via any mixture with equal probabilities of an orthonormal set for the space
  $endgroup$
  – glS
  Jun 9 at 14:58
  

  
  
  
  

add a comment | 

$begingroup$

If by "corresponding state vector" you mean a pure state $lvertpsirangle$ such that $lvertpsirangle!langlepsirvert$ is maximally mixed, then the answer is that there isn't one.

A density matrix $rho$ can be written as $rho=lvertpsirangle!langlepsirvert$ for some ket state $lvertpsirangle$ if and only if it is pure. One easy way to check for this is to compute $mathrmTr(rho^2)$, which is the so-called purity of the state, and equals $1$ if and only if the state is pure.

A maximally mixed state is as far away as possible from this situation, in the sense that it is the state (or one of the states) which corresponds to the minimal possible value of the purity.

You might also want to have a look at these related questions:

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


• Density matrices for pure states and mixed states

share|improve this answer

answered Jun 9 at 14:46

glSglS

5,24011 gold badge99 silver badges4545 bronze badges

$endgroup$

share|improve this answer

answered Jun 9 at 14:46

glSglS

5,24011 gold badge99 silver badges4545 bronze badges

answered Jun 9 at 14:46

glSglS

5,24011 gold badge99 silver badges4545 bronze badges

answered Jun 9 at 14:46

glSglS

5,24011 gold badge99 silver badges4545 bronze badges