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Questions tagged [density-matrix]

For questions about density matrices and related concepts and ideas, e.g. procedures for computing properties of quantum states from their density matrices.

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Understanding the classification of quantum states based on partial transposition: representations of the bipartite density matrix

I'm going through some slides on the PPT/NPT criteria along with Horodecki's paper, and I'm kind of stuck. Let's take this slide: Firstly, why can we write a bipartite density matrix as $\sum_{...
Sanchayan Dutta's user avatar
6 votes
1 answer
399 views

Equivalent determinant condition for Peres-Horodecki criteria

The Peres-Horodecki criteria for a 2*2 state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable. According to this paper (...
Mahathi Vempati's user avatar
8 votes
2 answers
3k views

Quantum fidelity simplified formula while both of the density matrices are single qubit states

I have a question while reading the quantum fidelity definition in Wikipedia Fidelity of quantum states, at the end of the Definition section of quantum fidelity formula, it says Explicit expression ...
tatakai's user avatar
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4 votes
1 answer
177 views

Quantum state where phase information is unknown

I'm trying to obtain a more intuitive understanding of the notion of quantum coherence and how to mathematically represent it. I know that coherence has to do with the interaction of phases between ...
Solarflare0's user avatar
2 votes
1 answer
341 views

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

The maximally mixed state on m qubits is defined to be the quantum state with associated density operator $\rho_m = \frac{1}{2^m} I$. Examples are On one qubit this is $\rho_1 = \frac{1}{2}(|0\...
gen's user avatar
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6 votes
1 answer
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Why do we use complex-conjugate instead of complex-conjugate-transpose when calculating the concurrence?

When we use the formula to calculate two-qubit entanglement, like these: $$ C(\rho)=\max \left\{\sqrt{e_{1}}-\sqrt{e_{2}}-\sqrt{e_{3}}-\sqrt{e_{4}}, 0\right\}\tag{18} $$ with the quantities $...
karry's user avatar
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2 votes
1 answer
130 views

What is the relevance of preservation of trace in completely postive trace preserving (CPTP) maps?

Why is the trace preserving part necessary? Is it not enough if it can take all matrices to matrices of trace 1?
Mahathi Vempati's user avatar
4 votes
0 answers
87 views

Is there a two-qudit Choi entanglement witness $W^{(+)}$?

Example 2 in arXiv:1811.09896 states that the "Choi EW (entanglement witness) $W^{(+)}$ obtained from the Choi map in $d=3$ $\ldots$ is given by \begin{equation} W^{(+)} = \frac{1}{6} \left( \...
Paul B. Slater's user avatar
4 votes
2 answers
680 views

Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?

Consider an arbitrary qudit state $\rho$ over $d$ modes. Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation: $$\rho=\frac{1}{d}\left(\mathbb I +\...
glS's user avatar
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How to show a density matrix is in a pure/mixed state?

Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
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3 votes
1 answer
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What are the ranges of the four $q$ parameters in the magic simplex of Bell states formula?

Equation (7) in the 2012 paper, "Complementarity Reveals Bound Entanglement of Two Twisted Photons" of B. C. Hiesmayr and W. Löffler for a state $\rho_d$ in the "magic simplex" of Bell states \begin{...
Paul B. Slater's user avatar
2 votes
2 answers
420 views

How does the probability of measurement turn out to be negative?

c) Compute $$\text{Prob}(\uparrow_\hat{n}\uparrow_\hat{m}) \equiv \text{tr}(\pmb{E}_A(\hat{n})\pmb{E}_B(\hat{n})\pmb{p}(\lambda)), \tag{4.164}$$ where $\pmb{E}_A(\hat{n})$ is the projection of Alice's ...
CFRedDemon's user avatar
6 votes
1 answer
797 views

Quantum teleportation with "noisy" entangled state

This is actually an exercise from Preskill (chapter 4, new version 4.4). So they are asking about the fidelity of teleporting a random pure quantum state from Bob to Alice, who both have one qubit of ...
CFRedDemon's user avatar
4 votes
0 answers
59 views

What proportions of certain sets of PPT-two-retrit states are bound entangled or separable?

For two particular (twelve-and thirteen-dimensional) sets of two-retrit states (corresponding to 9 x 9 density matrices with real off-diagonal entries), I have been able to calculate the Hilbert-...
Paul B. Slater's user avatar
1 vote
1 answer
4k views

How do I calculate the von Neumann entropy of a pure one-qubit density matrix?

Let's say I have a pure state of the form: $$\psi = \sqrt{\frac{3}{9}} \lvert 0 \rangle + \sqrt{\frac{6}{9}} \lvert 1 \rangle$$ Then the density matrix representation would be: $$\rho = \psi \otimes \...
QuestionEverything's user avatar
2 votes
1 answer
166 views

Decoherence in quantum systems always produces $\vert0\rangle$

I was recently asked two questions concerning error in quantum computing: Is it possible for quantum computers to exhibit behavior similar to flip errors in classical computers where a state $\vert0\...
Woody1193's user avatar
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4 votes
3 answers
1k views

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

@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
Sanchayan Dutta's user avatar
8 votes
2 answers
914 views

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

The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia: Quantum ...
Sanchayan Dutta's user avatar
2 votes
1 answer
244 views

Why is a density operator defined the way it's defined?

It's stated that the density operator is: $$\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.$$ But I don't understand why this is the way both in mixed state and pure state. ...
bilanush's user avatar
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1 answer
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How to find a separable decomposition for $|\Psi^+\rangle\!\langle\Psi^+|+|\Phi^+\rangle\!\langle\Phi^+|$?

The state $$ \frac{1}{2}\left(| \phi^+ \rangle \langle \phi^+ | + | \psi^+ \rangle \langle \psi^+ | \right) $$ where $$ | \phi^+ \rangle = \frac{1}{\sqrt2} \left(|00 \rangle + | 11 \rangle \right) $...
Mahathi Vempati's user avatar
5 votes
3 answers
1k views

How can pure state ensemble decompositions not be unique?

Apparently, the decomposition of a state into an ensemble of pure states is not unique. I can't understand why, as if I understood correctly a "pure state ensemble decomposition" is just the ...
user2723984's user avatar
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6 votes
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Is the set of classical-quantum states convex?

I read about the classical-quantum states in the textbook by Mark Wilde and there is an exercise that asks to show the set of classical-quantum states is not a convex set. But I have an argument to ...
qquery's user avatar
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3 answers
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$ I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
Mahathi Vempati's user avatar
1 vote
2 answers
772 views

How to write a post-measurement state, if we don't know the measurement result?

Once a state is measured, but we don't look at the result, is the state now written as a density matrix, that is, the probability that it could land on a measurement operator multiplied by the ...
Mahathi Vempati's user avatar
14 votes
3 answers
5k views

Density matrix after measurement on density matrix

Let's say Alice wants to send Bob a $|0\rangle$ with probability .5 and $|1\rangle$ also with probability .5. So after a qubit Alice prepares leaves her lab, the system could be represented by the ...
QuestionEverything's user avatar
7 votes
4 answers
1k views

Maximally mixed states for more than 1 qubit

For 1 qubit, the maximally mixed state is $\frac{\mathrm{I}}{2}$. So, for two qubits, I assume the maximally mixed state is the maximally mixed state is $\frac{\mathrm{I}}{4}$? Which is: $\frac{1}{...
Mahathi Vempati's user avatar
4 votes
1 answer
329 views

Suggest, partly based upon limited numerical results, possible “elegant” exact formulas for Bures two-qubit separability probability

Lovas and Andai (https://arxiv.org/abs/1610.01410) have recently established that the separability probability (ratio of separable volume to total volume) for the nine-dimensional convex set of two-...
Paul B. Slater's user avatar
4 votes
1 answer
887 views

Applying CNOT with local operations and two EPR pairs

Suppose Alice and Bob hold one qubit each of an arbitrary two-qubit state $|\psi \rangle$ that is possibly entangled. They can apply local operations and are allowed classical communication. Their ...
Joe's user avatar
  • 257
2 votes
2 answers
4k views

How do I construct a Density Matrix corresponding to a Hamiltonian?

I have a Hamiltonian and I want to know the corresponding density matrix. The matrix I'm interested in is the one in this question.
M00N KNIGHT's user avatar
1 vote
1 answer
1k views

Why is this Hamiltonian matrix diagonal?

I've only recently started using density matrices in my work but I am confused with the following code that I have whether I am getting the right matrix: ...
M00N KNIGHT's user avatar
7 votes
1 answer
186 views

Is there a relation between the factorisation of the joint conditional probability distribution and Bell inequality?

[I'm sorry, I've already posted the same question in the physics community, but I haven't received an answer yet.] I'm approaching the study of Bell's inequalities and I understood the reasoning ...
LadyOfShalott's user avatar
7 votes
1 answer
700 views

Difference between coherence transfer, polarization transfer and population transfer?

I asked a question on Physics Stack Exchange but no one answered the question and I didn't get enough views on it. I am asking it on QCSE because the question is related to experimental quantum ...
Jitendra's user avatar
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6 votes
2 answers
864 views

How do we derive the density operator of a subsystem?

The density operator can be used to represent uncertainty of quantum state from some perspective, aka a subsystem of the full quantum system. For example, given a Bell state: $|\psi\rangle = \frac{|...
ahelwer's user avatar
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5 votes
1 answer
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How is measurement modelled when using the density operator?

I've just learned about the density operator, and it seems like a fantastic way to represent the branching nature of measurement as simple algebraic manipulation. Unfortunately, I can't quite figure ...
ahelwer's user avatar
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4 votes
1 answer
369 views

How to calculate the off-diagonal elements of a density matrix using the measurement result?

For example, one have measured some states like $|0\rangle$ in the computational basis for many times and got the approximate probability of getting 0 and 1 ($P(0)$ and $P(1)$). Then how does he ...
raycosine's user avatar
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4 votes
1 answer
106 views

Modeling energy relaxation effects with density matrix formalism

I know there are measures that can be taken to mitigate the effects of dephasing (I'm referring here to Dynamic Decoupling and the other ideas it led to). I find it surprising that there is no ...
psitae's user avatar
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7 votes
2 answers
1k views

From Q# measurements to Bloch sphere

I would like to represent the state of a qubit on a Bloch sphere from the measurements made with Q#. According the documentation, it is possible to measure a qubit in the different Pauli bases (...
JRial95's user avatar
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10 votes
2 answers
802 views

What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?

For a Hilbert space $\mathcal{H}_A$, I have seen the phrase density matrices acting on $\mathcal{H}_A$ multiple times, e.g. here. It is clear to me that if $\mathcal{H}_A$ has finite Hilbert ...
Peter's user avatar
  • 509
22 votes
2 answers
21k views

Density matrices for pure states and mixed states

What is the motivation behind density matrices? And, what is the difference between the density matrices of pure states and density matrices of mixed states? This is a self-answered sequel to What&#...
Sanchayan Dutta's user avatar
8 votes
1 answer
4k views

How to find a density matrix of a qubit?

If we are given a state of a qubit, how do we construct its density matrix?
Archil Zhvania's user avatar
10 votes
2 answers
6k views

How to check if a matrix is a valid density matrix?

What conditions must a matrix hold to be considered a valid density matrix?
Archil Zhvania's user avatar
5 votes
2 answers
340 views

Only assuming the universe evolves according to a positive trace-preserving map, is there a proof that all subsystem evolution must be CPTP?

If we only assume that the wavefunction of the universe evolves according to $e^{-iHt}$, is there any proof that all subsystems of the universe (partial traces over parts of the universe) must evolve ...
user1271772 No more free time's user avatar

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