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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Proving $\langle j_2|\langle j_1| U(|0\rangle\!\langle0|\otimes\rho)U^\dagger|j_2\rangle|j_1\rangle =\operatorname{Tr}(M_j\rho)$
I'm trying to prove that:
$$
\langle j_2|\langle j_1| U(|0\rangle\!\langle0|\otimes\rho)U^\dagger|j_2\rangle|j_1\rangle
=\operatorname{Tr}(M_j\rho)
$$
where $\rho$ is the density operator, $M_j=\...
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Prove that the partial trace is equivalent to measuring and discarding
I'm trying to solve the following question:
"Prove that one way to compute $\mathrm Tr_B$ is to assume that someone has measured system
$B$ in any orthonormal basis but does not tell you the ...
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Trace of Hermitian Operator and Operator Function
I am having trouble understanding the following step. From:
$$\operatorname{trace}\left(\sum_z |z\rangle\langle z| \rho_A |z\rangle\langle z| \times \log( \sum_z |z\rangle\langle z| \sum_x |\langle x|...
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How do I compute the Relative Entropy between pure and mixed states?
Let
$$ \rho = \begin{bmatrix} .7738 & -.0556 \\ -.0556 & .0040 \end{bmatrix} ,
\sigma = \begin{bmatrix} .9454 & -.2273 \\ -.2273 & .0546 \end{bmatrix} \\$$
As you can ...
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Determining the quantum secret
I earlier posted I question Representing a Bell measurement on non adjacent qubits for which I got an excellent answer. Now I want to build upon that and do further analysis which is where I am stuck. ...
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How to find the reduced density matrix of a four-qubit system?
I have the state vector $|p\rangle$ made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to find the reduced density matrix ...
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Is there any meaning for a density operator if we omit the j-th row and column in quantum mechanics?
Assume we have a density operator (Hermitian, PSD, with trace 1, where PSD means positive semi-definite) called A for a particle. $v_i$ shows the i-th eigenvector of A and $\lambda_i$ shows the i-th ...
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What is the average value of $|c_i\bar c_j|$ for a random state $|\psi\rangle=\sum_i c_i|i\rangle$?
Consider the density matrix $\rho=|\psi\rangle\!\langle\psi|$ of a random pure state in an $N$-dimensional space (in other words, an $N$-dimensional qudit, $|\psi\rangle\in\mathbb C^N$), $\rho_{ij}=...
glS♦
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What's an easy way to determine a local density matrix?
In his lecture notes Scott Aaronson states:
Now, consider the $2$-qubit pure state $\frac{|00\rangle + |01\rangle + |10\rangle}{\sqrt{3}}$. We'll give the first qubit to Alice and the second to Bob....
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How to represent an ensemble of pure quantum states in Qiskit?
I was going through the Qiskit documentation to see if there was a way to represent a mixture of quantum states as a density matrix or otherwise. Is there a way to do it? If so, how?
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The meaning of the city plot in Qiskit
I read the documentation of qiskit and I can't understand the meaning of the city plot, like this:
Why do we need a 3D plot? Why can't we just use a 2D plot, where $ | 00\rangle$, $ | 01 \rangle$, $ ...
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Taking the two qubit reduced density matrix on a 5 qubit system
I am wanting to find the two qubit reduced density matrix on a 5 qubit system. I have the 5 qubit state in the form of a tensor product and I want to find the reduced density matrix of qubits 1 and 3. ...
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What is the probability that measurement finds it in the $|0\rangle$ state?
Suppose that there is an ensemble with 60% of the states prepared in
$$|a\rangle=\sqrt{\frac{2}{5}}|+\rangle-\sqrt{\frac{3}{5}}|-\rangle$$
and 40% in:
$$|b\rangle=\sqrt{\frac{5}{8}}|+\rangle+\sqrt{\...
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Which state describes carrier transport through channel? A mixed state or a pure state?
A pure quantum state is a state which can be described by a single ket vector. A mixed quantum state is a statistical ensemble of pure states. When carriers transport from source to drain in a Field ...
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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_{...
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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 (...
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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 ...
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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 ...
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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\...
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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}
$$
with the quantities $...
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What is the relevance of preservation of trace in completely postive trace preserving (CPTP) maps?
Why is the trace preserving part necessary? Is it not enough if it can take all matrices to matrices of trace 1?
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Is there a two-qudit Choi entanglement witness $W^{(+)}$?
Example 2 in arXiv:1811.09896 states that the "Choi EW (entanglement witness) $W^{(+)}$ obtained from the Choi map in $d=3$ $\ldots$ is given by
\begin{equation}
W^{(+)} = \frac{1}{6} \left( \...
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Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?
Consider an arbitrary qudit state $\rho$ over $d$ modes.
Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation:
$$\rho=\frac{1}{d}\left(\mathbb I +\...
glS♦
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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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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{...
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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 ...
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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 ...
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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-...
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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:
$$\rho = \psi \otimes \...
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Decoherence in quantum systems always produces $\vert0\rangle$
I was recently asked two questions concerning error in quantum computing:
Is it possible for quantum computers to exhibit behavior similar to flip errors in classical computers where a state $\vert0\...
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Purity of mixed states as a function of radial distance from origin of Bloch ball
@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere
The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia:
Quantum ...
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Why is a density operator defined the way it's defined?
It's stated that the density operator is:
$$\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.$$
But I don't understand why this is the way both in mixed state and pure state. ...
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How to find a separable decomposition for $|\Psi^+\rangle\!\langle\Psi^+|+|\Phi^+\rangle\!\langle\Phi^+|$?
The state
$$ \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) $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}{...
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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-...
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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 ...
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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.
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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:
...
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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 ...
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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 ...
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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{|...
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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 ...
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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 ...
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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 ...
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