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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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&#...
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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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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:
$$\rho=\frac{1}{d}\left(\mathbb I +\...
glS♦
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
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How to perform Quantum Process Tomography for three qubit gates?
I am trying to perform Quantum process tomography (QPT) on three qubit quantum gate. But I cannot find any relevant resource to follow and peform the experiment. I have checked Nielsen and Chuang's ...
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How to find a density matrix of a qubit?
If we are given a state of a qubit, how do we construct its density matrix?
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Density matrix after measurement on density matrix
Let's say Alice wants to send Bob a $|0\rangle$ with probability .5 and $|1\rangle$ also with probability .5. So after a qubit Alice prepares leaves her lab, the system could be represented by the ...
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Can one quantify entanglement between different parts of a system?
Consider some state $|\psi\rangle$ of $n$ qubits. One can take any subsystem $A$ and compute its density matrix $\rho_A =Tr_{B} |\psi\rangle \langle\psi|$. The entanglement between subsystem $A$ and ...
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Random quantum states and Schur-Weyl duality
Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator:
$$
\rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC.
$$
Let's ...
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?
I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following:
$$
F(\rho, \sigma) := \...
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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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Density matrices of multiples copies of a single Haar-Random state
In Pseudorandom States, Non-Cloning Theorems and Quantum Money, the authors state that:
Let $\rho_\mu^m$ be the density matrix of a random $|\psi\rangle^{\otimes m}$ for $|\psi\rangle$ chosen from ...
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Does partial tracing a system with three shared Bell states give the identity?
Suppose I share three Bell states among two participants Alice and Bob and Charlie in the following manner:
$$ |\psi\rangle=\left(\dfrac{|0\rangle_1|0\rangle_2+ |1\rangle_1|1\rangle_2}{\sqrt{2}}\...
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How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?
How can I calculate the eigenvalues of $\rho^{T_{B}}$ (PPT) of the following state
$$
\rho =\frac{1}{2}|0\rangle\langle0|\otimes|+\rangle\langle+|+\frac{1}{2}|+\rangle\langle+|\otimes|1\rangle\langle1|...
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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 ...
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Simulating a quantum circuit with decoherence and noise
Based on the answers given here and here, it is pretty clear that an arbitrary quantum circuit can be simulated with matrix algebra. The difficulty is that this assume perfect fidelity. I am unsure ...
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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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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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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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Show that if the Lindblad satisfy $\sum_\mu L_\mu L_\mu^\dagger=\sum_\mu L_\mu^\dagger L_\mu$ then the von Neumann entropy increases monotonically
How can we show that when the Lindblad operators satisfy the condition:
$$\sum_{\mu}L_{\mu} L_{\mu}^{\dagger} = \sum_{\mu} L_{\mu}^{\dagger}L_{\mu},\tag{1}$$
the master equation evolution ...
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Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$
I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
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Why are $d^2$ dimensions required to describe a density matrix?
A density matrix is defined as:
$$\sum p_i |\psi_i \rangle \langle \psi_i|$$
If the dimensionality of each $|\psi_i \rangle$ is $d$, why does it take $d^2$ dimensions to represent a density matrix? (...
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Semi-definite program for smooth min-entropy
The conditional min-entropy is defined as (wiki):
$$
H_{\min}(A|B)_{\rho} \equiv -\inf_{\sigma_B}\inf_{\lambda}\{\lambda \in \mathbb{R}:\rho_{AB} \leq 2^{\lambda} \mathbb{I} \otimes \sigma_B\}
$$
And ...
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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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How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived?
Let's say I have a density matrix of the following form:
$$
\rho := \frac{1}{2} (|a \rangle \langle a| + |b \rangle \langle b|),
$$
where $|a\rangle$ and $|b\rangle$ are quantum states. I saw that ...
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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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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 ...
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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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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
$$ | \phi^+ \rangle = \frac{1}{\sqrt2} \left(|00 \rangle + | 11 \rangle \right) $...
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What proportions of certain sets of PPT-two-retrit states are bound entangled or separable?
For two particular (twelve-and thirteen-dimensional) sets of two-retrit states (corresponding to 9 x 9 density matrices with real off-diagonal entries), I have been able to calculate the Hilbert-...
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Applying CNOT with local operations and two EPR pairs
Suppose Alice and Bob hold one qubit each of an arbitrary two-qubit state $|\psi \rangle$ that is possibly entangled. They can apply local operations and are allowed classical communication. Their ...
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Density matrix of a product of Bell states
Suppose I share two Bell states among two participants Alice and Bob in the following manner :
$$ |\psi\rangle=\left(\dfrac{|0\rangle_1|0\rangle_2+ |1\rangle_1|1\rangle_2}{\sqrt{2}}\right)\left(\...
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What is the matrix representation for $n$-qubit gates?
Let's say I have more than one qbits $|0\rangle|1\rangle$ and I want to perform a $H$ on both of them. I know the matrix representation for the Hadamard on a single qbit is
$$\frac{1}{\sqrt{2}}\begin{...
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Show that $I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$ for all states $\rho$
I am trying to show that for any qubit state p, the following holds:
$$I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$$
I have tried different manipulations,...
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Is the partial trace $\mathrm{Tr}_B(\rho)$ equal to $\sum_k \mathrm{Tr}[(\sigma_k\otimes I)^\dagger \rho]\sigma_k$?
Assume a composite quantum systes with state $|\psi_{AB}\rangle$ or better $\rho=|\psi_{AB}\rangle\langle\psi_{AB}|$. I want to know the state of system $A$ only, i.e. $\rho_A$.
Is there any ...
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Implementation of tomography on IBM Q
I wanted to ask how do you implement a circuit that finds the non-diagonal values of the density matrix of a quantum state on IBM Q?
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Finding Wigner function of four maximal entangled Bell state
How can we find a Wigner function for the four maximally entangled Bell states $(|00\rangle \pm |11\rangle)/\sqrt{2}$, $(|01\rangle \pm |10\rangle)/\sqrt{2}$?
I have used the basis where labels for ...
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Purification applied to indistinguishability
In Zhandry's compressed oracle paper, one can read the following:
Next, we note that the oracle $h$ being chosen at random is equivalent (from the adversary’s point of view) to $h$ being in uniform ...
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How to get the state of an individual qubit in a composite system?
Given a composite system with $N$ qubits represented by some $2^N$-dimensional vector, how would I get the quantum state of an individual qubit?
Note that I understand some states are not separable ...
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Why is a density operator defined the way it's defined?
It's stated that the density operator is:
$$\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.$$
But I don't understand why this is the way both in mixed state and pure state. ...
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How would I compute a density matrix of a 2 qubit mixed state?
I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this?
$$
|00> \;with \;probability \; 2/4 \\
...
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Is there an algorithm that can decide if a state is a mixed state or a pure state?
Given we are using the computational basis,is there a quantum algorithm that can decide if an arbitrary input state $\vert A\rangle$ ( using $N$ qubits) is a pure state or a mixed state? $\vert A\...
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Give an explicit derivation of the exact formula for the two-qubit absolute separability Hilbert-Schmidt probability $\approx 0.00365826$
The two-qubit eigenvalue ($\lambda_i$ >= 0, $i=1,\ldots,4$, $\lambda_4=1-\lambda_1-\lambda_2-\lambda_3$) condition of Verstraete, Audenaert, de Bie and de Moor
AbsoluteSeparability
(p. 6) for ...
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What experiments can distinguish between mixed and pure states?
To distinguish between a coherent and de-cohered stage of the same system
what experiments can provide the answer? The term Experiment is used here in the Bohr-Einstein-debate sense, a realizable ...
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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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Are quantum gates superoperators? How to write a quantum circuit as superoperator?
I have some questions related to superoperators:
What is the differences between quantum operators and superoperators? For instance quantum gates are also unitary operators but can we say quantum ...
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