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.
306
questions
20
votes
2
answers
17k
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&#...
- 16.9k
13
votes
3
answers
5k
views
What do the off-diagonal elements of a density matrix physically represent?
For simplicity, let's take a density matrix for a single qubit, written in the $\{|0\rangle,|1\rangle\}$ basis:
$$ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10}^* & 1-\rho_{00} \...
- 131
12
votes
3
answers
4k
views
Density matrix after measurement on density matrix
Let's say Alice wants to send Bob a $|0\rangle$ with probability .5 and $|1\rangle$ also with probability .5. So after a qubit Alice prepares leaves her lab, the system could be represented by the ...
- 1,725
10
votes
2
answers
5k
views
How to check if a matrix is a valid density matrix?
What conditions must a matrix hold to be considered a valid density matrix?
- 2,147
10
votes
2
answers
621
views
What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?
For a Hilbert space $\mathcal{H}_A$, I have seen the phrase
density matrices acting on $\mathcal{H}_A$
multiple times, e.g. here.
It is clear to me that if $\mathcal{H}_A$ has finite Hilbert ...
- 489
8
votes
2
answers
177
views
Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?
Suppose I have an arbitrary orthogonal projector $\Pi$ and two density operators $\rho, \sigma$. Is it true that:
$$ ||\Pi (\sigma - \rho) \Pi||_1 \le || \sigma - \rho ||_1 $$
where $||\cdot||_1$ ...
- 327
8
votes
2
answers
566
views
What's the 'physical consistency' in the partial trace scenario?
I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand:
Physical consistency requires that any ...
- 2,348
8
votes
1
answer
3k
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?
- 2,147
8
votes
0
answers
103
views
Query on Reduced Graph States
Reduced graph states are characterized as follows (from page 46 of this paper): Let $A \subseteq V$ be a subset of vertices of a graph $G = (V,E)$ and $B = V\setminus A$ the complement of $A$ in $V$. ...
- 747
7
votes
2
answers
3k
views
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 ...
- 869
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}{...
- 1,601
7
votes
3
answers
1k
views
How to prepare mixed states on a quantum computer?
I am a little bit confused by density matrix notation in quantum algorithms. While I am pretty confident with working with pure
states, I never had the need to work with algorithm using density ...
7
votes
4
answers
535
views
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 ...
- 673
7
votes
2
answers
729
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 ...
- 16.9k
7
votes
2
answers
897
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 (...
- 93
7
votes
1
answer
208
views
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 ...
- 1,385
7
votes
2
answers
2k
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 ...
- 75
7
votes
1
answer
179
views
Bob applies a projector - what happens to eigenvalues of Alice's reduced state?
Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global ...
- 2,367
7
votes
1
answer
606
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 ...
- 255
7
votes
1
answer
974
views
What is the intuition behind "states with support on orthogonal subspaces"?
I'm sure I don't fully understand support, but I am having trouble seeing how it connects to things like density operators. I have an idea that it means, according to Wikipedia:
"In mathematics, ...
- 73
7
votes
1
answer
173
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 ...
7
votes
1
answer
203
views
Closest quantum state with a fixed marginal: Analytical solution?
Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
- 2,367
6
votes
3
answers
2k
views
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)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
- 1,601
6
votes
2
answers
178
views
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
I've been trying to figure this out for a while and I'm totally lost.
My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$
So far ...
6
votes
3
answers
613
views
Interpretation of the unitaries involved in the eigenvalue decomposition of a density operator
If $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle \psi_{i}|$, this ensemble doesn't require $\langle \psi_{i}|\psi_{j}\rangle$=0. Given that $\rho$ is positive semi-definite, by the spectral theorem it ...
- 1,097
6
votes
2
answers
737
views
Quantum teleportation of a mixed state through a pure state?
Let's assume we have a register of qubits present in a mixed state
$$\rho = \sum_i^n p_i|\psi_i\rangle \langle \psi_i|$$
and we want to teleport $\rho$ through a random pure state $|\phi\rangle$. What ...
- 259
6
votes
2
answers
1k
views
What are the conditions ensuring a two-qubit density matrix is positive semidefinite?
I've seen some papers writing
$$\rho=\frac{1}{4}\left(\mathbb{I} \otimes \mathbb{I}+\sum_{k=1}^{3} a_{k} \sigma_{k} \otimes \mathbb{I}+\sum_{l=1}^{3} b_{l} \mathbb{I} \otimes \sigma_{l}+\sum_{k, l=1}^{...
- 2,772
6
votes
1
answer
392
views
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 $\sigma$ ...
- 1,331
6
votes
2
answers
665
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{|...
- 3,988
6
votes
1
answer
271
views
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 ...
6
votes
1
answer
517
views
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 ...
- 61
6
votes
2
answers
392
views
How does quantum teleportation work with mixed shared states?
I am given the scenario that instead of the two parties (A & B) sharing the bell state $|\phi_+\rangle$ they share the mixture $\rho_\lambda = \lambda|\phi_+\rangle\langle\phi_+|+(1-\lambda)\frac{\...
- 63
6
votes
2
answers
218
views
Can one always find purifications which preserve equality of statistical mixtures?
When pure states $|\psi_1⟩$, $|\psi_2⟩$ and $|\phi_1⟩$, $|\phi_2⟩$ in $\mathcal{H}_A \otimes \mathcal{H}_B$ have identical statistical mixtures
$$\frac{1}{2}(|\psi_1⟩⟨\psi_1| + |\psi_2⟩⟨\psi_2|) = \...
- 103
6
votes
1
answer
217
views
Are perfectly LOCC-indistinguishable states necessarily identical?
Let $\rho,\sigma\in\text{L}(\mathcal{H}_{XAB})$ be given by
$$ \rho = \sum_x |x\rangle\langle x|\otimes p_x\rho_x, \quad \sigma = \sum_x |x\rangle\langle x|\otimes q_x\sigma_x, $$
and consider ...
- 245
6
votes
1
answer
422
views
Why do we use complex-conjugate instead of complex-conjugate-transpose when calculating the concurrence?
When we use the formula to calculate two-qubit entanglement, like these:
$$
C(\rho)=\max \left\{\sqrt{e_{1}}-\sqrt{e_{2}}-\sqrt{e_{3}}-\sqrt{e_{4}}, 0\right\}\tag{18}
$$
with the quantities $...
- 457
6
votes
0
answers
81
views
Tripartite quantum marginal problem
Consider a tripartite quantum system with the three subsystems labeled $A, B,$ and $C$. Now take two states $\rho_{AB}$ on the joint system $AB$ and $\rho_{BC}$ on the joint system $BC$. Under what ...
- 926
5
votes
3
answers
4k
views
What is a "maximally mixed state"?
What is meant by maximally mixed states? Does this mean that there are partially mixed states?
For example, consider $\rho_{GHZ} = \left| {GHZ} \right\rangle \left\langle {GHZ} \right|$ and $\rho_W =...
- 255
5
votes
3
answers
438
views
How to find the distance between a given $\rho$ and the nearest pure state(s)?
I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state:
$$
\min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
- 5,715
5
votes
3
answers
869
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 ...
- 1,036
5
votes
3
answers
538
views
Why do purifications only differ by a local unitary?
Let's consider $\rho_A$ a density matrix. I introduce a space $B$ and an entangled state $|\Psi\rangle$ (the purification) so that:
$$\newcommand{\tr}{\operatorname{Tr}}\rho_A = \tr_B(|\Psi\rangle \...
- 1,142
5
votes
2
answers
199
views
Why are commuting density operators said to be "classical states"?
In quantum information it is commonly said that a set of states $S=\{ \rho_i \}_i$ is classical if $[\rho_m, \rho_n] = 0, \,\forall \rho_m,\rho_n \in S$. This is meant in the sense that all observed ...
- 53
5
votes
2
answers
1k
views
Why are $d^2$ dimensions required to describe a density matrix?
A density matrix is defined as:
$$\sum p_i |\psi_i \rangle \langle \psi_i|$$
If the dimensionality of each $|\psi_i \rangle$ is $d$, why does it take $d^2$ dimensions to represent a density matrix? (...
- 1,601
5
votes
2
answers
84
views
Does ${\rm tr}(\Pi \rho) = 1$ imply $\Pi\rho\Pi=\rho$?
Suppose I have a density matrix $\rho$ and an orthogonal projector $\Pi$. Is it true that, if $tr(\Pi \rho) = 1$ then it must hold that
$$\Pi \rho \Pi = \rho$$?
If yes, how can I prove it?
- 327
5
votes
2
answers
400
views
Proof for Cardinality of the Clifford Group
In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ...
- 1,331
5
votes
2
answers
166
views
How to calculate the spectral norm of the density operator used in Molina et al. 2012 paper?
In Molina et al (2012)'s article on quantum money, the proof of security of Wiesner's quantum money scheme depends on the fact that the density operator
$$Q = \frac{1}{4}\sum_{k \in \{0, 1, +, -\}}\...
- 153
5
votes
1
answer
951
views
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 ...
5
votes
1
answer
228
views
What is the superop simulator in Qiskit for?
I'm trying to understand what the use case of a superop simulator would be. My understanding is that density matrix is generally more resource intensive than state vector, but it has additional ...
- 153
5
votes
1
answer
240
views
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 ...
- 1,725
5
votes
1
answer
307
views
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 ...
- 1,119
5
votes
2
answers
170
views
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 ...
- 1,725