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.
64 questions
22
votes
2
answers
23k
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&#...
- 17.8k
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 ...
- 17.8k
5
votes
2
answers
768
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:
$$\rho=\frac{1}{d}\left(\mathbb I +\...
glS♦
- 26.9k
5
votes
1
answer
2k
views
How to perform Quantum Process Tomography for three qubit gates?
I am trying to perform Quantum process tomography (QPT) on three qubit quantum gate. But I cannot find any relevant resource to follow and peform the experiment. I have checked Nielsen and Chuang's ...
7
votes
3
answers
3k
views
Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
- 1,641
6
votes
1
answer
627
views
Random quantum states and Schur-Weyl duality
Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator:
$$
\rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC.
$$
Let's ...
- 1,515
3
votes
1
answer
591
views
Density matrices of multiples copies of a single Haar-Random state
In Pseudorandom States, Non-Cloning Theorems and Quantum Money, the authors state that:
Let $\rho_\mu^m$ be the density matrix of a random $|\psi\rangle^{\otimes m}$ for $|\psi\rangle$ chosen from ...
- 7,969
9
votes
1
answer
4k
views
How to find a density matrix of a qubit?
If we are given a state of a qubit, how do we construct its density matrix?
- 2,225
3
votes
1
answer
533
views
Averaging over a single Haar-random unitary applied $t$ times
I'm trying to compute the following integral:
$$\int U^{\otimes t}\left|x_1,\cdots,x_t\middle\rangle\middle\langle x'_1,\cdots,x_n'\right|\left(U^\dagger\right)^{\otimes t}\,\mathrm{d}\mu(U)$$
Where $\...
- 7,969
14
votes
3
answers
6k
views
Density matrix after measurement on density matrix
Let's say Alice wants to send Bob a $|0\rangle$ with probability .5 and $|1\rangle$ also with probability .5. So after a qubit Alice prepares leaves her lab, the system could be represented by the ...
- 1,785
8
votes
4
answers
1k
views
Simulating a quantum circuit with decoherence and noise
Based on the answers given here and here, it is pretty clear that an arbitrary quantum circuit can be simulated with matrix algebra. The difficulty is that this assume perfect fidelity. I am unsure ...
- 701
7
votes
1
answer
913
views
Quantum teleportation with "noisy" entangled state
This is actually an exercise from Preskill (chapter 4, new version 4.4). So they are asking about the fidelity of teleporting a random pure quantum state from Bob to Alice, who both have one qubit of ...
- 555
7
votes
2
answers
829
views
How does quantum teleportation work with mixed shared states?
I am given the scenario that instead of the two parties (A & B) sharing the bell state $|\phi_+\rangle$ they share the mixture $\rho_\lambda = \lambda|\phi_+\rangle\langle\phi_+|+(1-\lambda)\frac{\...
- 73
7
votes
1
answer
268
views
Can one quantify entanglement between different parts of a system?
Consider some state $|\psi\rangle$ of $n$ qubits. One can take any subsystem $A$ and compute its density matrix $\rho_A =Tr_{B} |\psi\rangle \langle\psi|$. The entanglement between subsystem $A$ and ...
- 1,695
6
votes
3
answers
590
views
Can a CPTP map increase the purity of a state?
I am wondering if there exist CPTP maps $T$ such that the purity of a quantum state $\rho$ can increase, i.e.
$$ \text{tr} ( T ( \rho )^2 ) \geq \text{tr} ( \rho ^2). $$
If so, what are the conditions ...
- 61
6
votes
1
answer
1k
views
How is measurement modelled when using the density operator?
I've just learned about the density operator, and it seems like a fantastic way to represent the branching nature of measurement as simple algebraic manipulation. Unfortunately, I can't quite figure ...
- 4,228
4
votes
1
answer
208
views
How to find a separable decomposition for $|\Psi^+\rangle\!\langle\Psi^+|+|\Phi^+\rangle\!\langle\Phi^+|$?
The state
$$ \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) $...
- 1,641
3
votes
2
answers
925
views
What's the idea behind the unitary freedom of ensemble decompositions for density matrices?
I was reading the book by Nielsen & Chuang. I got the part about why we use the density operators. And then I got to the section of theorem 2.6. It says roughly this thing:-
The sets $|{\tilde\...
- 1,005
2
votes
2
answers
493
views
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:
$$ |\psi\rangle=\left(\dfrac{|0\rangle_1|0\rangle_2+ |1\rangle_1|1\rangle_2}{\sqrt{2}}\...
- 1,470
-1
votes
1
answer
242
views
How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?
How can I calculate the eigenvalues of $\rho^{T_{B}}$ (PPT) of the following state
$$
\rho =\frac{1}{2}|0\rangle\langle0|\otimes|+\rangle\langle+|+\frac{1}{2}|+\rangle\langle+|\otimes|1\rangle\langle1|...
- 545
18
votes
3
answers
8k
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:
$$ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10}^* & 1-\rho_{00} \...
- 181
11
votes
2
answers
874
views
What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?
For a Hilbert space $\mathcal{H}_A$, I have seen the phrase
density matrices acting on $\mathcal{H}_A$
multiple times, e.g. here.
It is clear to me that if $\mathcal{H}_A$ has finite Hilbert ...
- 519
9
votes
2
answers
668
views
Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$
I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
- 1,459
8
votes
4
answers
2k
views
Maximally mixed states for more than 1 qubit
For 1 qubit, the maximally mixed state is $\frac{\mathrm{I}}{2}$.
So, for two qubits, I assume the maximally mixed state is the maximally mixed state is $\frac{\mathrm{I}}{4}$?
Which is:
$\frac{1}{...
- 1,641
8
votes
2
answers
992
views
Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere
The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia:
Quantum ...
- 17.8k
7
votes
2
answers
4k
views
How to show a density matrix is in a pure/mixed state?
Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
- 869
6
votes
2
answers
2k
views
What are the conditions ensuring a two-qubit density matrix is positive semidefinite?
I've seen some papers writing
$$\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}^{...
- 3,109
6
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 ...
- 1,136
6
votes
2
answers
2k
views
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 ...
- 435
6
votes
1
answer
388
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
2
answers
2k
views
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 ...
- 334
5
votes
2
answers
206
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 ...
- 1,785
5
votes
2
answers
223
views
Does the definition of separability of pure states require the components of the summands to be pure?
Does the definition of separability of pure states require the components of the summands to be pure? More precisely, let $\rho$ be a pure state (i.e., $\rho=|\phi\rangle\langle\phi|$) on the space $...
- 217
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? (...
- 1,641
5
votes
1
answer
302
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 ...
- 1,785
5
votes
2
answers
350
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 ...
5
votes
3
answers
146
views
How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
How can one prove that $tr(A\mid\psi\rangle\langle\psi\mid)=\langle\psi\mid A\mid\psi\rangle$? In Nielsen/Chuang they mention this is due to Gram-Schmidt decomposition but I can’t understand how.
- 101
5
votes
0
answers
66
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-...
- 947
4
votes
3
answers
233
views
How to show that a density matrix $\rho$ is extreme iff $\rho=|\psi\rangle\!\langle\psi|$?
A density matrix $ρ$ is called extreme if the only way to write $ρ$ as $ρ = p σ + (1 − p) τ$ ,
with $σ ∈ S_d$, $τ ∈ S_d$, and $p ∈ (0, 1)$ is to have $ρ = σ = τ$ .
I want to show that a density matrix ...
- 383
4
votes
1
answer
204
views
Is the closest diagonal state to a given state always the dephased original state?
This question is about the following optimization problem:
Given some density matrix $\rho\in\mathbb C^{n\times n}$ find the diagonal state which is closest to it in trace norm. More precisely, find
$...
- 3,474
4
votes
1
answer
262
views
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♦
- 26.9k
4
votes
1
answer
987
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 ...
- 257
4
votes
1
answer
1k
views
How do I determine if a given pure two-qubit state is separable?
I'm trying to self-study some topics about quantum computing and I came across a topic of state separability. Talking about that, I wanted to determine separability on the following state (from Qiskit ...
- 264
3
votes
2
answers
611
views
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?
3
votes
7
answers
496
views
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,...
3
votes
2
answers
1k
views
Density matrix of a product of Bell states
Suppose I share two Bell states among two participants Alice and Bob in the following manner :
$$ |\psi\rangle=\left(\dfrac{|0\rangle_1|0\rangle_2+ |1\rangle_1|1\rangle_2}{\sqrt{2}}\right)\left(\...
- 1,470
3
votes
2
answers
297
views
What is the matrix representation for $n$-qubit gates?
Let's say I have more than one qbits $|0\rangle|1\rangle$ and I want to perform a $H$ on both of them. I know the matrix representation for the Hadamard on a single qbit is
$$\frac{1}{\sqrt{2}}\begin{...
- 225
3
votes
1
answer
138
views
Are there quantum algorithms which initial state is a mixed one?
In this answer, it is stated that it is not yet known how to efficiently classically simulate separable mixed states, a statement supported in a comment to this answer.
However, I can't imagine an ...
- 7,969
3
votes
1
answer
275
views
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 ...
- 668
3
votes
1
answer
140
views
Give an explicit derivation of the exact formula for the two-qubit absolute separability Hilbert-Schmidt probability $\approx 0.00365826$
The two-qubit eigenvalue condition of Verstraete, Audenaert, de Bie, and de Moor (arXiv) (p. 6) for absolute separability is
\begin{equation}
\lambda_1-\lambda_3 < 2 \sqrt{\lambda_2 \lambda_4},\...
- 947