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.
71 questions from the last 365 days
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Prove that spectral decomposition is the minimal ensemble decomposition
I understand that the spectral decomposition of a density matrix
$\rho$ expresses it in terms of its eigenvalues and eigenvectors:
$$\rho=\sum_i\lambda_i\left|\psi_i\middle\rangle\!\middle\langle\psi\...
2
votes
2
answers
657
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A proof that 4 ≥ ∞ when using the Quantum One-Time Pad
A cryptographic scheme using a $n$-bit key to hide a $m$-bit plaintext is said to be perfectly secret when, without this key, we cannot get any information about the plaintext from the ciphertext. ...
0
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0
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25
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How do I analytically calculate the fidelity of a GHZ state produced by the photonic cluster state generation circuit?
In this paper, a GHZ state may be produced by a circuit where a spin qubit is entangled with N photon qubits by CNOTs.
How would I calculate the fidelity of a GHZ produced by this protocol, where an ...
- 143
0
votes
0
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60
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How to entangle two separate set into one?
I have two sets of qubits where the information is encoded in amplitude. How can I entangle them into one to save qubits.
The information $k_0,k_1,k_2,k_3$ and $k'_0,k'_1,k'_2,k'_3$ are encoding in ...
- 39
-1
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1
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48
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how to mix (or time average) two density matrix?
Given two density matrix $\rho_1,\rho_2$ with the same size, how to get a mix state of the two matrix,
$$
\rho = \frac12 (\rho_1+\rho_2)?
$$
e.g. there are two quantum channel both of them have 4 ...
- 39
0
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0
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40
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Can density matrices have transcendental off-diagonal elements?
Hello everyone,
I’ve been exploring parameterized density matrices and was curious about the conditions under which they remain valid. For example, consider the following 2x2 density matrix ...
1
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1
answer
45
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Computing the expected value of a spin - 1 particle component given density matrix
I have a density matrix $\rho$ where
$$\rho = \frac{1}{4} \cdot \begin{pmatrix} 2 & 1 & 1\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{pmatrix}$$
and the x component of a spin - 1 particle ...
- 33
2
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0
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30
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Set of `reachable` states from an initial density matrix with polynomial elements
I've been reading about the Bernoulli-factory problem and I'm particularly interested in deriving the results using the density matrix formalism, i.e., given required numbers of copies of the initial ...
- 21
0
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0
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31
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Discarding a Quantum register to extract kernel matrix in practical quantum computer
Question: How to extract a kernel matrix from a quantum state on a real quantum computer through discarding a register?
I am trying to understand the paper "Quantum Support Vector Machine for Big ...
- 9
2
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1
answer
144
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Does the trace distance between these averages of pure states satisfy $|\rho - \sigma|_1 \geq \frac{N_a - N_b}{N_a}$?
Consider two $n$ qubit density matrices:
$$
\rho = \frac{1}{N_a} \left(\sum_{i=1}^{N_a} |\psi_i\rangle \langle \psi_i| \right).
$$
$$
\sigma = \frac{1}{N_b} \left(\sum_{i=1}^{N_b} |\phi_i\rangle \...
- 1,515
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0
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36
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How to generate random quantum states in matlab?
I was wondering if there is some academic standard/any way of generating random n times n q-states/density matrices in Matlab without using any other package then QETLAB.
- 325
3
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1
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110
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Density Matrices for states $|+\rangle$ and state represented by $\rho = \frac{|0\rangle \langle0| + |1\rangle \langle1|}{2}$
As per my understanding, the first one is a "pure state" and represents a system with one qubit having equal probability of being measured as $|0\rangle$ or $|1\rangle$ (standard basis ...
3
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1
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101
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Finding a density matrix for a distribution of pure states
Let $\theta$ be a Gaussian variable with mean 0 and variance 1. Then for $t>0$, the variable $\theta \sqrt{t}$ is also Gaussian with mean $0$ and variance $t$. Let $|\psi_0\rangle$ be an arbitrary ...
- 3,284
4
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1
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204
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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
1
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1
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152
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Mathematical properties used to derive Kraus operators
In this answer, it was very well explained why Kraus operators are not numbers as it might seem when reading Nielsen and Chuang for the first time. I have a minor, purely technical and probably simple ...
- 53
2
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2
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186
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Fallacy of special significance of eigenvalues and eigenvectors of density operator
This question is an addition to the following question.
Nielsen and Chuang open the discussion of the unitary freedom on the ensemble for density matrices by pointing out the common fallacy to suppose ...
- 53
1
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0
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40
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Applying 2-qubit noise to a multi qubit system using local operations in Python
I am trying to create a simulation of 2-qubit dephasing noise that acts on the first two qubits of a multi qubit system. I would like to do this locally rather than expanding my kraus operators into $...
1
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1
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129
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Is a density matrix still positive semidefinite after applying a projection operator?
Suppose we have a density operator $\hat{\rho}$ and a projection operator $\hat{\Pi}$, are the matrices
$$\hat{\rho}'=\hat{\Pi}\hat{\rho}\hat{\Pi}^{\dagger}$$
and
$$\hat{\rho}''=(\hat{I}-\hat{\Pi})\...
- 391
2
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1
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114
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Can a quantum operation inflate the Bloch sphere?
The depolarizing noise channel uniformly deflates the Bloch sphere to a single point, which is $\mathbf{n}= (0,0,0)$ or in terms of quantum qubit states, we get a maximally mixed state $\rho = \frac{1}...
- 3,284
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0
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Help to understand a QFI derivation
Can anyone help me understand the QFI derivation being done in Appendix C of this paper?
The density matrix $\rho = \frac12(|\psi_1\rangle\langle\psi_1|+|\psi_2\rangle\langle\psi_2|)$. I understand ...
- 11
0
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1
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48
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Understanding how to calculate partial trace
W.r.t. this:
I would like to understand how equation 6.29 follows from 6.28. Can anyone explain this to me?
- 391
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0
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38
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Why is my mutual information negative in Python, and how can I accommodate that?
I'm calculating the mutual information between two 1 qubit subsystems in a quantum state using Python. Theoretically, mutual information should always be non-negative. However, I'm encountering very ...
- 47
1
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0
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42
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Alice and Bob play a Multi-Qubit game
Well I am quite new to this so excuse me if the question is absurd
Alice and Bob each can "measure" variables A and B respectively: Alice can use $a_1$ and $a_2$ methods of measurement while ...
- 11
4
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0
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57
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What is the minimum number of separable states (not necessarily pure) needed to decompose arbitrary separable states?
For a bipartite separable quantum state $\rho$ acting on Hilbert space $H\otimes H'$ with $\dim H=D$ and $\dim
H'=D'$, what is the minimum number of separable state needed for a decomposition? That ...
- 41
3
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3
answers
304
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Single-qubit quantum channel from the CNOT gate
I am studying quantum noise, chapter $8$ in Nielsen and Chuang. Section $8.2.2$ introduces an example for the definition of quantum operations, in particular the CX gate is introduced as an example. I ...
- 31
2
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0
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88
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Is there a theoretical method to achieve a positive semi-definite density matrix in QST?
The problem of encountering negative eigenvalues in the density matrix during Quantum State Tomography (QST) is well-explained in this Quantum Computing Stack Exchange post.
However, I am seeking ...
- 47
0
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1
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79
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How to convert from choi to chi matrix in qiskit
I have done a quantum process tomography experiment on a two qubit system.
...
0
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0
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42
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Tests of entanglement between one and many qubits
I have a 5-qubit state $|\psi \rangle$, which has a physical interpretation that the middle qubit is the "impurity" with spin $s_{imp} = 1/2$. The rest 4 qubits highlight the presence of ...
2
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2
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106
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Can non-linear operations be implemented as a circuit on a quantum computer?
Suppose I have a Quantum circuit, which gives an output state $|\psi \rangle$ let's say. I wish to obtain the reduced density matrix by tracing out subsystem B, i.e. $\rho = |\psi \rangle \langle \psi ...
2
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1
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69
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Density matrix $\rho = I/2$ implies an ensemble of orthonormal states
Suppose that a density matrix $\rho = I/2$ is obtained as a description of an ensemble of two pure states. How can I show that the ensemble must then be of the form:
$$
\{(|\psi\rangle, 1/2), (|\psi^\...
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4
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2
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93
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Why do minimal ensemble decompositions for $\rho$ contain $|\psi⟩\in{\rm supp}(\rho)$ with probability $1/\langle\psi|\rho^{-1}|\psi⟩?$
I came across the following exercise (2.73) in Nielsen & Chuang and am trying to understand it intuitively.
Here is my reasoning of what is going on:
The purpose of this exercise:
Let’s say we are ...
- 173
1
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2
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116
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What is meant with "different ensembles can give rise to the same density matrix?"
I am reading the Nielsen & Chuang section on density matrices and I don't understand the example given to demonstrate a concept. Here is what I am reading:
First, they said these two different ...
- 173
2
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0
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76
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Reasoning behind unitary freedom in the ensemble for density matrices theorem
Although my question has the same title of a different question, it is not a duplicate. I am asking a different question. I don't care why it made it into the book.
Here is a theorem from Nielsen &...
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2
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2
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516
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How to calculate the Schmidt decomposition of a state without SVD
I have this state of two qubits here:
$$
|\psi_{AB}\rangle = \frac{1}{2}(|0\rangle_A |0\rangle_B + |1\rangle_A |1\rangle_B + |1\rangle_A |0\rangle_B - |0\rangle_A |1\rangle_B)
$$
Which means that the ...
3
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3
answers
75
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Interpretation of a circuit that yields the same result for initializations $|+\rangle$ and $|-\rangle$
How can I interpret a quantum circuit that results in the same state for the initialization $\newcommand{\ket}[1]{|#1\rangle}\newcommand{\bra}[1]{\langle #1|}\ket{+}$ and $\ket{-}$?
For example, the ...
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1
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1
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What is the density matrix of a GHZ state when onle a qubit is in a decoherence channel?
Suppose Alice, Bob and Rob share a GHZ state. Now consider Rob's qubit is in a bit-flip channel. How to obtain the density matrix in this senario? Also i would be glad to get some articles adrresing ...
- 791
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0
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Particle number expectation value in QuTip
I am learning now to use QuTiP by going through their documentation site. I am trying to understand what does the argument - particle number expectation value in thermal density matrix do? How does it ...
- 33
2
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2
answers
133
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Can different density matrices have 100% fidelity with a given pure state?
I am trying to understand fidelity a bit better, to do so consider the bell state:
$$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle-|10\rangle\right),$$
the density matrix associated with this state ...
- 391
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0
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30
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Qiskit: Density matrix's dimension is too large
My circuit is with 18 qubits. I want to find density matrix by calling DensityMatrix.from_instruction(qc) but then I get an error "Maximum supported dimension for an ndarray is 32, found 36"....
- 157
1
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1
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382
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Exponential Quantum Speedup for the Traveling Salesman Problem - where is the catch?
Such an article claims that an NP-complete problem can be solved efficiently.
Is it real?
I noticed that they prepare a state $|0\rangle\langle0|+|1\rangle\langle1|$ on an ancilla, which is impossible ...
- 1,502
2
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1
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60
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Why is the trace distance between two density matrices not always $0$?
If $|A|_{tr}=Tr(\sqrt{A^\dagger A})$ then surely
$$
|\rho_1-\rho_2|_{tr}=Tr(\sqrt{(\rho_1-\rho_2)^\dagger (\rho_1-\rho_2)})
$$
$$
=Tr(\sqrt{(\rho_1^\dagger -\rho_2^\dagger)(\rho_1-\rho_2)})
$$
$$
=Tr(\...
- 123
3
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1
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140
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What's the Schmidt decomposition of $|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$?
$|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle) $
I absolutely cannot figure out the Schmidt decomposition of this state. I have looked at a ton of ...
- 31
2
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1
answer
55
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If states are close together does there always exist a channel close to the identity mapping one to the other?
Question: Given states $\rho,\omega\in\mathbb C^{n\times n}$ and $\varepsilon>0$ such that $\rho$ and $\omega$ are $\varepsilon$-close in trace norm does there exist a channel $\Phi$ with $\Phi(\...
- 3,474
3
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1
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59
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How to prove the inclusion relation $\text{Im} (\rho) \subseteq \text{Im} (\rho[X] \otimes \rho[Y])$ about density operators?
For $\rho \in \mathrm{D}(\mathcal{X} \otimes \mathcal{Y})$ denoting an arbitrary state of the pair $(\mathrm{X}, \mathrm{Y})$, how to prove the fact
$\text{Im} (\rho) \subseteq \text{Im} (\rho[X] \...
- 133
2
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1
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119
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Is $\rho = \sum_{j} p_j|n_j\rangle\langle n_j|$ a valid construction for any mixed state?
I have a mixed state $\rho$ and its hamiltonian $H$. Firstly, I find the eigenvalues $\{p_j\}$ of $\rho$, and orthonormal basis of $H$. I write $\rho$ in terms of $H$'s eigenstates and $\rho$'s ...
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1
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1
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156
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Efficient Clifford simulation and entropy of reduced density matrices
Suppose I have a Clifford circuit $C$ and I want to estimate the entanglement entropy of a subset of two qubits, say, $\{q_0, q_1\}$, i.e. the quantity $$S(\rho_{q_0 q_1}) = - \text{Tr}[\rho_{q_0 q_1} ...
- 368
1
vote
1
answer
53
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relationship between helstrom operators acting on different pairs of quantum states
Let $\rho_1, \rho_2, \rho_3, \rho_4$ be arbitrary single-qubit density matrices.
Let $A$ be an Hermitian operator and its spectral decomposition as $A = \sum_i \lambda_i \lvert i \rangle \langle i \...
- 127
0
votes
0
answers
44
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Better optimization of bounds on sums of Pauli strings?
I'm trying to bound a quantity $||\sum_i \alpha_i P_i ||$ where the $P_i$ are arbitrary Pauli strings, $||.||$ is the operator norm (max eigenvalue) and $\alpha_i$ are arbitrary real coefficients.
If ...
- 133
1
vote
0
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87
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State tomography in Qiskit on a subset of qubits of real QPU
Could anyone please explain how should I carry out a state tomography on a real device in Qiskit (version 0.43.2)?
I have access to devices with 127 qubits, but I want to perform a simulation using ...
- 11
0
votes
1
answer
69
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Density Matrix for a Quantum Circuit with Clifford Gates and a $T$ Gate in Qiskit
I am trying to analyze the impact of a single $T$ gate within a quantum circuit that primarily consists of Clifford gates. My goal is to understand the $T$ gate's role in $T$-design and Anti-...
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