Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now

Questions tagged [density-matrix]

A density matrix is a matrix that can be used to describe a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

Filter by
Sorted by
Tagged with
4
votes
1answer
80 views

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=\...
2
votes
1answer
29 views

Quantum Fisher information for pure states query

Assume that a density matrix is given in its eigenbasis as $$\rho = \sum_{k}\lambda_k |k \rangle \langle k|.$$ On page 19 of this paper, it states that the Quantum Fisher Information is given as $$F_{...
3
votes
2answers
40 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 ...
1
vote
1answer
65 views

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| * \log( \sum_z |z\rangle\langle z| \sum_x |\langle x|z \...
3
votes
2answers
63 views

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 ...
1
vote
0answers
42 views

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. ...
0
votes
1answer
51 views

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 ...
3
votes
1answer
121 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}=...
2
votes
2answers
149 views

Trace preserving condition in Choi's thorem

Choi's theorem states that any completely positive map $\Phi(\cdot) : C^*_{n\times n} \rightarrow C^*_{m \times m}$ can be expressed as $\Phi(\rho) = \sum_{j=1}^r F_j^\dagger \rho F_j$, for some $n \...
5
votes
1answer
70 views

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....
2
votes
2answers
67 views

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?
0
votes
1answer
82 views

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$, $ ...
0
votes
1answer
84 views

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. ...
1
vote
1answer
22 views

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{\...
0
votes
0answers
59 views

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 ...
4
votes
1answer
148 views

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_{...
5
votes
1answer
67 views

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 (...
4
votes
1answer
142 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 ...
4
votes
1answer
52 views

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 ...
2
votes
1answer
95 views

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\...
5
votes
1answer
85 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 $...
2
votes
1answer
39 views

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?
3
votes
0answers
26 views

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( \sum_{i=0}...
2
votes
2answers
80 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 +\...
4
votes
2answers
566 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 ...
2
votes
1answer
54 views

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{...
2
votes
2answers
134 views

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 ...
5
votes
1answer
120 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 ...
2
votes
0answers
29 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-...
1
vote
1answer
205 views

Computing von Neumann entropy of pure state in 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 \...
2
votes
1answer
62 views

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\...
1
vote
3answers
115 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 ...
4
votes
1answer
101 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 ...
1
vote
1answer
59 views

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. ...
4
votes
1answer
62 views

Convex Combination of Separable States

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) $...
4
votes
2answers
100 views

Non-uniqueness of pure states ensemble decomposition

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 ...
5
votes
1answer
108 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 ...
6
votes
2answers
480 views

Partial trace over a product of matrices - one factor is in tensor product form

$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$ 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
vote
2answers
577 views

Measuring but not looking at the 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 ...
7
votes
3answers
544 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 ...
3
votes
1answer
149 views

Total mutual information of a quantum system

In the discussions about quantum correlations, particularly beyond entanglement (discord, dissonance e.t.c), one can often meet two definitions of mutual information of a quantum system $\rho^{AB}$: ...
5
votes
4answers
400 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}{...
4
votes
1answer
51 views

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-...
4
votes
1answer
123 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 ...
2
votes
2answers
857 views

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.
1
vote
1answer
264 views

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: ...
7
votes
1answer
102 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 ...
6
votes
1answer
181 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 ...
6
votes
2answers
145 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
votes
1answer
180 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 ...