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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.

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9
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2answers
4k 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&#...
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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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How to check if a matrix is a valid density matrix?

What conditions must a matrix hold to be considered a valid density matrix?
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240 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 (...
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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
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2answers
118 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
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507 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 ...
6
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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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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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149 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{|...
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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 ...
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4answers
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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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1answer
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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 ...
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1answer
70 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 (...
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1answer
73 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....
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132 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 ...
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2answers
233 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 ...
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1answer
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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 $...
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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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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 ...
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1answer
154 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_{...
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1answer
115 views

How to calculate the off-diagonal elements of a density matrix using the measurement result?

For example, one have measured some states like $|0\rangle$ in the computational basis for many times and got the approximate probability of getting 0 and 1 ($P(0)$ and $P(1)$). Then how does he ...
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1answer
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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=\...
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1answer
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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 ...
4
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1answer
64 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
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1answer
129 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 ...
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1answer
174 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 ...
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1answer
53 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 ...
4
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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-...
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1answer
60 views

Modeling energy relaxation effects with density matrix formalism

I know there are measures that can be taken to mitigate the effects of dephasing (I'm referring here to Dynamic Decoupling and the other ideas it led to). I find it surprising that there is no ...
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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 for a graph $G = (V,E)$ and $B = V\setminus A$ the corresponding complement ...
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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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65 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 ...
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1answer
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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}=...
3
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1answer
99 views

What can we know about the eigenvalues of a reduced density matrix knowing the eigenvalues of the original matrix?

What information can we get out about the eigenvalues of a reduced density matrix knowing the eigenvalues of the original matrix? For example, it can be proved that if all the eigenvalues of a ...
3
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1answer
155 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 $\rho^{AB}$: ...
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1answer
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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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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}...
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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.
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2answers
150 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 \...
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2answers
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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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1answer
33 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_{...
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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 +\...
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2answers
150 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 ...
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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?
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1answer
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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\...
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1answer
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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{...
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1answer
65 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\...
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2answers
81 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?
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0answers
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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-...