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Density matrices for pure states and mixed states

Motivation The motivation behind density matrices is to represent a lack of knowledge about the state of a given quantum system, encapsulating within a single description of this system all the ...
DaftWullie's user avatar
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15 votes
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What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?

It is common that one refers to a density matrix (or, equivalently, a density operator) $\rho$ as acting on a particular space $\mathcal{H}$. This serves to establish the "type" of $\rho$ in computer ...
John Watrous's user avatar
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15 votes
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What is a "maximally mixed state"?

The maximally mixed state is a quantum state whose density matrix is proportional to the identity matrix. Physically, it may be interpreted as a uniform mixture of states in an orthonormal basis. The ...
Adam Zalcman's user avatar
14 votes
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Density matrix after measurement on density matrix

So, Bob is given the following state (also called the maximally-mixed state): $\rho = \frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1| = \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \...
ahelwer's user avatar
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13 votes
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How to check if a matrix is a valid density matrix?

If a matrix has unit trace and if it is positive semi-definite (and Hermitian) then it is a valid density matrix. More specifically check if the matrix is Hermitian; find the eigenvalues of the matrix ...
biryani's user avatar
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13 votes

What do the off-diagonal elements of a density matrix physically represent?

To put it very shortly, non-zero off-diagonal elements of the density matrix signify that your system features a quantum superposition between the elements of the basis that you chose to represent $\...
Carlos_San's user avatar
12 votes
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

The equation at the top of the question is not correct: there is a missing factor of $1/d$ on the right-hand side. Let's eliminate this factor from the left-hand side to make it simpler, so that the ...
John Watrous's user avatar
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11 votes

Density matrices for pure states and mixed states

The motivation behind density matrices[1]: In quantum mechanics, the state of a quantum system is represented by a state vector, denoted $|\psi\rangle$ (and pronounced ket). A quantum system with a ...
Sanchayan Dutta's user avatar
11 votes
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How to show a density matrix is in a pure/mixed state?

First, the example that you give is not a density matrix (they must have trace 1). Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are ...
DaftWullie's user avatar
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11 votes

What do the off-diagonal elements of a density matrix physically represent?

It can represent 'various' things depending upon the physical system and the context. For instance: (1) For a 'closed and isolated' system (let us say a qubit), it represents the 'coherence' between ...
Siddhant Singh's user avatar
10 votes
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How to perform Quantum Process Tomography for three qubit gates?

I am sure that since you are asking this question you probably already understand this, but for future & other's reference let me give a quick recap of what we are trying to achieve. Quantum ...
JSdJ's user avatar
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9 votes

How do I construct a Density Matrix corresponding to a Hamiltonian?

Your question remains very unclear as to what it actually is that you want to calculate. There is no direct correspondence between a system Hamiltonian and the quantum state of the system. No matter ...
DaftWullie's user avatar
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9 votes
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How to find a density matrix of a qubit?

If you're given $|\psi\rangle$, just calculate $\rho=|\psi\rangle\langle\psi|$. For example, $|\psi\rangle=\cos\theta|0\rangle+\sin\theta e^{i\phi}|1\rangle$, then $\langle\psi|=\cos\theta\langle 0|+\...
DaftWullie's user avatar
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9 votes

Maximally mixed states for more than 1 qubit

For two probability distributions, there is a clear notion how to say which one is more mixed: $\vec p$ is more mixed than $\vec q$ if it can be obtained from $\vec p$ by a mixing process, this is, a ...
Norbert Schuch's user avatar
9 votes

How to show a density matrix is in a pure/mixed state?

By spectral theorem density matrices are diagonizable, since they are hermitian (also they are positive semi-definite and have trace 1). That means that there is a set of $n$ non-negative eigenvalues $...
Danylo Y's user avatar
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9 votes
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What's the 'physical consistency' in the partial trace scenario?

Measurement average Measurement average $\langle M \rangle_\rho$ of observable (a Hermitian operator) $M$ on the state $\rho$ is the average of measurement outcomes $m$ in the limit of infinite number ...
Adam Zalcman's user avatar
9 votes

Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?

Yes, you can use Cauchy interlacing theorem to prove that. Let $M = \sigma - \rho$ and dimension of the space is $n$. In an appropriate basis $\Pi = I_k \oplus 0_{n-k}$. Assume $k=n-1$ (in general ...
Danylo Y's user avatar
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9 votes

Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?

Here is a slightly more general alternative to Danylo's answer. The trace norm satisfies the inequality $$ \| X Y Z \|_1 \leq \|X\| \|Y\|_1 \|Z\| $$ for all choices of operators $X$, $Y$, and $Z$ that ...
John Watrous's user avatar
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9 votes
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Collapse of the density operator

That's a very interesting observation you made. The issue boils down to the fact that your first method of computing a post-measurement state normalizes "too early". Effectively, in your ...
Rammus's user avatar
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9 votes

Can a CPTP map increase the purity of a state?

Yes, some quantum channels can increase purity. For example the preparation channel $$ T(X) = \mathrm{Tr}[X] |\psi\rangle \langle \psi| $$ that can be thought of as throwing away your system and ...
Rammus's user avatar
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8 votes

Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$

Marsl is correct, and his "hint" is really more a sketch of a solution than a hint. Rather than viewing the question or its solution as just formal algebra, you can also approach his same solution ...
Greg Kuperberg's user avatar
8 votes

Why are $d^2$ dimensions required to describe a density matrix?

It appears that you have some confusion regarding the basic notions of density operators and "dimension". Why $d^2$ dimensions "are required to describe" a density matrix isn't the right question to ...
Sanchayan Dutta's user avatar
8 votes
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How to prepare mixed states on a quantum computer?

I don't know if that's still useful but I've been asking this to myself recently and I've found a simple answer. If you want to prepare the mixed state $$\rho = \frac{1}{d}\sum_{i}^{d} |i\rangle\...
ikiga1's user avatar
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8 votes

What is a "maximally mixed state"?

Indeed, there are partially mixed states, although this terminology is seldom used. One can define measures of mixedness to make this quantitative. The best one is the von Neumann entropy, defined as $...
Mateus Araújo's user avatar
8 votes
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What is square of the square root of a density matrix?

If $\rho$ is a density matrix, then $\sqrt{\rho^2} = \rho$. To see why this is, let's start with the definition of the square root of a matrix. Ordinarily, if $A$ is a square matrix, there may be ...
John Watrous's user avatar
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8 votes

What is the density matrix of a pure state?

You can determine whether a state is pure or mixed by considering the purity $\gamma$ which is defined as the trace (i.e. the sum of diagonal entries) of the density matrix squared. \begin{equation} \...
Callum's user avatar
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7 votes

Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

Here the important fact is that the maximally mixed state is in fact an identity matrix. Let me rewrite the expression on the left in index notation (the summation sign is omitted according to the ...
Alexey Uvarov's user avatar
7 votes
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How can pure state ensemble decompositions not be unique?

Suppose that we're talking about $n\times n$ density operators, so that the rank will never exceed $n$. Now suppose that you choose $N$ to be much larger than $n$, and then arbitrarily pick a ...
John Watrous's user avatar
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7 votes
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Purity of mixed states as a function of radial distance from origin of Bloch ball

A density matrix $\rho$ has the properties of being Hermitian, non-negative and has trace 1. Any $2\times 2$ matrix can be written in the form $$ \rho=\frac{n_0\mathbb{I}+\vec{n}\cdot\vec{\sigma}}{2}....
DaftWullie's user avatar
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7 votes
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Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$

Hint: To make your induction work, write $$\eqalign{p^{\otimes n} - q^{\otimes n} & = & \left(p^{\otimes(n-1)}\otimes p \right)-\left(q^{\otimes (n-1)} \otimes q\right)\\ & = & \left(...
Marsl's user avatar
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