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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 ...
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12 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 & \...
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12 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 ...
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11 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 ...
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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 ...
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10 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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9 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 $\...
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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|+\...
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8 votes
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Partial trace over a product of matrices - one factor is in tensor product form

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 ...
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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 ...
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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 ...
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8 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 ...
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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 ...
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7 votes

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

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 ...
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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 ...
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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}....
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7 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 $...
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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(...
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6 votes

Density matrix after measurement on density matrix

So Alice sends Bob a qubit with the density matrix $$\rho = \frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1| = \begin{bmatrix} .5 & 0 \\ 0 & .5 \end{bmatrix}$$ as you said. (I'...
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6 votes

Why do purifications only differ by a local unitary?

$\newcommand{\ket}[1]{\vert#1\rangle}$ First, write $\ket\psi$ and $\ket{\tilde\psi}$ in their Schmidt decomposition: $$ \begin{aligned} \ket\psi &= \sum \lambda_i \ket{a_i}\ket{b_i}\ , \\ \ket{\...
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Semi-definite program for smooth min-entropy

You do not need a double objective function to solve this. Given $\rho_{AB}$ let $\rho_{ABC}$ be any purification of $\rho_{AB}$. Then we can write the smooth min-entropy as the following SDP \begin{...
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  • 3,902
6 votes

How can we upper bound the norm of a partial trace?

The $1$-norm decreases under partial trace and so we have an upper bound of $1$ when the states are normalized, $$ \|\mathrm{Tr}_B[|\psi_1\rangle \langle \psi_2|]\|_1 \leq \||\psi_1\rangle \langle \...
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  • 3,902
6 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 ...
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6 votes

Calculate the von Neumann Entropy of a two-qubit entangled state

You are calculating the entropy of one of the marginal states and so you would not expect the answer to be independent of $\theta$, except in the case that $|\psi\rangle = |\phi_A\rangle \otimes |\...
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6 votes
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What is the superop simulator in Qiskit for?

If you have an ideal quantum circuit, you can easily get its superoperator representation using qiskit.quantum_info.SuperOp as follows, ...
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6 votes
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How to find the distance between a given $\rho$ and the nearest pure state(s)?

Recall that for any Hermitian operator $A$ and any unit vector $|\psi\rangle$ the real number $\langle \psi|A|\psi\rangle$, known as the Rayleigh quotient, is bounded by the largest eigenvalue $\...
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