14
votes
Accepted
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 & \...
- 4,028
14
votes
Accepted
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 ...
- 5,083
14
votes
Accepted
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 ...
- 54.5k
13
votes
Accepted
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 ...
- 936
12
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 $\...
- 129
12
votes
Accepted
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 ...
- 20k
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 ...
- 17.1k
10
votes
Accepted
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 ...
- 5,083
10
votes
Accepted
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 ...
- 54.5k
10
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 ...
- 1,735
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 ...
- 54.5k
9
votes
Accepted
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|+\...
- 54.5k
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 ...
- 5,608
9
votes
Accepted
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 ...
- 20k
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 ...
- 6,611
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 ...
- 5,083
8
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 $...
- 6,611
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 ...
- 1,251
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 ...
- 17.1k
8
votes
Accepted
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
$$\rho = \frac{1}{d}\sum_{i}^{d} |i\rangle\...
- 126
8
votes
Accepted
How to perform Quantum Process Tomography for three qubit gates?
I am sure that since you are asking this question you probably already understand this, but for future & other's reference let me give a quick recap of what we are trying to achieve.
Quantum ...
- 5,289
8
votes
Accepted
Collapse of the density operator
That's a very interesting observation you made. The issue boils down to the fact that your first method of computing a post-measurement state normalizes "too early". Effectively, in your ...
- 5,201
8
votes
What is the density matrix of a pure state?
You can determine whether a state is pure or mixed by considering the purity $\gamma$ which is defined as the trace (i.e. the sum of diagonal entries) of the density matrix squared.
\begin{equation}
\...
- 683
7
votes
Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
Here the important fact is that the maximally mixed state is in fact an identity matrix.
Let me rewrite the expression on the left in index notation (the summation sign is omitted according to the ...
- 674
7
votes
Accepted
How can pure state ensemble decompositions not be unique?
Suppose that we're talking about $n\times n$ density operators, so that the rank will never exceed $n$. Now suppose that you choose $N$ to be much larger than $n$, and then arbitrarily pick a ...
- 5,083
7
votes
Accepted
Purity of mixed states as a function of radial distance from origin of Bloch ball
A density matrix $\rho$ has the properties of being Hermitian, non-negative and has trace 1.
Any $2\times 2$ matrix can be written in the form
$$
\rho=\frac{n_0\mathbb{I}+\vec{n}\cdot\vec{\sigma}}{2}....
- 54.5k
7
votes
Accepted
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
Hint: To make your induction work, write
$$\eqalign{p^{\otimes n} - q^{\otimes n} & = & \left(p^{\otimes(n-1)}\otimes p \right)-\left(q^{\otimes (n-1)} \otimes q\right)\\
& = & \left(...
- 869
7
votes
Accepted
Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$
Background
If $v_1, v_2, \dots, v_n$ is an orthonormal basis in the inner product space $V$, then any vector $u\in V$ can be expressed as a linear combination
$$
u = \alpha_1 v_1 + \alpha_2 v_2 + \...
- 20k
7
votes
Accepted
How to square a density matrix?
Squaring matrices works the same way as squaring numbers, i.e. you multiply the matrix by itself. Formally, $A^2=A\cdot A$. However, every density matrix $\rho$ is Hermitian, i.e. $\rho=\rho^\dagger$, ...
- 20k
7
votes
Accepted
what is square root of a density matrix power two?
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 ...
- 5,083
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