The density operator describes a quantum system in an (in general mixed) state.
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102 views
Why is this not a realisable operation on a quantum system?
Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
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2answers
80 views
Density Operator, Expectation Value, Coherent States
How would I go about evaluating expectation values like $\langle X \rangle$ and $\langle P \rangle$?
Work I've done:
I've done the integration over $\phi$ and rewrote $\rho$ as:
$\rho = ...
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2answers
399 views
What does the sum of two qubits tell about their correlations?
How much can I learn about correlations between two quits by measuring
the sum of their values? What is the best way to formalize such a
question?
Below is my original, longer formulation of ...
2
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1answer
98 views
Energy density of a quantum mechanical ensemble
How do we determine the energy density of a given system? I have seen that the density operator
$$\rho~=~\frac{\exp(-\beta \hat{H})}{\text{tr}(\exp(-\beta \hat{H}))}.$$
What does this mean exactly ...
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2answers
170 views
Does the reduced density matrix describes a real mixed state?
Suppose that we have two entangled particles A and B with pure state vector
$|\psi\rangle=a|0\rangle_A |1\rangle_B + b|1\rangle_A |0\rangle_B \hspace{1cm}(1)$
When we take the partial trace over the ...
3
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1answer
158 views
The effect of Quantum Decoherence on density operators
Suppose we have a qubit in state $| \Psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$
Suppose we expose this to decoherence, which we will express as the state $| R \rangle$ such that
$$| 0 ...
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votes
2answers
143 views
Sum of two density matrices: $\rho=p_1\rho_1+p_2\rho_2$
Suppose we have
$$\rho=p_1\rho_1+p_2\rho_2$$
Where $\rho_1$ and $\rho_2$ are density matrices with $p_1+p_2=1$
I'm trying to show this is also a density matrix
If we let
$$\rho_1=\sum_i^n ...
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2answers
167 views
Is it possible to write a Density Matrix in the following form?
Is it possible to write an arbitrary density matrix $\hat{\rho}$ in the following form ?
$$\hat{\rho} ~=~ \frac{1}{N} \sum_{\ell=1}^N \left|x_{\ell}\right\rangle \left\langle x_{\ell}\right|,$$
...
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0answers
57 views
What is the link between the density matrix and Hestenes' spinors in geometric algebra?
The density matrix (or state matrix) is a generalization of a wave function that is able to describe incoherent superpositions of an N-state system. It is often written as a matrix and observables are ...
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2answers
211 views
What is the physical interpretation of the density matrix in a double continuous basis $|\alpha\rangle$, $|\beta\rangle$?
(a) Any textbook gives the interpretation of the density matrix in a single continuous basis $|\alpha\rangle$:
The diagonal elements $\rho(\alpha, \alpha) = \langle \alpha |\hat{\rho}| \alpha ...
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3answers
251 views
Takhatajan's mathematical formulation of quantum mechanics
So I began skimming L. Takhatajan's Quantum Mechanics For Mathematicians, and saw the mathematical formulation of QM that he uses (page 51). (The PDF file is available here.)
I've only taken a basic ...
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1answer
129 views
Show that purity = 1 in a pure state
How can you show that for any pure state, the purity = 1?
Pure state: $\rho^2 = \rho$ and $Tr(\rho^2)=1$
Mixed state: $\rho^2 = \rho$ and $Tr(\rho^2)<1$
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6answers
551 views
Is the density operator a mathematical convenience or a 'fundamental' aspect of quantum mechanics?
In quantum mechanics, one makes the distinction between mixed states and pure states. A classic example of a mixed state is a beam of photons in which 50% have spin in the positive $z$-direction and ...
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67 views
Shape of the state space under different tensor products
I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this).
Recall: In a ...
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1answer
67 views
Krauss operators for random unitary
Suppose I have a density matrix $\rho$ and I act on it with a unitary matrix that is chosen randomly, and with even probability, from $S = \{ H_1, H_2 \ldots H_N \}$. I want to write the operation on ...
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1answer
154 views
Reduced density matrices for free fermions are thermal
Many recent papers study entanglement in eigenstates of fermionic free hamiltonians (normally on a lattice) using the basic assumption that the reduced density matrices are thermal (e.g. Peschel ...
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3answers
135 views
Hilbert-Schmidt basis for many qubits - reference
Every density matrix of $n$ qubits can be written in the following way
$$\hat{\rho}=\frac{1}{2^n}\sum_{i_1,i_2,\ldots,i_n=0}^3 t_{i_1i_2\ldots i_n} ...
11
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1answer
113 views
Majorana-like representation for mixed symmetric states?
Is there a generalization of the Majorana representation of pure symmetric $n$-qubit states to mixed states (made of pure symmetric $n$-qubit)?
By Majorana representation I mean the decomposition of ...
