# Tag Info

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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 ...

### Density matrices for pure states and mixed states

The motivation behind density matrices: 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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### 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 ...
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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 ...

### 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 ...

### 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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### 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 ...

### 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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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(... 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 + \...
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. $\rho=\rho^\dagger$, ...
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 ...