# Tag Info

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 ...
• 59.4k
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 ...
• 6,137
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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 ...
• 23k
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• 139
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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 ...
• 6,137

### 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 ...
• 17.6k
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 ...
• 59.4k

### 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 ...
• 1,765
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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 ...
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### 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 ...
• 59.4k
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• 7,452
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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 ...
• 23k

### 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 ...
• 6,137
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 ...
• 5,968

### 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 ...
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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 ...
• 1,436

### 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 ...
• 17.6k
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• 59.4k
### 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(...