# Questions tagged [density-matrix]

A density matrix is a matrix that can be used to describe a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state.

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### Query on Reduced Graph States

Reduced graph states are characterized as follows (from page 46 of this paper): Let $A \subseteq V$ be a subset of vertices of a graph $G = (V,E)$ and $B = V\setminus A$ the complement of $A$ in $V$. ...
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### What can we know about the eigenvalues of a reduced density matrix knowing the eigenvalues of the original matrix?

What information can we get out about the eigenvalues of a reduced density matrix knowing the eigenvalues of the original matrix? For example, it can be proved that if all the eigenvalues of a ...
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### About production and disagreements between density matrices

So let's say there are $2$ experimentalists who have density matrix systems $A$ and $B$. They both agree that for the experiment they need identical density matrices $\rho_A = \rho_B$ which is a mixed ...
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### Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$

I've been trying to figure this out for a while and I'm totally lost. My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$ So far ...
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### Equivalent determinant condition for Peres-Horodecki criteria

The Peres-Horodecki criteria for a 2*2 state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable. According to this paper (...
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### Quantum fidelity simplified formula while both of the density matrices are single qubit states

I have a question while reading the quantum fidelity definition in Wikipedia Fidelity of quantum states, at the end of the Definition section of quantum fidelity formula, it says Explicit expression ...
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### Quantum state where phase information is unknown

I'm trying to obtain a more intuitive understanding of the notion of quantum coherence and how to mathematically represent it. I know that coherence has to do with the interaction of phases between ...
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### What is the relevance of preservation of trace in completely postive trace preserving (CPTP) maps?

Why is the trace preserving part necessary? Is it not enough if it can take all matrices to matrices of trace 1?
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### Is there a two-qudit Choi entanglement witness $W^{(+)}$?

Example 2 in arXiv:1811.09896 states that the "Choi EW (entanglement witness) $W^{(+)}$ obtained from the Choi map in $d=3$ $\ldots$ is given by \begin{equation} W^{(+)} = \frac{1}{6} \left( \...
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I was recently asked two questions concerning error in quantum computing: Is it possible for quantum computers to exhibit behavior similar to flip errors in classical computers where a state $\vert0\... 3answers 498 views ### Purity of mixed states as a function of radial distance from origin of Bloch ball @AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ... 1answer 521 views ### Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection$\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia: Quantum ... 1answer 158 views ### Why is a density operator defined the way it's defined? It's stated that the density operator is: $$\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.$$ But I don't understand why this is the way both in mixed state and pure state. ... 1answer 125 views ### How to find a separable decomposition for$|\Psi^+\rangle\!\langle\Psi^+|+|\Phi^+\rangle\!\langle\Phi^+|$? The state $$\frac{1}{2}\left(| \phi^+ \rangle \langle \phi^+ | + | \psi^+ \rangle \langle \psi^+ | \right)$$ where $$| \phi^+ \rangle = \frac{1}{\sqrt2} \left(|00 \rangle + | 11 \rangle \right) ... 2answers 375 views ### Non-uniqueness of pure states ensemble decomposition Apparently, the decomposition of a state into an ensemble of pure states is not unique. I can't understand why, as if I understood correctly a "pure state ensemble decomposition" is just the ... 1answer 322 views ### Is the set of classical-quantum states convex? I read about the classical-quantum states in the textbook by Mark Wilde and there is an exercise that asks to show the set of classical-quantum states is not a convex set. But I have an argument to ... 2answers 1k views ### Partial trace over a product of matrices - one factor is in tensor product form$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ... 2answers 654 views ### How to write a post-measurement state, if we don't know the measurement result? Once a state is measured, but we don't look at the result, is the state now written as a density matrix, that is, the probability that it could land on a measurement operator multiplied by the ... 3answers 2k views ### Density matrix after measurement on density matrix Let's say Alice wants to send Bob a$|0\rangle$with probability .5 and$|1\rangle$also with probability .5. So after a qubit Alice prepares leaves her lab, the system could be represented by the ... 1answer 305 views ### Total mutual information of a quantum system In the discussions about quantum correlations, particularly beyond entanglement (discord, dissonance e.t.c), one can often meet two definitions of mutual information of a quantum system$\rho^{AB}$: ... 4answers 731 views ### Maximally mixed states for more than 1 qubit For 1 qubit, the maximally mixed state is$\frac{\mathrm{I}}{2}$. So, for two qubits, I assume the maximally mixed state is the maximally mixed state is$\frac{\mathrm{I}}{4}$? Which is:$\frac{1}{...
Suppose Alice and Bob hold one qubit each of an arbitrary two-qubit state $|\psi \rangle$ that is possibly entangled. They can apply local operations and are allowed classical communication. Their ...