Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

Questions tagged [density-matrix]

For questions about density matrices and related concepts and ideas, e.g. procedures for computing properties of quantum states from their density matrices.

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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Density matrices for pure states and mixed states

What is the motivation behind density matrices? And, what is the difference between the density matrices of pure states and density matrices of mixed states? This is a self-answered sequel to What's the difference between a pure and mixed quantum…
Sanchayan Dutta
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What do the off-diagonal elements of a density matrix physically represent?

For simplicity, let's take a density matrix for a single qubit, written in the $\{|0\rangle,|1\rangle\}$ basis: $$ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10}^* & 1-\rho_{00} \end{pmatrix} $$ The diagonal elements give us the…
KnightShuffler
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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 following density matrix: $$\rho = .5 |0\rangle…
QuestionEverything
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What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?

For a Hilbert space $\mathcal{H}_A$, I have seen the phrase density matrices acting on $\mathcal{H}_A$ multiple times, e.g. here. It is clear to me that if $\mathcal{H}_A$ has finite Hilbert dimension $n$, then this makes sense mathematically,…
Peter
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What is a "maximally mixed state"?

What is meant by maximally mixed states? Does this mean that there are partially mixed states? For example, consider $\rho_{GHZ} = \left| {GHZ} \right\rangle \left\langle {GHZ} \right|$ and $\rho_W = \left| {W} \right\rangle \left\langle {W}…
Bekaso
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How to check if a matrix is a valid density matrix?

What conditions must a matrix hold to be considered a valid density matrix?
PiMan
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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 mechanics is mathematically formulated in Hilbert…
Sanchayan Dutta
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Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?

Suppose I have an arbitrary orthogonal projector $\Pi$ and two density operators $\rho, \sigma$. Is it true that: $$ ||\Pi (\sigma - \rho) \Pi||_1 \le || \sigma - \rho ||_1 $$ where $||\cdot||_1$ denotes the trace norm?
NYG
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How to find a density matrix of a qubit?

If we are given a state of a qubit, how do we construct its density matrix?
PiMan
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Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ is $\mathrm{tr}(\rho\sigma)$. It then describes a…
Quantum Guy 123
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How to prepare mixed states on a quantum computer?

I am a little bit confused by density matrix notation in quantum algorithms. While I am pretty confident with working with pure states, I never had the need to work with algorithm using density matrices. I am aware we can have/create a quantum…
Nicholas Sathripa
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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 $V$. The reduced state $\rho_{G}^{A}:=…
John Doe
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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 I have constructed a weak inductive hypothesis over n…
Confused grad student
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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 for qubits. If rho and sigma are both qubit…
tatakai
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How to show a density matrix is in a pure/mixed state?

Say we have a single qubit with some density matrix, for example lets say we have the density matrix $$\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}.$$ I would like to know what is the procedure for checking whether this state is pure or…
bhapi
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