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. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

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Are there different orderings of the fifteen SU(4) generators in common use?

I've recently performed certain analyses (Archipelagos of Total Bound and Free Entanglement) pertaining to eq. (50) in Separable Decompositions of Bipartite Mixed States , that is \begin{equation} ...
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Why do purifications only differ by a local unitary?

Let's consider $\rho_A$ a density matrix. I introduce a space $B$ and an entangled state $|\Psi\rangle$ (the purification) so that: $$\newcommand{\tr}{\operatorname{Tr}}\rho_A = \tr_B(|\Psi\rangle \...
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Find the qutrit analogue of certain qubit and ququart formulas of Li and Qiao for testing separability

In eqs. (33), (43)-(46), (56) of their paper, "Separable Decompositions of Bipartite Mixed States" https://arxiv.org/abs/1708.05336, Li and Qiao present a number of formulas pertinent to testing the ...
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Simulating Classical Probabilistic Transitions with superoperators

I'm working on the following exercise: "Show how a classical probabilistic transition on an M -state system can be simulated by a quantum algorithm by adding an additional M -state ‘ancilla’ ...
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Why are $d^2$ dimensions required to describe a density matrix?

A density matrix is defined as: $$\sum p_i |\psi_i \rangle \langle \psi_i|$$ If the dimensionality of each $|\psi_i \rangle$ is $d$, why does it take $d^2$ dimensions to represent a density matrix? (...
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Density matrix for a diagonally polarized photon

I am struggling with the density matrix for diagonally polarized photons. Can I think of diagonally polarized photons as a mixture of vertically and horizontally polarized photons?
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What is “Lindblad Superoperator” in Stochastic Master Equation

I was reading a paper titled "Using a Recurrent Neural Network to Reconstruct Quantum Dynamics of a Superconducting Qubit from Physical Observations" and was confused about a stochastic master ...
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Does partial tracing a system with three shared Bell states give the identity?

Suppose I share three Bell states among two participants Alice and Bob and Charlie in the following manner: $$ |\psi\rangle=\left(\dfrac{|0\rangle_1|0\rangle_2+ |1\rangle_1|1\rangle_2}{\sqrt{2}}\...
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Density matrix of a product of Bell states

Suppose I share two Bell states among two participants Alice and Bob in the following manner : $$ |\psi\rangle=\left(\dfrac{|0\rangle_1|0\rangle_2+ |1\rangle_1|1\rangle_2}{\sqrt{2}}\right)\left(\...
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What is the matrix representation for $n$-qubit gates?

Let's say I have more than one qbits $|0\rangle|1\rangle$ and I want to perform a $H$ on both of them. I know the matrix representation for the Hadamard on a single qbit is $$\frac{1}{\sqrt{2}}\begin{...
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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 for a graph $G = (V,E)$ and $B = V\setminus A$ the corresponding complement ...
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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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Proving $\langle j_2|\langle j_1| U(|0\rangle\!\langle0|\otimes\rho)U^\dagger|j_2\rangle|j_1\rangle =\operatorname{Tr}(M_j\rho)$

I'm trying to prove that: $$ \langle j_2|\langle j_1| U(|0\rangle\!\langle0|\otimes\rho)U^\dagger|j_2\rangle|j_1\rangle =\operatorname{Tr}(M_j\rho) $$ where $\rho$ is the density operator, $M_j=\...
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Quantum Fisher information for pure states query

Assume that a density matrix is given in its eigenbasis as $$\rho = \sum_{k}\lambda_k |k \rangle \langle k|.$$ On page 19 of this paper, it states that the Quantum Fisher Information is given as $$F_{...
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Prove that the partial trace is equivalent to measuring and discarding

I'm trying to solve the following question: "Prove that one way to compute $\mathrm Tr_B$ is to assume that someone has measured system $B$ in any orthonormal basis but does not tell you the ...
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Trace of Hermitian Operator and Operator Function

I am having trouble understanding the following step. From: $$\operatorname{trace}\left(\sum_z |z\rangle\langle z| \rho_A |z\rangle\langle z| * \log( \sum_z |z\rangle\langle z| \sum_x |\langle x|z \...
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How do I compute the Relative Entropy between pure and mixed states?

Let $$ \rho = \begin{bmatrix} .7738 & -.0556 \\ -.0556 & .0040 \end{bmatrix} , \sigma = \begin{bmatrix} .9454 & -.2273 \\ -.2273 & .0546 \end{bmatrix} \\$$ As you can ...
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Determining the quantum secret

I earlier posted I question Representing a Bell measurement on non adjacent qubits for which I got an excellent answer. Now I want to build upon that and do further analysis which is where I am stuck. ...
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Is there any meaning for a density operator if we omit the j-th row and column in quantum mechanics?

Assume we have a density operator (Hermitian, PSD, with trace 1, where PSD means positive semi-definite) called A for a particle. $v_i$ shows the i-th eigenvector of A and $\lambda_i$ shows the i-th ...
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What is the average value of $|c_i\bar c_j|$ for a random state $|\psi\rangle=\sum_i c_i|i\rangle$?

Consider the density matrix $\rho=|\psi\rangle\!\langle\psi|$ of a random pure state in an $N$-dimensional space (in other words, an $N$-dimensional qudit, $|\psi\rangle\in\mathbb C^N$), $\rho_{ij}=...
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Trace preserving condition in Choi's thorem

Choi's theorem states that any completely positive map $\Phi(\cdot) : C^*_{n\times n} \rightarrow C^*_{m \times m}$ can be expressed as $\Phi(\rho) = \sum_{j=1}^r F_j^\dagger \rho F_j$, for some $n \...
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What's an easy way to determine a local density matrix?

In his lecture notes Scott Aaronson states: Now, consider the $2$-qubit pure state $\frac{|00\rangle + |01\rangle + |10\rangle}{\sqrt{3}}$. We'll give the first qubit to Alice and the second to Bob....
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How to represent an ensemble of pure quantum states in Qiskit?

I was going through the Qiskit documentation to see if there was a way to represent a mixture of quantum states as a density matrix or otherwise. Is there a way to do it? If so, how?
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The meaning of the city plot in Qiskit

I read the documentation of qiskit and I can't understand the meaning of the city plot, like this: Why do we need a 3D plot? Why can't we just use a 2D plot, where $ | 00\rangle$, $ | 01 \rangle$, $ ...
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Taking the two qubit reduced density matrix on a 5 qubit system

I am wanting to find the two qubit reduced density matrix on a 5 qubit system. I have the 5 qubit state in the form of a tensor product and I want to find the reduced density matrix of qubits 1 and 3. ...
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What is the probability that measurement finds it in the $|0\rangle$ state?

Suppose that there is an ensemble with 60% of the states prepared in $$|a\rangle=\sqrt{\frac{2}{5}}|+\rangle-\sqrt{\frac{3}{5}}|-\rangle$$ and 40% in: $$|b\rangle=\sqrt{\frac{5}{8}}|+\rangle+\sqrt{\...
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Which state describes carrier transport through channel? A mixed state or a pure state?

A pure quantum state is a state which can be described by a single ket vector. A mixed quantum state is a statistical ensemble of pure states. When carriers transport from source to drain in a Field ...
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Understanding the classification of quantum states based on partial transposition: representations of the bipartite density matrix

I'm going through some slides on the PPT/NPT criteria along with Horodecki's paper, and I'm kind of stuck. Let's take this slide: Firstly, why can we write a bipartite density matrix as $\sum_{...
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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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How can I write the maximally mixed state on m qubits as a linear combination of basis vectors?

The maximally mixed state on m qubits is defined to be the quantum state with associated density operator $\rho_m = \frac{1}{2^m} I$. Examples are On one qubit this is $\rho_1 = \frac{1}{2}(|0\...
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Why do we use complex-conjugate instead of complex-conjugate-transpose when calculating the concurrence?

When we use the formula to calculate two-qubit entanglement, like these: $$ C(\rho)=\max \left\{\sqrt{e_{1}}-\sqrt{e_{2}}-\sqrt{e_{3}}-\sqrt{e_{4}}, 0\right\}\tag{18} $$ with the quantities $...
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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( \sum_{i=0}...
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Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?

Consider an arbitrary qudit state $\rho$ over $d$ modes. Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation: $$\rho=\frac{1}{d}\left(\mathbb I +\...
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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 ...
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What are the ranges of the four $q$ parameters in the magic simplex of Bell states formula?

Equation (7) in the 2012 paper, "Complementarity Reveals Bound Entanglement of Two Twisted Photons" of B. C. Hiesmayr and W. Löffler for a state $\rho_d$ in the "magic simplex" of Bell states \begin{...
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How does the probability of measurement turn out to be negative?

c) Compute $$\text{Prob}(\uparrow_\hat{n}\uparrow_\hat{m}) \equiv \text{tr}(\pmb{E}_A(\hat{n})\pmb{E}_B(\hat{n})\pmb{p}(\lambda)), \tag{4.164}$$ where $\pmb{E}_A(\hat{n})$ is the projection of ...
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Quantum teleportation with “noisy” entangled state

This is actually an exercise from Preskill (chapter 4, new version 4.4). So they are asking about the fidelity of teleporting a random pure quantum state from Bob to Alice, who both have one qubit of ...
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What proportions of certain sets of PPT-two-retrit states are bound entangled or separable?

For two particular (twelve-and thirteen-dimensional) sets of two-retrit states (corresponding to 9 x 9 density matrices with real off-diagonal entries), I have been able to calculate the Hilbert-...
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Computing von Neumann entropy of pure state in density matrix

Let's say I have a pure state of the form: $$\psi = \sqrt{\frac{3}{9}} \lvert 0 \rangle + \sqrt{\frac{6}{9}} \lvert 1 \rangle$$ Then the density matrix representation would be: $$\rho = \psi \otimes \...
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Decoherence in quantum systems always produces $\vert0\rangle$

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\...
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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 ...
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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 ...
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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. ...
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Convex Combination of Separable States

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) $...
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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 ...