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.

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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&#...
Sanchayan Dutta's user avatar
4 votes
3 answers
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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 ...
Sanchayan Dutta's user avatar
5 votes
1 answer
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How to perform Quantum Process Tomography for three qubit gates?

I am trying to perform Quantum process tomography (QPT) on three qubit quantum gate. But I cannot find any relevant resource to follow and peform the experiment. I have checked Nielsen and Chuang's ...
Pralekh Dubey's user avatar
4 votes
2 answers
635 views

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 +\...
glS's user avatar
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6 votes
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Random quantum states and Schur-Weyl duality

Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator: $$ \rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC. $$ Let's ...
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7 votes
3 answers
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$ 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 ...
Mahathi Vempati's user avatar
8 votes
1 answer
4k views

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?
Archil Zhvania's user avatar
3 votes
1 answer
361 views

Averaging over a single Haar-random unitary applied $t$ times

I'm trying to compute the following integral: $$\int U^{\otimes t}\left|x_1,\cdots,x_t\middle\rangle\middle\langle x'_1,\cdots,x_n'\right|\left(U^\dagger\right)^{\otimes t}\,\mathrm{d}\mu(U)$$ Where $\...
Tristan Nemoz's user avatar
14 votes
3 answers
5k 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 ...
QuestionEverything's user avatar
8 votes
4 answers
843 views

Simulating a quantum circuit with decoherence and noise

Based on the answers given here and here, it is pretty clear that an arbitrary quantum circuit can be simulated with matrix algebra. The difficulty is that this assume perfect fidelity. I am unsure ...
Anna Naden's user avatar
7 votes
2 answers
592 views

How does quantum teleportation work with mixed shared states?

I am given the scenario that instead of the two parties (A & B) sharing the bell state $|\phi_+\rangle$ they share the mixture $\rho_\lambda = \lambda|\phi_+\rangle\langle\phi_+|+(1-\lambda)\frac{\...
Luca Ion's user avatar
7 votes
1 answer
243 views

Can one quantify entanglement between different parts of a system?

Consider some state $|\psi\rangle$ of $n$ qubits. One can take any subsystem $A$ and compute its density matrix $\rho_A =Tr_{B} |\psi\rangle \langle\psi|$. The entanglement between subsystem $A$ and ...
Nikita Nemkov's user avatar
5 votes
1 answer
892 views

How is measurement modelled when using the density operator?

I've just learned about the density operator, and it seems like a fantastic way to represent the branching nature of measurement as simple algebraic manipulation. Unfortunately, I can't quite figure ...
ahelwer's user avatar
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4 votes
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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) $...
Mahathi Vempati's user avatar
2 votes
2 answers
397 views

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}}\...
Upstart's user avatar
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Density matrices of multiples copies of a single Haar-Random state

In Pseudorandom States, Non-Cloning Theorems and Quantum Money, the authors state that: Let $\rho_\mu^m$ be the density matrix of a random $|\psi\rangle^{\otimes m}$ for $|\psi\rangle$ chosen from ...
Tristan Nemoz's user avatar
-1 votes
1 answer
196 views

How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?

How can I calculate the eigenvalues of $\rho^{T_{B}}$ (PPT) of the following state $$ \rho =\frac{1}{2}|0\rangle\langle0|\otimes|+\rangle\langle+|+\frac{1}{2}|+\rangle\langle+|\otimes|1\rangle\langle1|...
heromano's user avatar
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17 votes
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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} \...
KnightShuffler's user avatar
10 votes
2 answers
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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 ...
Peter's user avatar
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8 votes
2 answers
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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 ...
Sanchayan Dutta's user avatar
7 votes
4 answers
1k 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}{...
Mahathi Vempati's user avatar
7 votes
2 answers
4k views

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 ...
bhapi's user avatar
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6 votes
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How to find the reduced density matrix of a four-qubit system?

I have the state vector $|p\rangle$ made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to find the reduced density matrix ...
meelszz's user avatar
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1 answer
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Show that if the Lindblad satisfy $\sum_\mu L_\mu L_\mu^\dagger=\sum_\mu L_\mu^\dagger L_\mu$ then the von Neumann entropy increases monotonically

How can we show that when the Lindblad operators satisfy the condition: $$\sum_{\mu}L_{\mu} L_{\mu}^{\dagger} = \sum_{\mu} L_{\mu}^{\dagger}L_{\mu},\tag{1}$$ the master equation evolution ...
Sudhir Kumar Sahoo's user avatar
6 votes
1 answer
491 views

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 $\sigma$ ...
Quantum Guy 123's user avatar
5 votes
2 answers
1k views

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? (...
Mahathi Vempati's user avatar
5 votes
2 answers
189 views

How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived?

Let's say I have a density matrix of the following form: $$ \rho := \frac{1}{2} (|a \rangle \langle a| + |b \rangle \langle b|), $$ where $|a\rangle$ and $|b\rangle$ are quantum states. I saw that ...
QuestionEverything's user avatar
5 votes
2 answers
2k views

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 ...
dylan7's user avatar
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5 votes
2 answers
331 views

Only assuming the universe evolves according to a positive trace-preserving map, is there a proof that all subsystem evolution must be CPTP?

If we only assume that the wavefunction of the universe evolves according to $e^{-iHt}$, is there any proof that all subsystems of the universe (partial traces over parts of the universe) must evolve ...
user1271772's user avatar
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5 votes
1 answer
270 views

Semi-definite program for smooth min-entropy

The conditional min-entropy is defined as (wiki): $$ H_{\min}(A|B)_{\rho} \equiv -\inf_{\sigma_B}\inf_{\lambda}\{\lambda \in \mathbb{R}:\rho_{AB} \leq 2^{\lambda} \mathbb{I} \otimes \sigma_B\} $$ And ...
QuestionEverything's user avatar
5 votes
3 answers
106 views

How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?

How can one prove that $tr(A\mid\psi\rangle\langle\psi\mid)=\langle\psi\mid A\mid\psi\rangle$? In Nielsen/Chuang they mention this is due to Gram-Schmidt decomposition but I can’t understand how.
Alberto Zorzato's user avatar
5 votes
3 answers
1k views

How can pure state ensemble decompositions not be unique?

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 ...
user2723984's user avatar
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5 votes
2 answers
154 views

Does the definition of separability of pure states require the components of the summands to be pure?

Does the definition of separability of pure states require the components of the summands to be pure? More precisely, let $\rho$ be a pure state (i.e., $\rho=|\phi\rangle\langle\phi|$) on the space $...
Eric's user avatar
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4 votes
1 answer
902 views

How do I determine if a given pure two-qubit state is separable?

I'm trying to self-study some topics about quantum computing and I came across a topic of state separability. Talking about that, I wanted to determine separability on the following state (from Qiskit ...
Eenoku's user avatar
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4 votes
1 answer
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Applying CNOT with local operations and two EPR pairs

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 ...
Joe's user avatar
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4 votes
1 answer
250 views

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}=...
glS's user avatar
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4 votes
0 answers
57 views

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-...
Paul B. Slater's user avatar
3 votes
7 answers
422 views

Show that $I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$ for all states $\rho$

I am trying to show that for any qubit state p, the following holds: $$I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$$ I have tried different manipulations,...
evethompson123's user avatar
3 votes
0 answers
215 views

Finding Wigner function of four maximal entangled Bell state

How can we find a Wigner function for the four maximally entangled Bell states $(|00\rangle \pm |11\rangle)/\sqrt{2}$, $(|01\rangle \pm |10\rangle)/\sqrt{2}$? I have used the basis where labels for ...
Sudhir Kumar Sahoo's user avatar
3 votes
2 answers
546 views

Implementation of tomography on IBM Q

I wanted to ask how do you implement a circuit that finds the non-diagonal values of the density matrix of a quantum state on IBM Q?
Vladimir kozlov's user avatar
3 votes
2 answers
1k views

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(\...
Upstart's user avatar
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3 votes
2 answers
263 views

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{...
1ijk's user avatar
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3 votes
1 answer
170 views

Is the partial trace $\mathrm{Tr}_B(\rho)$ equal to $\sum_k \mathrm{Tr}[(\sigma_k\otimes I)^\dagger \rho]\sigma_k$?

Assume a composite quantum systes with state $|\psi_{AB}\rangle$ or better $\rho=|\psi_{AB}\rangle\langle\psi_{AB}|$. I want to know the state of system $A$ only, i.e. $\rho_A$. Is there any ...
draks ...'s user avatar
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3 votes
1 answer
111 views

Are there quantum algorithms which initial state is a mixed one?

In this answer, it is stated that it is not yet known how to efficiently classically simulate separable mixed states, a statement supported in a comment to this answer. However, I can't imagine an ...
Tristan Nemoz's user avatar
2 votes
1 answer
236 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. ...
bilanush's user avatar
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2 votes
1 answer
646 views

How to find the Kraus operators from the process matrix?

I am trying to find the Kraus operator from process matrix. For instance, suppose that for single qubit identity gate, I have the following process matrix: ...
quest's user avatar
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2 votes
1 answer
167 views

Purification applied to indistinguishability

In Zhandry's compressed oracle paper, one can read the following: Next, we note that the oracle $h$ being chosen at random is equivalent (from the adversary’s point of view) to $h$ being in uniform ...
Tristan Nemoz's user avatar
2 votes
1 answer
122 views

Give an explicit derivation of the exact formula for the two-qubit absolute separability Hilbert-Schmidt probability $\approx 0.00365826$

The two-qubit eigenvalue ($\lambda_i$ >= 0, $i=1,\ldots,4$, $\lambda_4=1-\lambda_1-\lambda_2-\lambda_3$) condition of Verstraete, Audenaert, de Bie and de Moor AbsoluteSeparability (p. 6) for ...
Paul B. Slater's user avatar
2 votes
1 answer
332 views

How would I compute a density matrix of a 2 qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this? $$ |00> \;with \;probability \; 2/4 \\ ...
lambda's user avatar
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2 votes
1 answer
244 views

How to get the state of an individual qubit in a composite system?

Given a composite system with $N$ qubits represented by some $2^N$-dimensional vector, how would I get the quantum state of an individual qubit? Note that I understand some states are not separable ...
dhjtricks's user avatar