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.

Filter by
Sorted by
Tagged with
2
votes
1answer
31 views

How can we prove that the covariance satisfies $\mathrm{Cov}_\rho(X,Y)=\mathrm{Cov}_\rho(Y,X)$?

While attempting to prove the Cauchy Schwarz Inequality I came across this problem. First of all, if we are given a $\rho$ density matrix and two matrix of obserables $X,Y$, after defining the ...
2
votes
3answers
135 views

Why does the trace of density operators need to be one?

Usually, the textbook starts with a few assumptions of what density operator $\rho$ has. One of them is $Tr(\rho) = 1$. Why is that?
1
vote
1answer
32 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 ...
3
votes
0answers
66 views

Analyzing the composition of a channel with its adjoint in relation with an identical composition obtained for the channel's complement

Let us consider two quantum channels $\Phi:M_d\rightarrow M_{d_1}$ and $\Phi_c:M_d\rightarrow M_{d_2}$ that are complementary to each other, i.e., there exists an isometry $V:\mathbb{C}^d\rightarrow \...
4
votes
1answer
54 views

Quantum indistinguishability using density operators

There is something that bugs me concerning the use of density matrices. For instance, to argue that quantum teleportation does not spread an information faster than light, Nielsen and Chuang state the ...
1
vote
1answer
30 views

Setting the Initial State of Density Matrix Simulation in Cirq

I am trying to use the DensityMatrixSimulator in Cirq. I want to append gates to the circuit conditional on measurements. In order to reduce computational resources, I am trying to run these ...
3
votes
2answers
139 views

Does a partial transpose always have real eigenvalues?

I am working with a tripartite system, but when I partially transpose the $8\times 8$ density matrix I get two complex eigenvalues. I know the criteria for the positive and negative eigenvalues, but ...
3
votes
2answers
57 views

How do you mix two pure states to obtain a mixed state?

If we have the following two states \begin{equation} |\psi\rangle_1 = \frac{1}{\sqrt{2}}|0\rangle_A|0\rangle_B + \frac{1}{\sqrt{2}} |1\rangle_A |1\rangle_B \end{equation} \begin{equation} |\psi\...
1
vote
1answer
146 views

Calculate the von Neumann Entropy of a two-qubit entangled state

After working through an exercise I got a confusion answer/solution that either may be because I've made a mistake or I'm not understanding von Neumann Entropy. I have the two qubit system $$ | \psi \...
0
votes
0answers
68 views

Stinespring dilation using ancilla in mixed state?

The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\...
1
vote
1answer
39 views

What is the conjugate transpose of $|0\rangle_{A}|1\rangle_{B}$?

Suppose a composite state is in $|0\rangle_{A}|1\rangle_{B}$. Then their conjugate transpose would be $\langle 0|_{A}\langle 1|_{B}$? Note: Why this question? Because I was checking MIT's "...
2
votes
1answer
39 views

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

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed state, $$ \frac{1}{9}\begin{bmatrix} 5 & 1 & −i \\ 1 & 2 &...
2
votes
1answer
44 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 \\ ...
0
votes
1answer
22 views

How many real numbers are required to describe density matrix for $n$ qubits?

(All of these coming from the topic of simulation of quantum systems) A density matrix $\rho$ Which describe state of $n$ qubits will have $2^{n} \times 2^{n}$ size. We have couple of conditions like ...
0
votes
1answer
64 views

Derivation of Equation $8.7$ in Nielsen Chuang [duplicate]

Equation \eqref{eq:sp1} represents the reduced state of the system after tracing over environment.(Page number 358) $$\mathcal{E}(\rho) = \mathrm{tr}_{env}(\lbrack U(\rho \otimes \rho_{env} )U^{\...
0
votes
0answers
38 views

What does “bipartite system” mean? [duplicate]

I have seen this term appearing multiple times when discussing density matrices. For example here is an excerpt from the lecture notes: We are now ready to introduce the idea of a reduced density ...
2
votes
1answer
137 views

How to express a probability distribution $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ in terms of a trace of a density matrix?

I have been given and expression for a probability distribution $$P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$$ and I have been asked to show that the above expression can be ...
1
vote
1answer
44 views

Is there an easy way to calculate the eigenvalues of the partial transpose of a given matrix? [duplicate]

Consider the state $$|\psi\rangle=(\cos\theta_A|0\rangle+\sin\theta_A|1\rangle)\otimes(\cos\theta_B|0\rangle+e^{i\phi_B}\sin\theta_B|1\rangle).$$ To calculate the $\rho^{T_B}$ I first calculate the $\...
2
votes
1answer
40 views

How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?

How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?
-1
votes
1answer
57 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|...
4
votes
1answer
144 views

What is the difference between $|0\rangle+|1\rangle$ and a balanced mixture of $|0\rangle$ and $|1\rangle$? [duplicate]

Suppose I have a quantum state $\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle$. Also I have a mixture of two quantum states $S_{1} = |0\rangle$ and $S_{2} = |1\rangle$. In this mixture $50\%$...
3
votes
1answer
79 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 ...
7
votes
3answers
488 views

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} \...
2
votes
1answer
51 views

$\rho_{SE}(0)=\rho_S(0)\otimes\rho_E(0)$: No coupling or no entanglement?

We know that the entangled states cannot be expressed like product state, e.g. $|\omega\rangle = |\psi\rangle\otimes|\phi\rangle$. In the density matrix describing the correlations between system $S$ ...
7
votes
2answers
383 views

What's the 'physical consistency' in the partial trace scenario?

I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand: Physical consistency requires that any ...
3
votes
1answer
76 views

What's the difference between $p(i|m)$ and $p(m|i)$ in measurement?

Suppose we perform a measurement described by measurement operators $M_m$. If the initial state is $|{\psi_i}\rangle$, then the probability of getting result $m$ is $$ \begin{align} p(m|i)=\| M_m|\...
-1
votes
1answer
59 views

Uniqueness of Density Operator

I have been reading "Introduction to Quantum Information Science" by Masahito Hayashi, Satoshi Ishizaka,Akinori Kawachi, Gen Kimura and Tomohiro Ogawa; Springer Publication. I'm currently in ...
4
votes
1answer
74 views

How is a qubit in superposition between $|0\rangle$ and $|1\rangle$ different from a mixture of $|0\rangle$ and $|1\rangle$? [duplicate]

Given that a qubit in equal superposition of $|0\rangle$ and $|1\rangle$ is represented by following wave function \begin{equation} \Psi = \frac{1}{\sqrt 2}(|0\rangle + |1\rangle) \end{equation} and ...
2
votes
1answer
57 views

What is the largest absolute value attainable by an off-diagonal real or complex component of a $4 \times 4$ density matrix?

To repeat the titular question: "What is the largest absolute value attainable by an off-diagonal real or complex component of a $4 \times 4$ density matrix?"
3
votes
0answers
33 views

Pure state ensembles achieving the Holevo $\chi$-quantity with at most $d^2$ states

Theorem 8.10 in Chapter 8 of Theory of Quantum Information asserts that the Holevo capacity of a quantum channel (between density operators on $\mathbb{C}^d$) can be achieved by an ensemble consisting ...
2
votes
1answer
101 views

In a bipartite system $AB$, why does the entanglement negativity $\mathcal{N}(\rho^{T_A})$ measure the entanglement between $A$ and $B$?

Consider a system composed of two subsystems $A$ and $B$ living in $\mathcal{H}=\mathcal{H}_A \otimes \mathcal{H}_B$. The density matrix of the system $AB$ is defined to be $\rho$. The entanglement ...
1
vote
1answer
107 views

Unitary freedom in the ensemble for density matrices

I was reading the book by Nielsen & Chuang. I got the part about why we use the density operators. And then I got to the section of theorem 2.6. It says roughly this thing:- The sets $|{\tilde\...
5
votes
2answers
128 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 ...
2
votes
1answer
79 views

Question about quantum error correction and density matrixs

I am now studying QEC and feel confused. If I have a density matrix before the correction: The circuit is: $\rho = p^0(1-p)^3|\varphi\rangle \langle\varphi|+p^1(1-p)^2\sum_{i=1}^3X_i|\varphi\rangle \...
2
votes
1answer
42 views

Two qubit state + Depolarizing channel = Bell diagonal state?

In multiple sources, e.g. RGK, KGR, it is stated (without proof) that if you take any two qubit state and send it through a depolarizing channel, the resulting state would be a Bell-diagonal state. ...
3
votes
1answer
61 views

How can we upper bound the norm of a partial trace?

Suppose we have the normalised states $|\phi_{1}\rangle,|\phi_{2}\rangle \in A \otimes B$ where $A$ and $B$ are $d$-dimensional complex vector spaces. Suppose $|\langle\phi_{2}|\phi_{1}\rangle| < ...
0
votes
1answer
62 views

Produce a quantum state with its density matrix an identity matrix up to an constant

For a n-qubit quantum state $|\psi\rangle=\displaystyle\sum_{i=0}^{2^N-1}|i\rangle$, by definition it's density matrix is $|\psi\rangle\langle\psi|=\displaystyle\sum_{i,j=0}^{2^N-1}|j\rangle\langle i|$...
4
votes
0answers
98 views

Is the set of two-qubit absolutely separable states convex?

Companion question on MathOverflow Let us order the four nonnegative eigenvalues, summing to 1, of a two-qubit density matrix ($\rho$) as \begin{equation} 1 \geq x \geq y \geq z \geq (1-x-y-z) \geq 0. ...
2
votes
1answer
73 views

Depolarization of density operator with zeros in diagonal

I suppose a quantum state with density matrix like the following is not valid. $$ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. $$ Now, let's say I have a valid density operator representing ...
1
vote
1answer
57 views

Trace distance of two classical-quantum state with hashing

Let's say I have a classical-quantum(cq) state $\rho_{XE}$, where the classical part $(X)$ is orthogonal. It's trace distance from another uniform density operator is defined to be: $$ \frac{1}{2}||\...
4
votes
1answer
55 views

Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?

The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum ...
3
votes
2answers
64 views

Proof of the no-communication theorem

Let $A, B$ be (finite-dimensional) Hilbert spaces, and $\rho$ some mixed state of $A \otimes B$. I am trying to show that a measurement performed on the '$A$-subsystem' does not affect $\rho^B = \text{...
2
votes
0answers
66 views

What is the relation between density matrices and phase-space probability distributions?

According to its tag description, a density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical ...
1
vote
1answer
58 views

Computation of qubits with quantum gates using density matrix form

I'm making a quantum circuit with qubits and quantum gates. While I'm doing it, I have some problem with it. My calculation process is below. As you can see, start qubit is $|0 \rangle$ and after 'X' ...
1
vote
2answers
57 views

How to represent the state vector form of a qubit in density matrix representation? [duplicate]

While I'm studying state vector and density matrix. I wonder how to write qubit state as density matrix. qubit state can be represented with state vector form. But how about density matrix?
2
votes
0answers
42 views

Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?

Below is a question and an answer. How does quantum information relate to, diverge from or reduce to Shannon information, which used log probabilities? What people are more often interested in are ...
1
vote
1answer
53 views

Trace distance bound after partial trace

Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.} I know that the trace distance ...
6
votes
1answer
147 views

Bob applies a projector - what happens to eigenvalues of Alice's reduced state?

Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global ...
4
votes
1answer
119 views

What is the intuition behind “states with support on orthogonal subspaces”?

I'm sure I don't fully understand support, but I am having trouble seeing how it connects to things like density operators. I have an idea that it means, according to Wikipedia: "In mathematics, ...
2
votes
1answer
59 views

How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?

Let's say I have 2 density operators $A$ and $B$. Now, here is what I am trying to calculate: $$\newcommand{\tr}{\operatorname{trace}} \tr(A \sqrt{B} A \sqrt{B}). $$ I saw that this trace can be ...