12
votes
Accepted
Proof of an Holevo information inequality for a classical-classical-quantum channel
It appears that the statement is not true in general. Suppose $X = Y = \{0,1\}$, $\mathcal{H}$ is the Hilbert space corresponding to a single qubit, and $W$ is defined as
\begin{align}
W(0,0) & = |...
- 5,083
12
votes
Accepted
What is a "maximally mixed state"?
The maximally mixed state is a quantum state whose density matrix is proportional to the identity matrix. Physically, it may be interpreted as a uniform mixture of states in an orthonormal basis.
The ...
- 20k
11
votes
Accepted
Is the set of all states with negative conditional Von Neumann entropy convex?
The conditional von Neumann entropy is a concave function: if $\rho$ and $\sigma$ are states of a pair of registers $(\mathsf{X},\mathsf{Y})$ and $\lambda\in[0,1]$ is a real number, then
$$
\mathrm{H}(...
- 5,083
9
votes
Accepted
Degradable channels and their quantum capacity
A channel $\Phi$ is said to be degradable if there exists another channel $\Xi$ such that $\Xi\Phi$ is complementary to $\Phi$.
The idea here is as follows. Suppose $\Phi$ is a channel and $\Psi$ is ...
- 5,083
9
votes
Is the set of all states with negative conditional Von Neumann entropy convex?
Geometric characterization (as any other characterization) of subsets of the quantum state space in relation with their locality and entanglement properties becomes very complicated as the number of ...
- 2,485
9
votes
Maximally mixed states for more than 1 qubit
For two probability distributions, there is a clear notion how to say which one is more mixed: $\vec p$ is more mixed than $\vec q$ if it can be obtained from $\vec p$ by a mixing process, this is, a ...
- 5,608
8
votes
What is entropy quantum computing?
I've never heard about "entropy quantum computing" before. Both the links you provided only give extremely generic information, mostly devoid of details, together with rather bold claims. A ...
glS♦
- 23k
7
votes
How does the conditional min-entropy $H_{\rm min}(A|B)_\rho$ relate to the conditional entropy $H(X|Y)_\rho$?
The conditional min-entropy $\text{H}_{\text{min}}(A | B)_{\rho}$ can be defined for an arbitrary state $\rho$ of a pair of registers $(A,B)$ as
$$
- \inf_{\sigma} \,\text{D}_{\text{max}}(\rho \| \...
- 5,083
7
votes
Accepted
What is the Von Neumann entropy of $\rho = \sum_ip_i|i\rangle\langle i| \otimes \rho_i$?
Operator $\rho$ is not a tensor product, it's a sum of tensor products
$$
p_1|1\rangle\langle 1| \otimes \rho_1 + p_2|2\rangle\langle 2| \otimes \rho_2 + \dots + p_d|d\rangle\langle d| \otimes \rho_d.
...
- 6,611
7
votes
Accepted
How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?
We can bound the amount of information that can be retrieved from $|\psi\rangle$ using Holevo's bound.
Alice and Bob
Let us first reformulate the situation in the terms usually employed in the context ...
- 20k
7
votes
What is "linear" in linear entropy?
$S_L$ called linear because it's obtained from the usual definition of von Neumann entropy $S = -\mathrm{Tr}(\rho \ln \rho)$ by taking a linear approximation for the natural log $\ln \rho = \rho - 1$. ...
- 17.1k
6
votes
Accepted
Does computing the quantum mutual information $I(\rho^{AB})$ require full tomographic information of $\rho^{AB}$?
The mutual information can be written in terms of the relative entropy, please see
Nielsen and Chuang (the entropy Venn diagram figure 11.2). I am writing the equation in the question's notation:
$$I(...
- 2,485
6
votes
Accepted
Accessible information of system vs system, apparatus and environment
For density matrices $\rho_A$ and $\rho_B$ having eigenvalues $\lambda^{\left(A\right)}$ and $\lambda^{\left(B\right)}$, \begin{align}S\left(\rho_A\otimes\rho_B\right) &= -\rho_A\otimes\rho_B\ln\...
- 3,647
6
votes
Calculate the von Neumann Entropy of a two-qubit entangled state
You are calculating the entropy of one of the marginal states and so you would not expect the answer to be independent of $\theta$, except in the case that $|\psi\rangle = |\phi_A\rangle \otimes |\...
- 5,201
6
votes
What is a "maximally mixed state"?
If you want to describe one part of a many-qubit state, such as a GHZ state, then usually, that one qubit cannot be said to be in a state $|\phi\rangle$. That only works if the overall state is ...
- 54.5k
6
votes
What is a "maximally mixed state"?
Indeed, there are partially mixed states, although this terminology is seldom used. One can define measures of mixedness to make this quantitative. The best one is the von Neumann entropy, defined as
$...
- 2,312
5
votes
What is Landauer’s principle?
It means that if you lose information from your system, that information must have been transferred to the system's surroundings. This shows up as an increase in the entropy in the surroundings. This ...
- 579
5
votes
Accepted
Where does the Xmon simulator from Googles cirq framework its entropy from?
Cirq uses numpy's pseudo random number generator to pick measurement results, e.g. here is code from XmonStepper.simulate_measurement:
...
- 32.5k
5
votes
Accepted
Can an isometry leave entropy invariant?
You don't need any additional conditions beyond those already stated in the question. That is, for any isometry $V: A \rightarrow A\otimes B$ and any unit vector $|\psi\rangle_B$, there will always be ...
- 5,083
5
votes
Accepted
Understanding classical vs. quantum channel capacities
These are not really the definitions of classical and quantum capacity, as I will explain. Before doing that, let me adjust the notation being used slightly: let $\Phi:\text{L}(\mathcal{X}) \...
- 5,083
5
votes
Accepted
Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?
That quantity appears to be identical to Holevo information, which turns out to be the upper bound on how much classical information you can transmit using a quantum channel [1].
More generally the ...
- 6,024
5
votes
Accepted
Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$
This only holds if the two distributions are independent. In this case
$$
\begin{aligned}
H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\
&= \...
- 5,201
5
votes
Measuring entanglement entropy using a stabilizer circuit simulator
I think stim is the right tool for the job here, because it gives you access to the stabilizer generators and also it defines stim.PauliString which you can use to ...
- 32.5k
5
votes
Understanding the definition of entropy in the joint entropy theorem derivation
It is not just the binary entropy that is denoted $H(p_i)$. The quantity that is relevant here is the Shannon entropy of the distribution $\{p_i\}$ which is defined as
$$
H(p_i) = - \sum_i p_i \log ...
- 5,201
4
votes
Shannon entropy is least when Measurement basis = Mixture basis
My favourite way of proving that the Shannon entropy is minimized for a measurement in the qubit basis is through the notion of majorizaion (see Nielsen and Chuang or the book on Matrix Analyis by ...
4
votes
Accepted
Maximally mixed states for more than 1 qubit
The Von Neumann entropy of $1/2 (|00\rangle \langle 00| + |11\rangle \langle 11|)$ is one bit. For $I/4$ it's two bits of entropy instead.
The entropy of states that only have entries on the ...
- 32.5k
4
votes
Accepted
Building Intuition for Relative Von Neumann Entropy
Posting an answer because I realised what my issue was:
What I didn't realise then: When a density matrix is written in any basis, the diagonal elements correspond to the probabilities of the density ...
- 1,611
4
votes
Prove that quantum channels cannot increase the Holevo information of an ensemble
Suppose that $\mathsf{X}$ is a register that can store each possible choice for $x$, as a classical state, while $\mathsf{Y}$ is a register that can store each possible state $\rho_x$. It is then ...
- 5,083
4
votes
Accepted
In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?
I don't understand how they go from the first to the second line
So, you're starting from $-\sum_{ij}p_i\lambda^j_i\log(p_i\lambda^j_i)$. Remember that $\log(ab)=\log(a)+\log(b)$, so this is the same ...
- 54.5k
4
votes
Can an isometry leave entropy invariant?
You can always do that.
Subspaces $\text{Im}(V)$ and $A\otimes |0\rangle$ have the same dimension, so there must be some unitary that translates one subspace to another. That is, $\exists W \in \text{...
- 6,611
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