21
votes
Accepted
How can classical bits be copied if qubits cannot be copied?
TL;DR: The ban on copying is not nearly as universal as it might seem. No-cloning theorem actually allows copying as long as it is limited to orthogonal states. Classical information is the type of ...
- 19.1k
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) & = |...
- 4,958
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
$$
\mathrm{H}(...
- 4,958
10
votes
Accepted
Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
The equation at the top of the question is not correct: there is a missing factor of $1/d$ on the right-hand side. Let's eliminate this factor from the left-hand side to make it simpler, so that the ...
- 4,958
9
votes
Is the set of all states with negative conditional Von Neumann entropy convex?
Geometric characterization (as any other characterization) of subsets of the quantum state space in relation with their locality and entanglement properties becomes very complicated as the number of ...
- 2,455
8
votes
Violation of the Quantum Hamming bound
You may be interested in the answers to this question. One example of a degenerate code beating the quantum Hamming bound is here. I also have a numerical example of a small violation in my own work, ...
- 52.3k
8
votes
Accepted
Degradable channels and their quantum capacity
A channel $\Phi$ is said to be degradable if there exists another channel $\Xi$ such that $\Xi\Phi$ is complementary to $\Phi$.
The idea here is as follows. Suppose $\Phi$ is a channel and $\Psi$ is ...
- 4,958
8
votes
Accepted
What does it mean to take the Choi-Jamiolkowski of a quantum channel?
Let me quote my answer from over at physics.SE:
The intuition
Let us consider a channel $\mathcal E$, which we want to apply to a state $\rho$. (This could equally well be part of a larger system.) ...
- 5,443
8
votes
Accepted
Closeness of purifications of states
No dimension-independent bound is possible.
Consider states $\rho_A$ and $\sigma_A$ that are close in $p$-norm (for $p>1$) but have relatively low fidelity. Specifically, assume
$$
\|\rho_A - \...
- 4,958
8
votes
Accepted
Can $2^n$ bits be sent with $n$ instances of quantum teleportation?
Quantum teleportation can send a single qubit from Alice to Bob, with two classical bits
Correct, on the condition that Alice and Bob also have an entangled qubit pair shared between them. This ...
- 1,235
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
$$
- \inf_{\sigma} \,\text{D}_{\text{max}}(\rho \| \...
- 4,958
7
votes
Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
Here the important fact is that the maximally mixed state is in fact an identity matrix.
Let me rewrite the expression on the left in index notation (the summation sign is omitted according to the ...
- 664
7
votes
Accepted
How many classical bits are needed to represent a qubit?
There are two types of information in physics:
Classical information
Quantum information
Physics doesn't answer the question "What is (classical or quantum) information?". This is philosophic ...
- 3,174
7
votes
Accepted
What is the Von Neumann entropy of $\rho = \sum_ip_i|i\rangle\langle i| \otimes \rho_i$?
Operator $\rho$ is not a tensor product, it's a sum of tensor products
$$
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.
...
- 6,443
7
votes
Accepted
Does $\mathcal E^{\otimes n}$ admit a more efficient Stinespring dilation than the one used for $\mathcal E$?
No.
The minimal size of the environment is just the rank of the Choi matrix of $\mathcal E$, call it $J(\mathcal E)$. Since $J(\mathcal E^{\otimes n}) = \big(J(\mathcal E)\big)^{\otimes n}$ and $\text{...
- 2,258
7
votes
Accepted
Does the no-hiding theorem suggest that quantum information is never destroyed?
It is true that unitary evolution cannot destroy information. This is the content of the no-cloning theorem and its time reversal - the no-deleting theorem. The no-hiding theorem says something ...
- 19.1k
7
votes
How to describe a known quantum state using classical information?
Generally speaking, in order to describe elements of a set $A$ using classical information we need two ingredients: a non-empty finite alphabet $\Sigma$ and an encoding $E: A\to\Sigma^\omega$ which ...
- 19.1k
7
votes
Accepted
Schmidt decomposition for tripartite system $ABC$ with vanishing mutual information between $A$ and $C$
TL;DR: The key observation is that Schmidt basis on a subsystem consists of eigenvectors of the reduced state of that subsystem. Consequently, if the reduced state is a product state then its Schmidt ...
- 19.1k
7
votes
Accepted
Existence of a perturbed channel that achieves a perturbed output state
Yes, the channel $\tilde{N}$ necessarily exists.
Notice first that the state $\rho_B$ is the completely mixed state $\mathbb{1}/d$. So, in order for $\tilde{\rho}_{A'B}$ to be contained in $S$, three ...
- 4,958
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\...
- 3,577
6
votes
Accepted
What exactly is the relation between the Holevo quantity and the mutual information?
Right, they are quite similar. The Holevo bound is a bound on the amount of accessible information between your quantum system and your classical system. The I(X;B) object written in the HSW theorem ...
- 76
6
votes
Accepted
Why is the quantum Fisher information for pure states $F_Q[\rho,A]=4(\Delta A)^2$?
Suppose $\lambda_0 = 1$ and the rest are $0$.
$$
F_Q [\rho,A] = 2 \sum_{k,l} \frac{(\lambda_k-\lambda_l)^2}{\lambda_k + \lambda_l} | \langle k |A| l \rangle |^2\\
= 2 \sum_{k=0,l \neq 0} \frac{(1-0)^...
- 3,603
6
votes
Accepted
If quantum computing always return random measurement (or uncertain measurement), why do we still need it?
Short answer:
Assuming you are measuring in the computational basis (Z basis), $\{|0\rangle, |1\rangle \}$, there is no randomness upon measurement in the following quantum circuit (you will always ...
- 13.3k
6
votes
Accepted
Can one quantify entanglement between different parts of a system?
Of course you can. Take any entanglement measure that can be applied to a system whose overall description is a density matrix, and you can apply that to the density matrix describing your subsystem. (...
- 52.3k
6
votes
Accepted
Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?
Given $\rho$ and a fixed ensemble $\{ |\psi_i \rangle \}$ it might not be possible to write $\rho$ as $\sum_i p_i |\psi_i \rangle \langle \psi_i |$. For example, let $| + \rangle = \frac{1}{\sqrt{2}} ...
- 926
6
votes
Accepted
No-cloning theorem and distinguishing between two non-orthogonal quantum states revisited
TL;DR: The assumption of non-orthogonality is implicitly used by the linked answer. It is needed due to a "loophole" in the no-cloning theorem that allows cloning of known orthogonal states.
...
- 19.1k
6
votes
Accepted
What is the root of the non-trace-preserving bit-flip map
Assuming w.l.o.g. that $p\in\mathbb{R}$, the linear map in the question may be rewritten as
$$
\mathcal{E}(\rho) = p^2\rho+p^2X\rho X = 2p^2\left(\frac12\rho + \frac12 X\rho X\right)
$$
where $X$ is ...
- 19.1k
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
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 ...
- 4,958
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}) \...
- 4,958
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