# Tag Info

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 ...
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### Counterexamples in quantum information theory

Quantum Channels Quantum channels: general properties Not every positive map is completely positive. One may argue that this is the mother of all counterexamples in quantum information: the ...
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### Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

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 ...
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### 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) & = |...
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### 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, ...
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### 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.) ...
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### 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 ...
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### 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 ...
• 23.2k
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### 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 ...
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### 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\...
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### 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 ...
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### 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)^...
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### Can classical linear algebra solvers implement quantum algorithms with similar speed-ups?

The complexity that you are glossing over is that in the general case you need to store $2^n$ complex amplitudes to even represent an $n$ qubit system classically. Therefore, for a quantum computer of ...
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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 ...