Questions tagged [entropy]
For questions about the various kinds of entropies --- as defined in the context of quantum information theory and quantum statistical mechanics.
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What is the Bregman projection of $\rho+\epsilon X$ onto the space of states?
Notation:
Let $S(X)=\operatorname{tr}[X\log X]$ for any $X\geq0$ be the von Neumann entropy and let $D_S(X,Y)=\operatorname{tr}[X\log X-X\log Y -X+Y]$ be the corresponding Bregman divergence (which is ...
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Why does not Shannon Chain Rule work in Quantum Mechanics
Shannon Entropy conditional variant is $H(A|B)=H(AB)-H(B)$. Its chain rule implies: $H(A|BC)=H(AB|C)-H(B|C)$. The Quantum conditional min entropy does not satisfy the chain rule, but satisfies the ...
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Efficient Clifford simulation and entropy of reduced density matrices
Suppose I have a Clifford circuit $C$ and I want to estimate the entanglement entropy of a subset of two qubits, say, $\{q_0, q_1\}$, i.e. the quantity $$S(\rho_{q_0 q_1}) = - \text{Tr}[\rho_{q_0 q_1} ...
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Proof that the relative entropy satisfies $S(\rho\|\sigma)=S(T\rho\|T\sigma)$ iff $\hat TT\rho=\rho$, $\hat TT\sigma=\sigma$ for some $\hat T$
To prove the saturation condition for the strong subadditivity of the von Neumann entropy, the authors of [HJPW2004] make use of the following characterisation of when the monotonicity of the ...
glS♦
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What are examples of states saturating the strong subadditivity of the von Neumann entropy?
A well-known property of classical distribution is that they satisfy strong subadditivity, meaning that for any tripartite joint probability distribution $p(x,y,z)$, we have the inequality
$$H(AB)+H(...
glS♦
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On the use of $\log(P\otimes Q)= \log P\otimes I+I\otimes\log Q$ for relations between entropic quantities. What if $P,Q$ are only semidefinite?
Many properties of entropic quantities are shown by resorting to related properties of the relative entropy of suitable quantities. For instance, subadditivity of entropy may follow from non ...
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Maximum entanglement entropy of a random circuit
Consider the random quantum circuit below where the gates are randomly taken from SU(4) accordingly with the Haar measure. I am looking to determine an upper bound on the entanglement entropy between ...
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Exercise 11.7 in Nielsen & Chuang and basic properties of Shannon entropy
I apologize in advance if this question is trivial, I'm aware I'm a total beginner in this field. This is the exercise I would like to solve:
As to the first point, what I get is that one should ...
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Is entanglement trainable?
There exists a famous result from Google that the gradients of the parameters of quantum neural networks (QNN) vanish exponentially with the number of qubits in the quantum circuit. Their result ...
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What is the classical cost of simulating an arbitrary quantum state?
The past couple of years has seen various groups claim quantum advantage/utility only to have their experiments efficiently simulated with classical methods, notably using tensor networks.
My question ...
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Minimizing $1 - \text{Tr}(\Phi(\rho,U)^2)$
I am looking for a computationally efficient way to minimize the following function. Let
$$\Phi(\rho, U) = \text{Tr}_2(U\rho U^\dagger)$$
be a reduced density matrix where $\rho = \overline{\rho}_1 \...
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Mutual information between Alice and Eve in a BB84 intercept resend attack
I'm new to information theory and i need to calculate $I(A,E)$.
To calculate it I need conditional entropy $H(A|E)$.
I assume the BB84 protocol standard states $\{ |0\rangle,|1\rangle \},\{|+\rangle,|-...
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Can post-measurement states have entropy larger than the original state?
Given a set of measurement operators $\{M_i\}$ that sum to unity, consider the post-measurement states on some $\rho$ as $\rho_i:=(\sqrt{M_i}\rho\sqrt{M_i})/p_i$ and $p_i:=\mathrm{Tr}(M_i\rho)$.
It's ...
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Entropic uncertainty relations with measurements and memory [duplicate]
In entropic uncertainty relations involving measurements and memory, one has a quantum state $\rho_{AB}$. Alice holds register $A$ and performs one of two measurements denoted by observables $R$ and $...
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Does quantum mutual information encompass information only about quantum correlations, or does it encompass both classical and quantum correlations?
I am confused about what quantum mutual information gives us.
Does it give all kinds of quantum correlations? Or does it give all kinds of quantum and classical correlations?
If it consists of ...
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