Questions tagged [partial-trace]

For questions centred around or involving the notion of partial trace. The partial trace is a generalization of the notion of trace defined for multipartite systems.

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What is the technique for calculating $\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$?

I am stuck on calculating $\mathcal{E}(\rho)=\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$. For example, in the case when $U$ is the CNOT matrix $$U=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 &...
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What does equality of partial traces, ${\rm Tr}_1\rho={\rm Tr}_1\sigma$, say about a pair of states $\rho,\sigma$?

Let $\rho,\sigma$ be a pair of bipartite quantum states such that ${\rm Tr}_1\rho={\rm Tr}_1\sigma$. What does this tell us about $\rho,\sigma$? More precisely, is there a way to write more explicitly ...
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Partial trace formulation

I was looking into the partial trace and ran into the following equation shown below. Though I understand the general notion of the partial trace, I cannot derive how these two are equal in the ...
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How to write the joint action of a CP map that acts on a single qubit of a bipartite state?

The question Say I have a completely-positive (CP) map $\mathcal{A}_{ij}$ defined in terms of two projectors $\Pi_i = |i\rangle \langle i |$ and $\Pi_j = |j\rangle \langle j |$ that acts on a density ...
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Why does $\rho^A=\mathrm{tr}_B(\rho^{AB})$ guarantee that $\mathrm{tr}(M\rho^A)=\mathrm{tr}((M\otimes I_B)\rho^{AB})$?

Niesen and Chuang, 2nd edition, page 107, Box 2.6, in its motivation for partial trace, says that if M is an observable on system A and $\tilde{M}$ is the corresponding observable on system AB, then ...
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Mathematics of Measurement then Partial Trace

Say we have the following quantum state: $$ |\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle +|10\rangle)$$ To measure the first qubit and then further trace out the first qubit, my notes have the ...
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How to compute the partial trace on the first register of the state $|\chi\rangle=\frac{1}{||A||}\sum_{i=0}^{m-1}||A_i|||A_i\rangle|i\rangle$?

Given the quantum state $$|\chi\rangle=\dfrac{1}{||A||}\sum_{i=0}^{m-1}||A_i|||A_i\rangle|i\rangle,$$ how can we obtain the partial trace operation on the first register, i.e., $$\begin{align}\text{tr}...
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Prove that $\rho_{AB} \leq |B|(\rho_A\otimes I_B)$ for any bipartite state $\rho_{AB}$

I'm trying to prove the following statement but am lost on how to show it. For a quantum state $\rho_{AB}$ with marginal $\rho_A$, how can one show that $$ \rho_{AB} \leq|B|(\rho_A\otimes I_B)$$ where ...
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Closest quantum state with a fixed marginal: Analytical solution?

Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
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How to compute derivatives of partial traces of the form $\frac{\partial \operatorname{Tr}_B(F(\mathbf{X}))}{\partial \mathbf{X}}$?

The Matrix Cookbook says that for any differentiable matrix function $F(\cdot)$, it holds that $$\frac{\partial \operatorname{Tr}(F(\mathbf{X}))}{\partial \mathbf{X}}=f(\mathbf{X})^{T},$$ where $f(\...
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What do normalization term and partial measurement represent when tracing out ancillary qubits?

I am reading a paper and I am having trouble following some equations. The system in this paper has $N$ qubits, with $N_A$ ancillary and the rest ($N - N_A$) as data qubits. For the purpose of this ...
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Tripartite quantum marginal problem

Consider a tripartite quantum system with the three subsystems labeled $A, B,$ and $C$. Now take two states $\rho_{AB}$ on the joint system $AB$ and $\rho_{BC}$ on the joint system $BC$. Under what ...
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What does the partial trace of $|W\rangle$ states represent physically?

Given the W-state $|W\rangle = |001\rangle + |010\rangle + |100\rangle$, where $|ijk \rangle $implies $|i\rangle_A \otimes |j\rangle_B \otimes |k \rangle_C$, the partial trace over first qubit turns ...
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Classical versus quantum correlations and partial traces

Given a bipartite state $\rho_{AB}$ living in the Hilbert space $\mathcal H(A\otimes B)$ we can always define two local states on $A$ and $B$ respectively by taking the appropriate partial traces: $$\...
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Can one always find purifications which preserve equality of statistical mixtures?

When pure states $|\psi_1⟩$, $|\psi_2⟩$ and $|\phi_1⟩$, $|\phi_2⟩$ in $\mathcal{H}_A \otimes \mathcal{H}_B$ have identical statistical mixtures $$\frac{1}{2}(|\psi_1⟩⟨\psi_1| + |\psi_2⟩⟨\psi_2|) = \...
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If the partial traces $\rho_A,\rho_B$ are pure, does it imply that $\rho$ is a product state?

Suppose $\rho$ is some bipartite state such that the partial traces $\rho_A={\rm Tr}_B\rho$ and $\rho_B={\rm Tr}_A\rho$ are both pure. Does this necessarily imply that $\rho$ is a product state? This ...
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Permutation invariant states have permutation invariant purifications - proof?

I don't remember where I came across the statement but I'm pretty sure it is true and am interested in understanding why it holds. For any $n-$ register state $\rho^n \in H^{\otimes n}$ that is ...
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Does the entangibility of density operators rely on what component spaces are being specified?

Is the entangibility of density operators relied on what component spaces are being specified? More precisely, let $H$ be a Hilbert space, $\rho$ be a density operator on $H$. Suppose we were not ...
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Trace distance between mixed state and pure state vs trace distance between their purifications

Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that $$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$ where $\|A\...
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How to trace out qubits from a multipartite density matrix [duplicate]

I have a density made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to trace out 2nd and 3rd qubits. Is there any reference ...
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Partial trace instead of trace in definition of entropy

For a bipartite quantum state $\rho_{AB}$, we have that the von Neumann entropy is $$S(\rho_{AB}) = -\text{Tr}(\rho_{AB}\log\rho_{AB})$$ If instead, one took the partial trace above and obtained $$\...
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Is there a characterization for the set of states with given marginals?

Let $\rho_A,\rho_B$ be two states. Is there any way to characterise the set of bipartite states $\rho$ such that $\mathrm{Tr}_B(\rho)=\rho_A$ and $\mathrm{Tr}_A(\rho)=\rho_B$? If I assume $\rho$ to be ...
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What is the physical intuition behind taking the partial trace of a state?

I want to confirm my understanding of a partial trace. Essentially, we have a system that $H_a \otimes H_b$. When we trace out system $b$, what we are doing is basically reducing the system down to as ...
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2 votes
3 answers
241 views

Why are entanglement breaking channels, defined as $\Phi(\rho)=\sum_a \operatorname{Tr}(\mu(a)\rho)\sigma_a$, entanglement breaking?

Define an entanglement breaking channel $\Phi$ as a channel (CPTP map) of the form $$\Phi(\rho) = \sum_a \operatorname{Tr}(\mu(a)\rho) \sigma_a\tag A$$ for some POVM $\{\mu(a)\}_a$ and states $\...
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Constructing a state with constraints on reduced states

Suppose $\rho'_{AB} \approx_\varepsilon \rho_{AB}$ in trace distance. Is there an explicit construction of some state $\tilde{\rho}_{AB}$ using $\rho'_{AB}, \rho'_A, \rho'_B, \rho_A$ and $\rho_B$ (but ...
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Can we express $\mathrm{tr}_A((A\otimes B)\rho_{AB})$ in terms of $A$, $B$, $\rho_A$ and $\rho_B$?

For a density matrix $\rho_{AB}$ and some operators $A, B$, is there a way to express $$\text{Tr}_A((A\otimes B)\rho_{AB})$$ using the reduced states $\rho_A$ and $\rho_B$ and operators $A$ and $B$? ...
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2 votes
1 answer
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What is the most general quantum operation that preserves the marginal?

Suppose I have two states $\rho_{AB}$ and $\sigma_{AB}$ such that the marginals $\rho_A = \sigma_A$. What is the most general operation that could have acted on $\rho$ to output $\sigma$? For example, ...
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If $\text{supp}(\rho_{AB})\subset \text{supp}(\sigma_{AB})$, is $\text{supp}(\rho_{A})\subset \text{supp}(\sigma_{A})$?

For any linear operator $A$, the support of $A$ is the orthogonal complement of its kernel. Hence when we say, $supp(A)\subset supp(B)$, we have that for any vector $v$ in the kernel of $B$ i.e. $Bv = ...
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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 "...
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Does $\frac{\Pi_A\otimes I_B}{\text{Tr}((\Pi_A\otimes I_B)\rho_{AB})}\rho_{AB}=\rho_{AB}$ hold for a state $\rho_{AB}$ and projector $\Pi_A$?

For some projector $\Pi_A$ and state $\rho_{AB}$, let $$\sigma_{AB} = \frac{\Pi_A\otimes I_B}{\text{Tr}((\Pi_A\otimes I_B)\rho_{AB})}\rho_{AB}$$ Is it the case that $\sigma_B = \rho_B$? It seems ...
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Does $\mathrm{tr}(A \otimes B) = \mathrm{tr} (A) \otimes \mathrm{tr}(B)$ hold for partial trace?

I was reading this question from this site answered by DaftWullie. I would like to request you to read the question there. The answer says However, in this particular case, the calculation is much ...
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Partial trace and SWAP in the basis of subsystems

I'm trying to derive equation $(1)$ on p.2 in Lloyd et al, 2013 which reads $$ \text{Tr}_A\left[\exp(-i\theta S_{AB}) (\rho_A \otimes \sigma_B) \exp(i\theta S_{AB}) \right] = (\cos^2 \theta) \sigma_B +...
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2 votes
1 answer
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How can I understand these two equations about the indirect measurement?

I'm reading an article about environmental monitoring and information transfer. Suppose $S$ represents a quantum system and $E$ is the environment. Assume at time $t=0$ there are no correlations ...
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8 votes
2 answers
479 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 ...
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3 votes
3 answers
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Where does the term $|\psi\rangle\langle\psi|$ come from while calculating the expectation value?

Suppose there're two systems $A$ and $B$, if we're in a pure state $|\psi\rangle\in\mathbb{H}_a\otimes\mathbb{H}_b$. Let $\hat A$ be an operator acting on $\mathbb{H}_a$ and $|\psi\rangle=\sum_{i,j}\...
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How do I trace out the second qubit to find the reduced density operator? [duplicate]

I'm doing an exercise to trace out the second qubit to find the reduced density operator for the first qubit: $tr_2|11\rangle\langle00| = |1\rangle\langle0|\langle0|1\rangle$ I'm just wondering if I ...
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4 votes
0 answers
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Calculating the entropy of a quantum state

Let $\rho_{AR}$ be some $d-$dimensional pure quantum state. Consider a channel $N_{A\rightarrow B}$ that outputs a constant state in $B$. We now consider the Stinespring dilation of this channel with ...
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3 votes
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Partial Trace of Werner State

I am trying to trace out the second qubit of the Werner State: \begin{align} W &=\frac{1-s}{4}I_{4}+\frac{s}{2}(|00\rangle\langle{00}|+|11\rangle\langle11|+|11\rangle \langle00|+|00\rangle \langle ...
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7 votes
1 answer
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Positive semidefinite relationship after partial trace

Let $\rho_{ABC}$ and $\sigma_{C}$ be arbitrary quantum states and $\lambda\in \mathbb{R}$ be minimal such that $$\rho_{ABC}\leq \lambda \rho_{AB}\otimes\sigma_C$$ We assume there are no issues with ...
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  • 395
3 votes
1 answer
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Fidelity of extensions of states

Given two states $\rho_A, \sigma_A$, Uhlmann's theorem states that the fidelity between them is achieved in the following way $$F(\rho_A, \sigma_A) = \max_{U_{R'}}F(\rho_{AR'}, (I\otimes U_{R'})\...
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4 votes
2 answers
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Is the trace distance between multipartite states invariant under permutations?

Consider two multipartite states $\rho_{A_1A_2..A_L}$ and $\sigma_{A_1A_2..A_L}$ in $\mathcal{H}_{A_1} \otimes\mathcal{H}_{A_2} \otimes...\mathcal{H}_{A_L} $. For an arbitrary permutation $\pi$ over $\...
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3 votes
1 answer
100 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| < ...
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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}||\...
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1 vote
1 answer
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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 ...
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3 votes
1 answer
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Quantum marginal problem - constructing a global state from reduced states

Consider a bipartite state $\rho_{AB}$ with reduced states $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Suppose one obtains states $\rho'_{A}$ and $\rho'_{B}$ such that $\|\...
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6 votes
1 answer
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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 ...
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  • 2,008
4 votes
1 answer
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Non-lockability of quantum max-entropy

Lockability and non-lockability are explained in this paper. A real valued function of a quantum state is called non-lockable if its value does not change by too much after discarding a subsystem. The ...
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  • 2,008
4 votes
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Eigenvalues of a quantum state after partial tracing

I am interested in the smallest nonzero eigenvalue of a quantum state. Does this eigenvalue always increasing after a partial trace i.e. the smallest nonzero eigenvalue of $\rho_A$ is always larger ...
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Implementing Partial Trace in IBMs quantum computer

I am trying to implement the partial trace operation on IBMs quantum computer. I am simulating the depolarising channel with the following code ...
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1 vote
1 answer
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Given $\rho,\sigma$ such that $F(\rho,\sigma)=0$, what can we say about $F(\text{Tr}_A(\rho),\text{Tr}_B(\sigma))$?

Lets say $\rho,\sigma$ satisfy $F(\rho,\sigma)=0$, i.e., they are quantum states living on orthogonal supports. What can we say about $F(\text{Tr}_A(\rho),\text{Tr}_B(\sigma))$? I am looking for upper ...
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