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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61 views

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: $$\...
6
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2answers
149 views

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|) = \...
3
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1answer
73 views

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 ...
3
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0answers
34 views

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 ...
3
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2answers
50 views

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 ...
4
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1answer
62 views

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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0answers
24 views

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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0answers
37 views

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 $$\...
3
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1answer
119 views

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 ...
5
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2answers
115 views

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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3answers
150 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 $\...
2
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1answer
54 views

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 ...
3
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2answers
100 views

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 $B$? ...
2
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1answer
88 views

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, ...
5
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1answer
49 views

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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1answer
40 views

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 "...
4
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1answer
28 views

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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1answer
65 views

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 ...
4
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2answers
145 views

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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1answer
50 views

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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2answers
408 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 ...
3
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2answers
98 views

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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2answers
75 views

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 ...
4
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0answers
109 views

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 ...
3
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2answers
75 views

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 ...
7
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1answer
111 views

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 ...
3
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1answer
98 views

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'})\...
4
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2answers
140 views

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 $\...
3
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1answer
68 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| < ...
1
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1answer
62 views

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}||\...
1
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1answer
80 views

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 ...
3
votes
1answer
109 views

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 $\|\...
6
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1answer
154 views

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 ...
4
votes
1answer
67 views

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 ...
4
votes
1answer
159 views

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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2answers
167 views

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 ...
1
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1answer
45 views

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 ...
1
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1answer
166 views

Can SWAP operators change trace of a product state? [closed]

I am currently reading https://arxiv.org/abs/1501.03099. In the third part of the paper, "Measuring and detecting quantumness", the authors define the SWAP operators, use them on the initial ...
3
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1answer
68 views

Is the partial trace $\mathrm{Tr}_B(\rho)$ equal to $\sum_k \mathrm{Tr}[(\sigma_k\otimes I)^\dagger \rho]\sigma_k$?

Assume a composite quantum systes with state $|\psi_{AB}\rangle$ or better $\rho=|\psi_{AB}\rangle\langle\psi_{AB}|$. I want to know the state of system $A$ only, i.e. $\rho_A$. Is there any ...
1
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0answers
143 views

Is it possible to partial trace the $\chi$-matrix of $4$ qubits $q_0,q_1,q_2,q_3$ to obtain a description of what happens to $q_1$?

Considering a $\chi$-matrix of a circuit with, say, 4 qubits, is it possible to trace out 3 of them from $\chi$ - for example qubits $q_0$, $q_2$ and $q_3$ - thus gaining the process matrix describing ...
1
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1answer
65 views

What can be inferred about the closeness of reduced qubit states from the closeness of the bipartite quantum state?

Given a qubit state $|\psi\rangle \in \mathcal{H}$, and two bipartite general mixed states $\rho$ and $\sigma$, such that, $$\langle \psi|\otimes \langle \psi|\rho - \sigma |\psi\rangle \otimes |\psi \...
3
votes
1answer
41 views

Does the trace distance between marginals bound the distance between the overall states?

If the quantum states of the subsystems of two systems are close (for example: in terms of trace distance), are the states of the larger systems also close, i.e., if $$ ||\rho_A - \rho_{A^\prime}||\...
1
vote
2answers
57 views

Prove that the partial trace is a quantum operation, finding its Kraus representation

I am referring to Nielsen and Chuang Quantum Computation and Quantum Information 10th Anniversary Edition Textbook, Chapter 8.3. A linear operator $E_i:H_{QR}\longrightarrow H_Q $ is defined by: $$...
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2answers
101 views

Prove that $\operatorname{Tr}_B(O_A M)=O_A\operatorname{Tr}_B(M)$

Can someone help me with the following question? Let $M$ be a general operator on the composite system $\mathcal{H}_A\otimes \mathcal{H}_B$ and let $O_A$ be an operator on $\mathcal{H}_A$. Using ...
2
votes
1answer
97 views

Fidelity With Bell State Calculation

Let's say I have the following state: $$ |\psi\rangle = \sqrt{\frac{2}{3}} |0000\rangle_{a_1b_1a_2b_2} + \sqrt{\frac{1}{6}} \big( |0011\rangle_{a_1b_1a_2b_2} + |1100\rangle_{a_1b_1a_2b_2} \big). $$ I ...
2
votes
1answer
49 views

Given partial states, can one construct the best estimate of the full state?

Given some partial states $\rho_{AB}$ and $\sigma_{AC}$, is there a general procedure to construct a state $\delta_{ABC}$ such that the following sum of trace distances $$||\text{Tr}_C(\delta_{ABC}) -...
0
votes
1answer
56 views

Which simple property of partial trace are we using here?

I would like to know which property is being used in this example. For $Tr_1$ the partial trace on the first system: $$Tr_1[(|0\rangle\otimes|0\rangle)(\langle 0| \otimes \langle0|)] =|0\rangle\...
3
votes
2answers
74 views

Simulating Classical Probabilistic Transitions with superoperators

I'm working on the following exercise: "Show how a classical probabilistic transition on an M -state system can be simulated by a quantum algorithm by adding an additional M -state ‘ancilla’ ...
5
votes
3answers
465 views

How is the partial trace related to the operator sum representation? [duplicate]

In Quantum Computation and Quantum Information by Nielsen and Chuang, the authors introduce operator sum representation in Section 8.2.3. They denote the evolution of a density matrix, when given an ...
4
votes
2answers
520 views

Prove that the partial trace is equivalent to measuring and discarding

I'm trying to solve the following question: "Prove that one way to compute $\mathrm Tr_B$ is to assume that someone has measured system $B$ in any orthonormal basis but does not tell you the ...