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 ...
- 4,853
9
votes
Accepted
Positive semidefinite relationship after partial trace
No, not necessarily. For example, take $\rho$ to be a GHZ state and let $\sigma$ be the completely mixed state of one qubit. We then have $\lambda=4$ and $\mu=2$.
- 4,853
9
votes
Accepted
What's the 'physical consistency' in the partial trace scenario?
Measurement average
Measurement average $\langle M \rangle_\rho$ of observable (a Hermitian operator) $M$ on the state $\rho$ is the average of measurement outcomes $m$ in the limit of infinite number ...
- 18.2k
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 ...
- 614
6
votes
How can we upper bound the norm of a partial trace?
The $1$-norm decreases under partial trace and so we have an upper bound of $1$ when the states are normalized,
$$
\|\mathrm{Tr}_B[|\psi_1\rangle \langle \psi_2|]\|_1 \leq \||\psi_1\rangle \langle \...
- 4,516
6
votes
Accepted
Is the trace distance between multipartite states invariant under permutations?
A permutation of the qubits is a unitary operation. The trace distance is invariant under unitaries (https://en.wikipedia.org/wiki/Trace_distance#Properties). Thus, statement 1 is true.
- 51.1k
6
votes
Accepted
If the partial traces $\rho_A,\rho_B$ are pure, does it imply that $\rho$ is a product state?
Yes; in fact, $\rho$ is both separable and pure.
We can start by writing any state $\rho$ in its eigenbasis
$$\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|,$$ where $p_i$ are probabilities (i.e., ...
- 2,684
5
votes
Is the trace distance between multipartite states invariant under permutations?
I'd like to add a small addition to the answer of @DaftWullie about why you would expect this operationally to be true -- without knowing permutations correspond to unitary matrices.
It boils down to ...
- 4,516
5
votes
Accepted
Fidelity of extensions of states
Let's start with the second question. There is nothing special about an extension $\sigma_{AR}^{\ast}$ that allows it to be optimal for the right-hand side of (1); any extension $\sigma_{AR}$ of $\...
- 4,853
5
votes
Accepted
How do I trace out the second qubit to find the reduced density operator?
Suppose you have the state $|\psi\rangle = \dfrac{|00\rangle + |11\rangle}{\sqrt{2}} = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} $ then its density matrix representation is
$$ \...
- 13.1k
5
votes
What's the 'physical consistency' in the partial trace scenario?
The point of physical consistency is about how we can define the operator $M$ as acting on system $\rho^A$, or we can define the operator $M \otimes \mathbb{1}_B$ as acting on the system $\rho^{AB}$, ...
5
votes
Accepted
Can we express $\mathrm{tr}_A((A\otimes B)\rho_{AB})$ in terms of $A$, $B$, $\rho_A$ and $\rho_B$?
In general, the knowledge of the marginals $\rho_A$ and $\rho_B$ and the operators $A$ and $B$ is insufficient to compute $\mathrm{tr}_A((A\otimes B)\rho_{AB})$. Indeed, we can find two different ...
- 18.2k
5
votes
Accepted
Prove that a channel is close to acting on only one system
I suppose you're asking the following: for any $\epsilon\ge0$ and $\Phi$ that satisfy conditions, is there $\delta_\epsilon \rightarrow 0$ when $\epsilon \rightarrow 0$, such that there exists a ...
- 6,353
4
votes
Accepted
How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
Let's start with a general state
$$
\rho\otimes\rho_0=\sum_{x,y\in\{0,1\}}\langle x|\rho|y\rangle|x\rangle\langle y|\otimes |0\rangle\langle 0|.
$$
If we apply the controlled-not, we have
$$
\...
- 51.1k
4
votes
How many Kraus operators are required to characterise a channel with different start and end dimensions?
The answer is yes, you need $d_1 d_2$ operators, as already pointed out in the other answer. Here I'll show explicitly why this is the case.
Let $\Phi\in\mathrm{T}(\mathcal X,\mathcal Y)$ be a CPTP ...
glS♦
- 21.2k
4
votes
Accepted
How is the partial trace related to the operator sum representation?
I think it helps here to write things explicitly.
Suppose $\mathcal E(\rho)=\operatorname{Tr}_E[U(\rho\otimes|\mathbf e_0\rangle\!\langle\mathbf e_0|)U^\dagger]$.
Pick a basis for the environment in ...
glS♦
- 21.2k
4
votes
Accepted
Does the trace distance between marginals bound the distance between the overall states?
No. Just take two Bell states. They have identical reduced density matrices yet are orthogonal, that is, as distant from each other as it gets.
- 5,292
4
votes
Accepted
Prove that the partial trace is a quantum operation, finding its Kraus representation
I think the presentation in N&C is a little confusing because $\rho$ is used in two contexts. I'll substitute one of those for a $\sigma$.
You can define
$$
E_i=I\otimes\langle i|,
$$
which will ...
- 51.1k
4
votes
Can SWAP operators change trace of a product state?
The SWAP operator has been widely used to ``linearize'' polynomial functions of the density matrix. To understand this carefully, consider the following setup: Let $\mathcal{H} = \mathcal{H}_{A} \...
- 1,753
4
votes
Accepted
Is the partial trace $\mathrm{Tr}_B(\rho)$ equal to $\sum_k \mathrm{Tr}[(\sigma_k\otimes I)^\dagger \rho]\sigma_k$?
They are exactly the same. Remember that you can write
$$
\rho=\sum_{i,j}\rho_{ij}\sigma_i\otimes\sigma_j.
$$
If you take the partial trace, you have
$$
\rho_A=\sum_{i,j}\rho_{ij}\sigma_i \text{Tr}(\...
- 51.1k
4
votes
Accepted
Non-lockability of quantum max-entropy
Let $D_{\alpha}(\rho\|\sigma):= \frac{1}{\alpha - 1} \log \mathrm{Tr}[\rho^\alpha \sigma^{1-\alpha}]$ be the Petz-Rényi divergence for $\alpha \in (0,1)\cup(1,\infty)$. Note that for $\alpha \in (0,1)\...
- 4,516
4
votes
Accepted
Bob applies a projector - what happens to eigenvalues of Alice's reduced state?
Just notice that
$$
\text{Tr}(\rho'_{AB}) = \text{Tr}(\Pi_B\rho_{B}).
$$
One way to see this is to consider any decomposition
$$
\rho_{AB} = \sum_i A_i \otimes B_i,
$$
where $A_i, B_i$ just some ...
- 6,353
4
votes
Accepted
Partial Trace of Werner State
$\frac{s}{2}|0\rangle\langle1|\otimes|0\rangle\langle1|+\frac{s}{2}|1\rangle\langle0|\otimes|1\rangle\langle0|$ should disappear when you take the trace over them, as $\langle0|1\rangle$ and $\langle1|...
- 1,087
4
votes
Partial trace and SWAP in the basis of subsystems
Using the fact $ e^{i\theta S} = \text{cos}(\theta) \cdot I + i \cdot \text{sin}(\theta) \cdot S $, we calculate
\begin{align*}
e^{-i\theta S} (\rho \otimes \sigma) e^{i\theta S}
& =
(\text{cos}(...
- 1,366
4
votes
Accepted
Partial trace and SWAP in the basis of subsystems
For any $\theta\in\mathbb{R}$ and any operator $T$ such that $T^2=I$ we have
$$
\exp(i\theta T) = I\cos\theta + i T\sin\theta
$$
(c.f. exercise 4.2 on p.175 in Nielsen & Chuang). Therefore,
$$
\...
- 18.2k
4
votes
Accepted
If $\text{supp}(\rho_{AB})\subset \text{supp}(\sigma_{AB})$, is $\text{supp}(\rho_{A})\subset \text{supp}(\sigma_{A})$?
If $\rho_{AB}, \sigma_{AB} \ge 0$, i.e. they are positive semi-definite then yes (otherwise no).
To prove it observe that if $\text{supp}(\rho_{AB})\subset \text{supp}(\sigma_{AB})$ then we can write ...
- 6,353
4
votes
Accepted
What is the physical intuition behind taking the partial trace of a state?
TL;DR Tracing out a subsystem corresponds to discarding it.
Suppose Alice has subsystem $A$ and Bob has subsystem $B$ of the composite system $AB$ in state $\rho_{AB}$. Tracing out subsystem $B$ ...
- 18.2k
4
votes
Why does $\rho^A=\mathrm{tr}_B(\rho^{AB})$ guarantee that $\mathrm{tr}(M\rho^A)=\mathrm{tr}((M\otimes I_B)\rho^{AB})$?
The equality follows by hitting both sides of
$$
X_A\mathrm{tr}_B(Y_{AB})=\mathrm{tr}_B((X_A\otimes I_B)Y_{AB})\tag1
$$
with the trace and setting $X_A=M$ and $Y_{AB}=\rho^{AB}$.
We can establish $(1)...
- 18.2k
4
votes
Accepted
What is the technique for calculating $\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$?
The partial trace of a bipartite state $\sigma_{ab}$ of two qubits $a$ and $b$ is
$$
\mathrm{tr}_b(\sigma_{ab}) = \langle 0_b|\sigma_{ab}|0_b\rangle+\langle 1_b|\sigma_{ab}|1_b\rangle.\tag1
$$
...
- 18.2k
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