Questions tagged [trace-distance]
In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions. (Wikipedia)
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Closeness of $\rho$ such that $\text{Tr}(|\psi\rangle\langle\psi|\rho)\le1/2^n+{\cal O}(2^{-2n} )$ for all $|\psi\rangle$ to the maximally mixed state
Consider an $n$ qubit density matrix $\rho$ such that
$$\text{Tr}(|\psi\rangle\langle \psi| ~\rho) \leq \frac{1}{2^{n}} + \mathcal{O}\left(\frac{1}{2^{2n}} \right), $$
for every $n$ qubit pure state $|...
4
votes
1answer
57 views
Closeness of unitary dilations of CPTP maps
Let $\Phi_1,\Phi_2 \colon S(\mathcal{H}) \to S(\mathcal{H})$ be CPTP maps on the same Hilbert space $\mathcal{H}$ which are $\varepsilon$-close in diamond norm, and let $U_1,U_2$ be respective unitary ...
2
votes
1answer
50 views
Does ${\rm tr}(\Pi_z\rho\Pi_z)\le p$ imply $\cal E(\rho)$ and $\cal E(\Pi_{-z}\rho\Pi_{-z})$ are close in trace distance?
Suppose I have a quantum operation $\mathcal{E}$ and a state $\rho$ such that:
$$ \operatorname{tr}(\Pi_z \rho \Pi_z) \le p $$
for some probability $p$ and some projection $\Pi_z$ onto some subspace ...
4
votes
1answer
135 views
How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?
I create a quantum state $| \psi \rangle = \frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ for a randomly chosen $x_0,x_1$ of 50 bits. I give this quantum state $|\psi \rangle$ to you and you return ...
2
votes
1answer
65 views
Is the trace norm monotone with respect to quantum operations?
The trace norm is defined to be $$\| K \| = \mathrm{tr}\sqrt{K^\dagger K}.$$
Is it true that we have
$$\| \mathcal E(K) \|\leq \|K\|,$$
for any quantum operation $\mathcal{E}: A\otimes B \to A\otimes ...
5
votes
2answers
59 views
If $\rho,\sigma$ are classical-quantum states, can the fidelity $F(\rho,\sigma)$ be expressed in terms of $F(\rho_i,\sigma_i)$?
Let $\rho = \sum_i \vert i\rangle\langle i\vert \otimes \rho_i$ and $\sigma = \sum_i\vert i\rangle\langle i\vert\otimes\sigma_i$ where we are using the same orthonormal basis indexed by $\vert i\...
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0answers
42 views
Minmax theorem for optimization over isometries and states
I have the following minmax problem and I am wondering if the order of the minimum and maximum can be interchanged and if yes, why?
Let $\|\cdot\|_1$ be the trace norm defined as $\|\rho\|_1 = \text{...
5
votes
1answer
117 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
59 views
General expression for a state that is close in trace distance to a pure state
Suppose we are given that a quantum state $\rho$ is close in trace distance to a pure state $\vert\psi\rangle\langle\psi\vert$. That is
$$\|\rho - \vert\psi\rangle\langle\psi\vert\|_1 \leq \varepsilon,...
4
votes
3answers
112 views
How to find the distance between a given $\rho$ and the nearest pure state(s)?
I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state:
$$
\min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
2
votes
1answer
56 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 ...
2
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0answers
21 views
Approximating an ensemble with an orthogonal ensemble
Consider an arbitrary ensemble $\{p_x\rho_x\}_{x\in X}$ and define the state
$$ \rho = \sum_{x\in X} \vert x \rangle\langle x \vert \otimes p_x\rho_x. $$
I am interested in understanding the quantity
$...
4
votes
0answers
46 views
If $\rho \approx_{\varepsilon}\sigma$, how to find $\Pi\rho\Pi$ to ensure that $\text{supp}(\Pi\rho\Pi)\subset\text{supp}(\sigma)$?
Let $\rho$ and $\sigma$ be positive semidefinite operators with trace less than or equal to 1. Let $\rho\approx_{\varepsilon}\sigma$ i.e. they are close in some distance, such as the trace distance.
...
4
votes
1answer
53 views
Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance?
Can anyone explain why the $l_1$ distance has the property that probability distributions $P,Q$ with orthogonal support (meaning that the product $p_iq_i$ vanishes for each value of $i$) are at a ...
4
votes
2answers
148 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
votes
1answer
79 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| < ...
5
votes
1answer
165 views
Closeness of purifications of states
Uhlmann's theorem states that if two states $\rho_A, \sigma_A$ satisfy $F(\rho_A, \sigma_A)\geq 1 - \varepsilon$, then there for any purification $\Psi_{AR}$ of $\rho_A$, one can find a purification $\...
2
votes
2answers
83 views
Properties of composition of isometry and a perturbed adjoint
Suppose $\vert\Phi\rangle_{AR} = \frac{1}{\sqrt{|D|}}\sum_{i\in D} \vert ii\rangle_{AR}$ is the maximally entangled state. Let $V_{A\rightarrow BE}$ and $\tilde{V}_{A\rightarrow BE}$ be two isometries ...
1
vote
1answer
86 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:
$$
\frac{1}{2}||\...
2
votes
2answers
295 views
Prove that the trace norm is dual to the spectral norm
Suppose $A\in L(X,Y)$. $||\cdot||$ denotes spectral norm and denotes the largest singular value of a matrix, i.e. the largest eigenvalue of $\sqrt{A^*A}$.
$||\cdot||_{tr}$ denotes trace norm. We have ...
1
vote
1answer
109 views
Trace distance bound after partial trace
Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.}
I know that the trace distance ...
3
votes
1answer
120 views
Quantum marginal problem - constructing a global state from reduced states
Consider a bipartite state $\rho_{AB}$ with reduced states $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$.
Suppose one obtains states $\rho'_{A}$ and $\rho'_{B}$ such that $\|\...
3
votes
1answer
66 views
How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?
Let's say I have 2 density operators $A$ and $B$. Now, here is what I am trying to calculate:
$$\newcommand{\tr}{\operatorname{trace}}
\tr(A \sqrt{B} A \sqrt{B}).
$$
I saw that this trace can be ...
2
votes
1answer
186 views
Relationship between trace distance and total variation distance
Consider two quantum states $\rho$ and $\sigma$ and the probability distributions induced by measuring both of them in the standard basis. Let’s call the probability distributions $p_{\rho}$ and $p_{\...
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vote
2answers
277 views
Relation between trace distance and inner product between pure states
Let $|\phi\rangle,|\psi\rangle$ be two state vectors, and let $d=\frac{1}{2}\mathrm{Tr}(\sqrt{(|\phi\rangle\langle\phi|-|\psi\rangle\langle\psi|)^2})$ be their trace distance. Then it will always hold ...
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vote
1answer
44 views
Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)
This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
1
vote
1answer
154 views
Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?
The motivation for this question comes from trace distance. For any two states $\rho, \sigma$, the trace distance $T(\rho, \sigma)$ is given by
$$T(\rho, \sigma) = |\rho - \sigma|_1,$$
where $|\cdot|...
4
votes
1answer
277 views
Saturating the Fuchs-van de Graaf inequality
It is well-known that one side of the Fuchs-van de Graaf inequality is saturated for pure states, i.e. $F(\rho,\sigma)^2 = 1-d(\rho,\sigma)^2$ when $\rho$ and $\sigma$ are pure (here we are using the ...
2
votes
2answers
62 views
What can be said about the closeness of two states if the difference of their fidelity measured with respect to a fixed state is close to 0?
Suppose I have two states $\rho$ and $\sigma$. We are given that,
$$Tr((\rho - \sigma)|\psi\rangle\langle\psi|) \geq \epsilon$$
where $|\psi\rangle$ is a fixed state and $\epsilon \rightarrow 0$,
...
3
votes
1answer
48 views
Does the trace distance between marginals bound the distance between the overall states?
If the quantum states of the subsystems of two systems are close (for example: in terms of trace distance), are the states of the larger systems also close, i.e., if
$$ ||\rho_A - \rho_{A^\prime}||\...
5
votes
2answers
60 views
Prove that for one-qubit unitaries $\text{Tr}|U-V|=2\max_\psi\|(U-V)|\psi\rangle\|$
Given two 1-qubit rotations $U=R_n (\theta)$ and $V=R_m(\phi)$ with $n$ and $m$ vectors defining a rotation and $\theta, \phi$ angles, define $D(U,V)=Tr(|U-V|)$ where $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
0
votes
2answers
106 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 ...
4
votes
1answer
160 views
Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?
I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following:
$$
F(\rho, \sigma) := \...
2
votes
1answer
108 views
Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
2
votes
1answer
76 views
It two unitaries are delta apart in trace norm, then what is the trace norm of outputs states when the same input state is applied to two unitaries?
Suppose we are given two unitary matrices $U$ and $V$, with the following guarantee,
$$||U - V||_1 \geqslant \delta$$
for some $\delta \geqslant 0$.
We apply an input density state $\rho$ ...
9
votes
1answer
417 views
Does the trace distance have a geometric interpretation?
Consider the trace distance between two quantum states $\rho,\sigma$, defined via
$$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$
where $|A|\equiv\sqrt{A^\dagger A}$.
When $\rho$ and $\sigma$...
6
votes
2answers
157 views
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
I've been trying to figure this out for a while and I'm totally lost.
My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$
So far ...
6
votes
1answer
231 views
Is the diamond norm subadditive under composition?
The diamond norm distance between two operations is the maximum trace distance between their outputs for any input (including inputs entangled with qubits not being operated on).
Is it the case that ...
8
votes
2answers
1k views
What is intuition for the trace distance between quantum states?
Given two mixed states $\rho$ and $\sigma$, the trace distance between the states is defined by $\sum_{i=1}^n |\lambda_i|$, where $\lambda_i$'s are eigenvalues of $\rho - \sigma$.
I know the ...
3
votes
1answer
183 views
Trace distance of two classical-quantum states
I have these two classical-quantum states:
$$\rho = \sum_{a} \lvert a\rangle \langle a\lvert \otimes q^a \\
\mu = \sum_{a} \lvert a\rangle \langle a\lvert \otimes r^a $$
Where $a$ are the classical ...