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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1answer
40 views

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
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1answer
55 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
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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\...
3
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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
107 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\...
1
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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
111 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
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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
51 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
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2answers
147 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
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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
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1answer
82 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
285 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
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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
183 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_{\...
1
vote
2answers
249 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 ...
1
vote
1answer
43 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
152 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
263 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
47 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
105 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
150 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
106 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
411 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
156 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
180 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 ...