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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How to prove the strong convexity of the trace distance?
On page $408$ of Nielsen & Chuang in the step going from equation $(9.48)$ to $(9.49)$, I don't see how:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i) \leq D(p_i, q_i)$$
I proceed as follows:
$$\sum\...
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How many measurements are needed to distinguish two fixed density matrices?
Suppose there are two fixed density matrices $\rho_1$ and $\rho_2$ are prepared for equal probability. Can we say something about the minimum number of measurements required to distinguish the two ...
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Bounds relating min-fidelity and induced one-norm
Consider two CPTP maps $M_{A\rightarrow B}$ and $N_{A\rightarrow B}$. Let $\Phi = M - N$.
To distinguish between the two maps, there are several measures but here I want to compare two of them. The ...
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Question regarding the trace-preserving quantum operator trace distance
In Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, the proof of Theorem 9.2 (Trace-preserving quantum operations are contractive) on ...
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Prove that the trace distance is upper-bounded by the Hilbert-Schmidt distance
In (Haah et al. 2015), in the third page, second column, the authors use the following result: given a pair of states $\rho,\sigma$, we have
$$
\|\rho-\sigma\|_1 \le 2\sqrt{\min(\operatorname{rank}(\...
glS♦
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Why is state discrimination possible to infidelity $\delta$ using $n=\Theta(1/\delta)$ states?
In (Haah et al. 2015), in the first section, the authors study the asymptotic behaviours of fidelity and trace distance between $\rho^{\otimes n}$ and $\sigma^{\otimes n}$ for some given pair of ...
glS♦
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Prove the triangle inequality for the trace norm: $\|M+N\|_1\le \|M\|_1+\|N\|_1$
I have been trying to show that $$||M+N|| \le ||M|| + ||N||$$ However, I seem to be missing some fundamental property of either how the trace or square root acts on these sums of matrices, or how the ...
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Prove that a channel is close to acting on only one system
Background
Suppose I have a quantum channel $\Phi:B(\mathcal{H}_1)\rightarrow B(\mathcal{H}_1)\otimes B(\mathcal{H}_2)$, such that there is some small $\epsilon$ such that for any two input states $\...
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Why is the fidelity, rather than the trace distance, the standard choice to compare quantum states?
I don't think it's particularly controversial to say that the "standard" way people use to compare quantum states is via the fidelity. Yes, sometimes the trace distance is used as well, but ...
glS♦
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What does distinguishability mean in this case?
In a lecture, we were given the following example to explain the operational significance of the trace distance. Suppose that Alice prepares one of two (known) states $\rho_0$ or $\rho_1$ with equal ...
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Prove that the Bures metric satisfies a contractive property and has unitary invariance
In this paper, the authors assert that the Bures metric satisfies a contractive property and has unitary invariance. These terms aren't defined or proved in the paper, nor is any reference given for a ...
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Understanding conditional $L_2$ distances
I see that conditional $L_2$ distances from uniform are defined in the following way:
$L_2(\rho_{AB}\vert \sigma_B)= \text{tr}\left(((\rho_{AB}- \mu_{A} \otimes \rho_{B}) (\mathbb{I}_A \otimes \...
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Game formulation of Quantum GAN
Quantum Generative Adversarial Network (QuGAN) generates a desired quantum state via a minimax game between generator and discriminator (equivalently, it's optimizing a trace distance between ...
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maximization of trace between two operators with respect to different norm constraints
I want to maximize $\text{Tr}(XY)$ over $X$ for fixed $Y$, where $X$ and $Y$ are both hermitian (but doesn't necessarily positive) operators, and $X$ is constrained by its p-norm bounded by $1$, i.e. $...
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Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?
Suppose I have an arbitrary orthogonal projector $\Pi$ and two density operators $\rho, \sigma$. Is it true that:
$$ ||\Pi (\sigma - \rho) \Pi||_1 \le || \sigma - \rho ||_1 $$
where $||\cdot||_1$ ...
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Helstrom Measurement when two quantum states are close
I've been reading a paper about Entangled-quantum GAN (see this PDF) and wondering why descriptions below Eq.(3) in the paper are in fact true.
To summarize the description, suppose we have two ...
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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 $|...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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{...
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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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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
$$\|\rho - \vert\psi\rangle\langle\psi\vert\|_1 \leq \varepsilon,...
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How to find the distance between a given $\rho$ and the nearest pure state(s)?
I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state:
$$
\min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
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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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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
$...
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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.
...
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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 ...
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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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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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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 $\...
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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 ...
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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:
$$
\frac{1}{2}||\...
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Prove that the trace norm is dual to the spectral norm
Suppose $A\in L(X,Y)$. $||\cdot||$ denotes spectral norm and denotes the largest singular value of a matrix, i.e. the largest eigenvalue of $\sqrt{A^*A}$.
$||\cdot||_{tr}$ denotes trace norm. We have ...
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Trace distance bound after partial trace
Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.}
I know that the trace distance ...
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Quantum marginal problem - constructing a global state from reduced states
Consider a bipartite state $\rho_{AB}$ with reduced states $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$.
Suppose one obtains states $\rho'_{A}$ and $\rho'_{B}$ such that $\|\...
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How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?
Let's say I have 2 density operators $A$ and $B$. Now, here is what I am trying to calculate:
$$\newcommand{\tr}{\operatorname{trace}}
\tr(A \sqrt{B} A \sqrt{B}).
$$
I saw that this trace can be ...
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Relationship between trace distance and total variation distance
Consider two quantum states $\rho$ and $\sigma$ and the probability distributions induced by measuring both of them in the standard basis. Let’s call the probability distributions $p_{\rho}$ and $p_{\...
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Relation between trace distance and inner product between pure states
Let $|\phi\rangle,|\psi\rangle$ be two state vectors, and let $d=\frac{1}{2}\mathrm{Tr}(\sqrt{(|\phi\rangle\langle\phi|-|\psi\rangle\langle\psi|)^2})$ be their trace distance. Then it will always hold ...
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Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)
This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
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Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?
The motivation for this question comes from trace distance. For any two states $\rho, \sigma$, the trace distance $T(\rho, \sigma)$ is given by
$$T(\rho, \sigma) = |\rho - \sigma|_1,$$
where $|\cdot|...
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Saturating the Fuchs-van de Graaf inequality
It is well-known that one side of the Fuchs-van de Graaf inequality is saturated for pure states, i.e. $F(\rho,\sigma)^2 = 1-d(\rho,\sigma)^2$ when $\rho$ and $\sigma$ are pure (here we are using the ...
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What can be said about the closeness of two states if the difference of their fidelity measured with respect to a fixed state is close to 0?
Suppose I have two states $\rho$ and $\sigma$. We are given that,
$$Tr((\rho - \sigma)|\psi\rangle\langle\psi|) \geq \epsilon$$
where $|\psi\rangle$ is a fixed state and $\epsilon \rightarrow 0$,
...
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Does the trace distance between marginals bound the distance between the overall states?
If the quantum states of the subsystems of two systems are close (for example: in terms of trace distance), are the states of the larger systems also close, i.e., if
$$ ||\rho_A - \rho_{A^\prime}||\...
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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 $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
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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 ...
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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:
$$
F(\rho, \sigma) := \...
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Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
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