AIP conference proceedings

Randomizing quantum states to Shatten p -norm for all p ≥ 1

Kabgyun Jeong

Published Jan 1, 2014• 1 citation

Abstract

We formularize a method for randomizing quantum states with respect to the Shatten p-norms (p ≥ 1) in trace class. In particular, this work includes the operator norm, p = ∞, and the trace norm, p = 1, simultaneously in a single statement via McDiarmid's inequality and a net construction.

DOI: 10.1063/1.4903127

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AIP conference proceedings

Randomizing quantum states to Shatten p -norm for all p ≥ 1

Kabgyun Jeong

Published Jan 1, 2014• 1 citation

Publisher

Abstract

DOI: 10.1063/1.4903127

Related papers

Randomizing quantum states in Shatten $p$ -norms

Kabgyun Jeong

In this paper, we formularize a method for randomizing quantum states with respect to the Schatten $p$ -norm ( $p\ge1$ ). Our theorem includes the Lemma 2.2 of Hayden and Winter [Commun. Math. Phys. {\bf 284}, 263--280 (2008)] for the norm case of $p>1$ . We exploit the methods of a net construction on the unit sphere and McDiard's inequality in probability theory, and then we prove certain general cases (all $p$ ) of randomization tool for quantum states, which includes the operator norm and trace norm simultaneously in a single statement.In this paper, we formularize a method for randomizing quantum states with respect to the Schatten $p$-norm ($p\ge1$). Our theorem includes the Lemma 2.2 of Hayden and Winter [Commun. Math. Phys. {\bf 284}, 263--280 (2008)] for the norm case of $p>1$. We exploit the methods of a net construction on the unit sphere and McDiard's inequality in probability theory, and then we prove certain general cases (all $p$) of randomization tool for quantum states, which includes the operator norm and trace norm simultaneously in a single statement.

arXiv:1204.1813Apr 9, 20120 citations

Randomizing Channels in Quantum Information Theory

정갑균

The randomization of quantum states in Quantum Information Theory is an important issue and has many applications on information processing. Such a method for randomizing quantum states is given by a completely positive and trace-preserving map known as an epsilon-randomizing map or a random unitary channel.\nIn this thesis, we represent a general formula for the\nepsilon-randomizing maps with respect to the p-norm of\nd dimensional quantum states and introduce some applications. Recently, the epsilon-randomizing channel together with its complementary channel gives a key technique for the proof of the additivity conjecture, that is, the additivity violation for the classical\ncapacity of quantum channel, which is the motivation of our work.\n\nThere are plenty of applications of the epsilon-randomizing channels in the view of reducing communication resources as well as the approximate encryption and decryption of quantum states. We explicitly investigate several applications such as\napproximate quantum state sharing, approximate state merging, and quantum sequential transmission protocol.The randomization of quantum states in Quantum Information Theory is an important issue and has many applications on information processing. Such a method for randomizing quantum states is given by a completely positive and trace-preserving map known as an epsilon-randomizing map or a random unitary channel.\nIn this thesis, we represent a general formula for the\nepsilon-randomizing maps with respect to the p-norm of\nd dimensional quantum states and introduce some applications. Recently, the epsilon-randomizing channel together with its complementary channel gives a key technique for the proof of the additivity conjecture, that is, the additivity violation for the classical\ncapacity of quantum channel, which is the motivation of our work.\n\nThere are plenty of applications of the epsilon-randomizing channels in the view of reducing communication resources as well as the approximate encryption and decryption of quantum states. We explicitly investigate several applications such as\napproximate quantum state sharing, approximate state merging, and quantum sequential transmission protocol.

Jan 1, 20121 citation

A remark on the paper ``Randomizing quantum states: Constructions and applications''

Guillaume Aubrun

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Approximate Private Quantum Channels on Fermionic Gaussian Systems

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The private quantum channel (PQC) maps any quantum state to the maximally mixed state for the discrete as well as the bosonic Gaussian quantum systems, and it has fundamental meaning on the quantum cryptographic tasks and the quantum channel capacity problems. In this paper, we primally introduce a notion of approximate private quantum channel (ε-PQC) on fermionic Gaussian systems (i.e., ε-FPQC), and construct its explicit form of the fermionic (Gaussian) private quantum channel. First of all, we suggest a general structure for ε-FPQC on the fermionic Gaussian systems with respect to the Schatten p-norm class, and then we give an explicit proof of the statement in the trace norm case. In addition, we study that the cardinality of a set of fermionic unitary operators agrees on the ε-FPQC condition in the trace norm case. This result may give birth to intuition on the construction of emerging fermionic Gaussian quantum communication or computing systems.The private quantum channel (PQC) maps any quantum state to the maximally mixed state for the discrete as well as the bosonic Gaussian quantum systems, and it has fundamental meaning on the quantum cryptographic tasks and the quantum channel capacity problems. In this paper, we primally introduce a notion of approximate private quantum channel (ε-PQC) on fermionic Gaussian systems (i.e., ε-FPQC), and construct its explicit form of the fermionic (Gaussian) private quantum channel. First of all, we suggest a general structure for ε-FPQC on the fermionic Gaussian systems with respect to the Schatten p-norm class, and then we give an explicit proof of the statement in the trace norm case. In addition, we study that the cardinality of a set of fermionic unitary operators agrees on the ε-FPQC condition in the trace norm case. This result may give birth to intuition on the construction of emerging fermionic Gaussian quantum communication or computing systems.

arXiv:2003.11907Journal of Quantum Information ScienceJan 1, 20211 citation

Global vs. Product Observables in Bipartite Quantum Systems: The Sharp Bound

Zhi Li, Xiaofei Shi

To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every $z\in M_n\otimes M_m$ , we prove $\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon$ , where $\|\cdot\|_1$ is the trace norm and $\|\cdot\|_\varepsilon$ is the injective tensor norm associated with the trace norms on $M_n$ and $M_m$ . To prove the upper bound, we establish an $L_1$ noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient $\sqrt{2}$ is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every $z\in M_n\otimes M_m$, we prove $\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon$, where $\|\cdot\|_1$ is the trace norm and $\|\cdot\|_\varepsilon$ is the injective tensor norm associated with the trace norms on $M_n$ and $M_m$. To prove the upper bound, we establish an $L_1$ noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient $\sqrt{2}$ is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.

quant-pharXiv:2608.06235Aug 7, 20260 citations

In this paper, we formularize a method for randomizing quantum states with respect to the Schatten

p

-norm (

p\ge1

). Our theorem includes the Lemma 2.2 of Hayden and Winter [Commun. Math. Phys. {\bf 284}, 263--280 (2008)] for the norm case of

p>1

. We exploit the methods of a net construction on the unit sphere and McDiard's inequality in probability theory, and then we prove certain general cases (all

p

) of randomization tool for quantum states, which includes the operator norm and trace norm simultaneously in a single statement.

d \log d

random unitary operators provide examples of such channels on $\C^d$. We show that a simple trick improves the efficiency of the argument and reduces the number of unitary operators to roughly

d

z\in M_n\otimes M_m

, we prove

\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon

, where

\|\cdot\|_1

is the trace norm and

\|\cdot\|_\varepsilon

is the injective tensor norm associated with the trace norms on

M_n

and

M_m

. To prove the upper bound, we establish an

L_1

noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient

\sqrt{2}

is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.