Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1610.05122 · Physical Review A

Symmetrizing cost of quantum states

Eyuri Wakakuwa

Published Mar 27, 2017• 13 citations
arXivHTMLPDF

Abstract

We introduce and analyze a task that we call symmetrization, in which a state of a quantum system, associated with a symmetry group, is transformed by a random unitary operation to a symmetric state. Each element of the unitary ensemble is required to be symmetry preserving, in the sense that it keeps the set of symmetric states invariant. We consider an asymptotic limit of infinitely many copies and vanishingly small error, and analyze the symmetrizing cost, that is the minimum cost of randomness per copy required for symmetrization. We prove that the symmetrizing cost of an arbitrary quantum state is equal to the relative entropy of frameness, thereby providing it with a direct operational meaning.

DOI
10.1103/physreva.95.032328

Related papers

The Deimginarity Cost of Quantum States

Xian Shi

Here we address a task denoted as deimaginarity, which is to transform a state into a real state with the aid of random covariant-free unitary operations. We consider the minimum cost of randomness required for deimaginarity in the scenario of infinite copies and suiciently small error, and we prove that the deimaginarity cost of a state $ρ$ is equal to its regularized relative entropy of imaginarity, which can be seen as an operational interpretation of the relative entropy of imaginarity.

arXiv:2501.12891•Jan 22, 2025•0 citations

Markovianizing cost of tripartite quantum states

Eyuri Wakakuwa, Akihito Soeda, Mio Murao

We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed to a quantum Markov chain conditioned by Bnwith a small error, by a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is an approximate quantum Markov chain. Our results have an application for distributed quantum computation.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2015•2 citations

Markovianizing Cost of Tripartite Quantum States

Eyuri Wakakuwa, Akihito Soeda, Mio Murao

We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of n copies of a tripartite state ρ ABC, and is transformed to a quantum Markov chain conditioned by Bn with a small error, using a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a singleletter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.

arXiv:1504.05805•IEEE Transactions on Information Theory•Dec 21, 2016•16 citations

The Cost of Randomness for Converting a Tripartite Quantum State to be Approximately Recoverable

Eyuri Wakakuwa, Akihito Soeda, Mio Murao

We introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed by a random unitary operation on Anto another state, which is approximately recoverable from its reduced state on AnBn(Case 1) or BnCn (Case 2). We analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly small error of recovery, mainly focusing on the case of pure states. We prove that the minimum cost in Case 1 is equal to the Markovianizing cost of the state, for which a single-letter formula is known. With an additional requirement on the convergence speed of the recovery error, we prove that the minimum cost in Case 2 is also equal to the Markovianizing cost. Our results have an application for distributed quantum computation.

arXiv:1512.06920•IEEE Transactions on Information Theory•Apr 14, 2017•7 citations

The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations

Iman Marvian, Robert W Spekkens

If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state's ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state psi relative to the symmetry group G are completely specified by the characteristic function of the state, defined as chi_psi(g)= where g\in G and U is the unitary representation of interest. For a symmetry described by a compact Lie group G, we show that two pure states can be reversibly interconverted one to the other by symmetric operations if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. Characteristic functions also allow us to easily identify the conditions for one pure state to be converted to another by symmetric operations (in general irreversibly) for the various paradigms of single-copy transformations: deterministic, state-to-ensemble, stochastic and catalyzed.

arXiv:1104.0018•New Journal of Physics•Mar 1, 2013•227 citations

Coherence cost for violating conservation laws

Hiroyasu Tajima, Naoto Shiraishi, Keiji Saito

In isolated quantum systems that have conserved quantities, such as energy and total magnetization, the conserved quantities cannot vary their values with their own dynamics.To violate conservation laws of the system keeping the unitary nature in the system's dynamics, quantum coherence must be supplied from the outer environment.This provides a general and abstract interpretation of the underlying mechanism of unitary operations on systems of interest.In this paper, we consider the amount of quantum coherence cost to realize the desired unitary operation.We derive upper and lower bounds for the coherence cost to implement arbitrary unitary operation within the desired error.These two bounds asymptotically match each other in a small error regime, and give an asymptotic equality.We present applications of our theories to several physical situations such as quantum heat engines and entanglement erasure.

arXiv:1906.04076•Physical Review Research•Dec 15, 2020•32 citations
We introduce and analyze a task that we call symmetrization, in which a state of a quantum system, associated with a symmetry group, is transformed by a random unitary operation to a symmetric state. Each element of the unitary ensemble is required to be symmetry preserving, in the sense that it keeps the set of symmetric states invariant. We consider an asymptotic limit of infinitely many copies and vanishingly small error, and analyze the symmetrizing cost, that is the minimum cost of randomness per copy required for symmetrization. We prove that the symmetrizing cost of an arbitrary quantum state is equal to the relative entropy of frameness, thereby providing it with a direct operational meaning.
Here we address a task denoted as deimaginarity, which is to transform a state into a real state with the aid of random covariant-free unitary operations. We consider the minimum cost of randomness required for deimaginarity in the scenario of infinite copies and suiciently small error, and we prove that the deimaginarity cost of a state ρρρ is equal to its regularized relative entropy of imaginarity, which can be seen as an operational interpretation of the relative entropy of imaginarity.
We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed to a quantum Markov chain conditioned by Bnwith a small error, by a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is an approximate quantum Markov chain. Our results have an application for distributed quantum computation.
We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of n copies of a tripartite state ρ ABC, and is transformed to a quantum Markov chain conditioned by Bn with a small error, using a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a singleletter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.
We introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed by a random unitary operation on Anto another state, which is approximately recoverable from its reduced state on AnBn(Case 1) or BnCn (Case 2). We analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly small error of recovery, mainly focusing on the case of pure states. We prove that the minimum cost in Case 1 is equal to the Markovianizing cost of the state, for which a single-letter formula is known. With an additional requirement on the convergence speed of the recovery error, we prove that the minimum cost in Case 2 is also equal to the Markovianizing cost. Our results have an application for distributed quantum computation.
If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state's ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state psi relative to the symmetry group G are completely specified by the characteristic function of the state, defined as chi_psi(g)= where g\in G and U is the unitary representation of interest. For a symmetry described by a compact Lie group G, we show that two pure states can be reversibly interconverted one to the other by symmetric operations if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. Characteristic functions also allow us to easily identify the conditions for one pure state to be converted to another by symmetric operations (in general irreversibly) for the various paradigms of single-copy transformations: deterministic, state-to-ensemble, stochastic and catalyzed.
In isolated quantum systems that have conserved quantities, such as energy and total magnetization, the conserved quantities cannot vary their values with their own dynamics.To violate conservation laws of the system keeping the unitary nature in the system's dynamics, quantum coherence must be supplied from the outer environment.This provides a general and abstract interpretation of the underlying mechanism of unitary operations on systems of interest.In this paper, we consider the amount of quantum coherence cost to realize the desired unitary operation.We derive upper and lower bounds for the coherence cost to implement arbitrary unitary operation within the desired error.These two bounds asymptotically match each other in a small error regime, and give an asymptotic equality.We present applications of our theories to several physical situations such as quantum heat engines and entanglement erasure.