In quantum information theory, it is widely believed that entanglement concentration for bipartite pure states is asymptotically reversible. In order to examine this, we give a precise formulation of the problem, and show a trade-off relation between performance and reversibility, which implies the irreversibility of entanglement concentration. Then, we regard entanglement concentration as entangled state compression in an entanglement storage with lower dimension. Because of the irreversibility of entanglement concentration, an initial state can not be completely recovered after the compression process and a loss inevitably arises in the process. We numerically calculate this loss and also derive for it a highly accurate analytical approximation.In quantum information theory, it is widely believed that entanglement concentration for bipartite pure states is asymptotically reversible. In order to examine this, we give a precise formulation of the problem, and show a trade-off relation between performance and reversibility, which implies the irreversibility of entanglement concentration. Then, we regard entanglement concentration as entangled state compression in an entanglement storage with lower dimension. Because of the irreversibility of entanglement concentration, an initial state can not be completely recovered after the compression process and a loss inevitably arises in the process. We numerically calculate this loss and also derive for it a highly accurate analytical approximation.

arXiv:1305.6250May 27, 20134 citations

Irreversibility of Entanglement Concentration for Pure State

Wataru Kumagai, Masahito Hayashi

For a pure state $ψ$ on a composite system $\mathcal{H}_A\otimes\mathcal{H}_B$ , both the entanglement cost $E_C(ψ)$ and the distillable entanglement $E_D(ψ)$ coincide with the von Neumann entropy $H(\mathrm{Tr}_{B}ψ)$ . Therefore, the entanglement concentration from the multiple state $ψ^{\otimes n}$ of a pure state $ψ$ to the multiple state $Φ^{\otimes L_n}$ of the EPR state $Φ$ seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate ${L_n}/{n}$ goes to $H(\mathrm{Tr}_{B}ψ)$ . In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state $ψ^{\otimes M_n}$ after the concentration for $ψ^{\otimes n}$ , we evaluate the asymptotic behavior of the loss number $n-M_n$ of $ψ$ . This evaluation is thought to be closely related to the entanglement compression in distant parties.For a pure state $ψ$ on a composite system $\mathcal{H}_A\otimes\mathcal{H}_B$, both the entanglement cost $E_C(ψ)$ and the distillable entanglement $E_D(ψ)$ coincide with the von Neumann entropy $H(\mathrm{Tr}_{B}ψ)$. Therefore, the entanglement concentration from the multiple state $ψ^{\otimes n}$ of a pure state $ψ$ to the multiple state $Φ^{\otimes L_n}$ of the EPR state $Φ$ seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate ${L_n}/{n}$ goes to $H(\mathrm{Tr}_{B}ψ)$. In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state $ψ^{\otimes M_n}$ after the concentration for $ψ^{\otimes n}$, we evaluate the asymptotic behavior of the loss number $n-M_n$ of $ψ$. This evaluation is thought to be closely related to the entanglement compression in distant parties.

arXiv:1205.4370May 20, 20122 citations

Asymptotic reversibility of LOCC conversions

Kosuke Ito, Wataru Kumagai, Masahito Hayashi

Recently, two of the authors showed that entanglement concentration is irreversible. However, it is still not clear what kind of LOCC conversion is reversible. We derive the necessary and sufficient condition for reversibility of LOCC conversion between two bipartite pure entangled states in an asymptotic setting. Simultaneously, we evaluate how many copies of the initial state is to be lost to overcome irreversibility of LOCC conversion. Our result is useful for designing how to store entangled states via LOCC operations without error.Recently, two of the authors showed that entanglement concentration is irreversible. However, it is still not clear what kind of LOCC conversion is reversible. We derive the necessary and sufficient condition for reversibility of LOCC conversion between two bipartite pure entangled states in an asymptotic setting. Simultaneously, we evaluate how many copies of the initial state is to be lost to overcome irreversibility of LOCC conversion. Our result is useful for designing how to store entangled states via LOCC operations without error.

IEEE International Symposium on Information Theory (ISIT)Jun 1, 20140 citations

Very Strong Irreversibility of Quantum Entanglement

Tulja Varun Kondra, Raphael Brinster, Hermann Kampermann +2 more

The manipulation of quantum entanglement is fundamentally irreversible: some mixed entangled states require pure entanglement for their preparation, although no pure entanglement can be recovered from them by local operations and classical communication. This irreversibility is known to persist even under the maximal class of operations that do not generate entanglement, revealing a fundamental distinction between entanglement theory and thermodynamics. We construct cases for which any attempt to restore reversibility necessarily incurs an error that increases exponentially with the number of copies. Technically, we demonstrate a strict separation between the exponential strong-converse distillable entanglement and the exponential strong-converse entanglement cost. Our result resolves a conjecture posed by Lami and Regula (Nat. Phys. 19, 184-189 (2023)) and strengthens it by showing that the irreversibility of entanglement persists even at the level of polynomially (in the number of copies) growing error. We further derive a semidefinite-programming lower bound on the exponential strong-converse cost under non-entangling operations. Finally, for the class of completely PPT-preserving operations, we construct analytically solvable families of antisymmetric states exhibiting the exponential strong-converse irreversibility. Remarkably, to our knowledge, no analogous separation between exponential strong converse cost and the analogous distillable entanglement is currently known even under the more restrictive class of LOCC operations.The manipulation of quantum entanglement is fundamentally irreversible: some mixed entangled states require pure entanglement for their preparation, although no pure entanglement can be recovered from them by local operations and classical communication. This irreversibility is known to persist even under the maximal class of operations that do not generate entanglement, revealing a fundamental distinction between entanglement theory and thermodynamics. We construct cases for which any attempt to restore reversibility necessarily incurs an error that increases exponentially with the number of copies. Technically, we demonstrate a strict separation between the exponential strong-converse distillable entanglement and the exponential strong-converse entanglement cost. Our result resolves a conjecture posed by Lami and Regula (Nat. Phys. 19, 184-189 (2023)) and strengthens it by showing that the irreversibility of entanglement persists even at the level of polynomially (in the number of copies) growing error. We further derive a semidefinite-programming lower bound on the exponential strong-converse cost under non-entangling operations. Finally, for the class of completely PPT-preserving operations, we construct analytically solvable families of antisymmetric states exhibiting the exponential strong-converse irreversibility. Remarkably, to our knowledge, no analogous separation between exponential strong converse cost and the analogous distillable entanglement is currently known even under the more restrictive class of LOCC operations.

quant-pharXiv:2607.27195Jul 30, 20260 citations

Entanglement Concentration

Mark M. Wilde

Entanglement is one of the most useful resources in quantum information processing. If a sender and receiver share noiseless entanglement in the form of maximally entangled states, then Chapter 6 showed how they can teleport quantum bits between each other with the help of classical communication, or they can double the capacity of a noiseless qubit channel for transmitting classical information. We will see further applications in Chapter 20 where they can exploit noiseless entanglement to assist in the transmission of classical or quantum data over a noisy quantum channel. Given the utility of maximal entanglement, a reasonable question is to ask what a sender and receiver can accomplish if they share pure entangled states that are not maximally entangled. In the quantum Shannon-theoretic setting, we make the further assumption that the sender and receiver can share many copies of these pure entangled states. We find out in this chapter that they can “concentrate” these non-maximally entangled states to maximally entangled ebits, and the optimal rate at which they can do so in the asymptotic limit is equal to the “entropy of entanglement” (the von Neumann entropy of half of one copy of the original state). Entanglement concentration is thus another fundamental task in noiseless quantum Shannon theory, and it gives a different operational interpretation to the von Neumann entropy. Entanglement concentration is perhaps complementary to Schumacher compression in the sense that it gives a firm quantum information-theoretic interpretation of the term “ebit” (just as Schumacher compression did for the term “qubit”), and it plays a part in demonstrating how the entropy of entanglement is the unique measure of entanglement for pure bipartite states.Entanglement is one of the most useful resources in quantum information processing. If a sender and receiver share noiseless entanglement in the form of maximally entangled states, then Chapter 6 showed how they can teleport quantum bits between each other with the help of classical communication, or they can double the capacity of a noiseless qubit channel for transmitting classical information. We will see further applications in Chapter 20 where they can exploit noiseless entanglement to assist in the transmission of classical or quantum data over a noisy quantum channel. Given the utility of maximal entanglement, a reasonable question is to ask what a sender and receiver can accomplish if they share pure entangled states that are not maximally entangled. In the quantum Shannon-theoretic setting, we make the further assumption that the sender and receiver can share many copies of these pure entangled states. We find out in this chapter that they can “concentrate” these non-maximally entangled states to maximally entangled ebits, and the optimal rate at which they can do so in the asymptotic limit is equal to the “entropy of entanglement” (the von Neumann entropy of half of one copy of the original state). Entanglement concentration is thus another fundamental task in noiseless quantum Shannon theory, and it gives a different operational interpretation to the von Neumann entropy. Entanglement concentration is perhaps complementary to Schumacher compression in the sense that it gives a firm quantum information-theoretic interpretation of the term “ebit” (just as Schumacher compression did for the term “qubit”), and it plays a part in demonstrating how the entropy of entanglement is the unique measure of entanglement for pure bipartite states.

Cambridge University Press eBooksApr 18, 20130 citations

Irreversibility of entanglement distillation for a class of symmetric states

K. G. H. Vollbrecht, Reinhard F. Werner, Michael M. Wolf

We investigate the irreversibility of entanglement distillation for a symmetric $(d+1)$ -parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension $d\ifmmode\times\else\texttimes\fi{}d$. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, so that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure-state entanglement and every operation is allowed, which preserves the positivity of the partial transpose. It is shown that reversibility occurs only in cases where the state is a tagged mixture. The reversible cases are shown to be completely characterized by minimal uncertainty vectors for entropic uncertainty relations.We investigate the irreversibility of entanglement distillation for a symmetric $(d+1)$-parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension $d\ifmmode\times\else\texttimes\fi{}d$. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, so that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure-state entanglement and every operation is allowed, which preserves the positivity of the partial transpose. It is shown that reversibility occurs only in cases where the state is a tagged mixture. The reversible cases are shown to be completely characterized by minimal uncertainty vectors for entropic uncertainty relations.

arXiv:quant-ph/0301072Physical Review AJun 3, 200419 citations

For a pure state

ψ

on a composite system

\mathcal{H}_A\otimes\mathcal{H}_B

, both the entanglement cost

E_C(ψ)

and the distillable entanglement

E_D(ψ)

coincide with the von Neumann entropy

H(\mathrm{Tr}_{B}ψ)

. Therefore, the entanglement concentration from the multiple state

ψ^{\otimes n}

of a pure state

ψ

to the multiple state

Φ^{\otimes L_n}

of the EPR state

Φ

seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate

{L_n}/{n}

goes to

H(\mathrm{Tr}_{B}ψ)

. In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state

ψ^{\otimes M_n}

after the concentration for

ψ^{\otimes n}

, we evaluate the asymptotic behavior of the loss number

n-M_n

ψ

. This evaluation is thought to be closely related to the entanglement compression in distant parties.

We investigate the irreversibility of entanglement distillation for a symmetric

(d+1)

-parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension $d\ifmmode\times\else\texttimes\fi{}d$. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, so that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure-state entanglement and every operation is allowed, which preserves the positivity of the partial transpose. It is shown that reversibility occurs only in cases where the state is a tagged mixture. The reversible cases are shown to be completely characterized by minimal uncertainty vectors for entropic uncertainty relations.