Hereunder, we study the class of irreducible private states that are private states from which all the secret content is accessible via measuring their key part. We provide the first protocol which distills key not only from the key part, but also from the shield if only the state is reducible. We prove also a tighter upper bound on its performance. This implies in particular that the irreducible private states are all strictly irreducible if and only if the entangled but key-undistillable states (bound key states) exist. In turn, all the irreducible private states of the dimension $4\otimes 4$ are strictly irreducible, that is, after an attack on the key part they become separable states. Providing the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we show a possible way to apply our findings in the context of quantum key repeaters. As a step towards this direction, we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions, yet not large enough to reach the goal of application. However this bound leads to the proof of no-key swapping for some private states in the single copy case without using the fact that they are strictly irreducible.Hereunder, we study the class of irreducible private states that are private states from which all the secret content is accessible via measuring their key part. We provide the first protocol which distills key not only from the key part, but also from the shield if only the state is reducible. We prove also a tighter upper bound on its performance. This implies in particular that the irreducible private states are all strictly irreducible if and only if the entangled but key-undistillable states (bound key states) exist. In turn, all the irreducible private states of the dimension $4\otimes 4$ are strictly irreducible, that is, after an attack on the key part they become separable states. Providing the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we show a possible way to apply our findings in the context of quantum key repeaters. As a step towards this direction, we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions, yet not large enough to reach the goal of application. However this bound leads to the proof of no-key swapping for some private states in the single copy case without using the fact that they are strictly irreducible.

Dec 28, 20161 citation

General Paradigm for Distilling Classical Key From Quantum States

Karol Horodecki, Michał Horodecki, Paweł Horodecki +1 more

In this paper, we develop a formalism for distilling a classical key from a quantum state in a systematic way, expanding on our previous work on a secure key from bound entanglement (Horodecki, 2005). More detailed proofs, discussion, and examples are provided of the main results. Namely, we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory, with the only change being that instead of distilling Einstein–Podolsky–Rosen (EPR) pairs, the parties distill private states. The form of these general private states are given, and we show that there are a number of useful ways of expressing them. Some of the private states can be approximated by certain states, which are bound entangled. Thus, distillable entanglement is not a requirement for a private key. We find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel (PQC). We also find a general class of states, which have negative partial transpose (are NPT), but which appear to be bound entangled. The relative entropy distance is shown to be an upper bound on the rate of a key. This allows us to compute theexactvalue of a distillable key for a certain class of private states.In this paper, we develop a formalism for distilling a classical key from a quantum state in a systematic way, expanding on our previous work on a secure key from bound entanglement (Horodecki, 2005). More detailed proofs, discussion, and examples are provided of the main results. Namely, we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory, with the only change being that instead of distilling Einstein–Podolsky–Rosen (EPR) pairs, the parties distill private states. The form of these general private states are given, and we show that there are a number of useful ways of expressing them. Some of the private states can be approximated by certain states, which are bound entangled. Thus, distillable entanglement is not a requirement for a private key. We find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel (PQC). We also find a general class of states, which have negative partial transpose (are NPT), but which appear to be bound entangled. The relative entropy distance is shown to be an upper bound on the rate of a key. This allows us to compute theexactvalue of a distillable key for a certain class of private states.

arXiv:quant-ph/0506189IEEE Transactions on Information TheoryMar 19, 2009147 citations

Secure Key from Bound Entanglement

Karol Horodecki, Michał Horodecki, Paweł Horodecki +1 more

We characterize the set of shared quantum states which contain a cryptographically private key. This allows us to recast the theory of privacy as a paradigm closely related to that used in entanglement manipulation. It is shown that one can distill an arbitrarily secure key from bound entangled states. There are also states that have less distillable private keys than the entanglement cost of the state. In general, the amount of distillable key is bounded from above by the relative entropy of entanglement. Relationships between distillability and distinguishability are found for a class of states which have Bell states correlated to separable hiding states. We also describe a technique for finding states exhibiting irreversibility in entanglement distillation.We characterize the set of shared quantum states which contain a cryptographically private key. This allows us to recast the theory of privacy as a paradigm closely related to that used in entanglement manipulation. It is shown that one can distill an arbitrarily secure key from bound entangled states. There are also states that have less distillable private keys than the entanglement cost of the state. In general, the amount of distillable key is bounded from above by the relative entropy of entanglement. Relationships between distillability and distinguishability are found for a class of states which have Bell states correlated to separable hiding states. We also describe a technique for finding states exhibiting irreversibility in entanglement distillation.

arXiv:quant-ph/0309110Physical Review LettersApr 26, 2005340 citations

Bound Entangled States with a Private Key and their Classical Counterpart

Māris Ozols, Graeme Smith, John A. Smolin

Entanglement is a fundamental resource for quantum information processing. In its pure form, it allows quantum teleportation and sharing classical secrets. Realistic quantum states are noisy and their usefulness is only partially understood. Bound-entangled states are central to this question--they have no distillable entanglement, yet sometimes still have a private classical key. We present a construction of bound-entangled states with a private key based on classical probability distributions. From this emerge states possessing a new classical analogue of bound entanglement, distinct from the long-sought bound information. We also find states of smaller dimensions and higher key rates than previously known. Our construction has implications for classical cryptography: we show that existing protocols are insufficient for extracting private key from our distributions due to their "bound-entangled" nature. We propose a simple extension of existing protocols that can extract a key from them.Entanglement is a fundamental resource for quantum information processing. In its pure form, it allows quantum teleportation and sharing classical secrets. Realistic quantum states are noisy and their usefulness is only partially understood. Bound-entangled states are central to this question--they have no distillable entanglement, yet sometimes still have a private classical key. We present a construction of bound-entangled states with a private key based on classical probability distributions. From this emerge states possessing a new classical analogue of bound entanglement, distinct from the long-sought bound information. We also find states of smaller dimensions and higher key rates than previously known. Our construction has implications for classical cryptography: we show that existing protocols are insufficient for extracting private key from our distributions due to their "bound-entangled" nature. We propose a simple extension of existing protocols that can extract a key from them.

arXiv:1305.0848Physical Review LettersMar 18, 201425 citations

Limitations on quantum key repeaters for all key correlated states

Sikorski, Leonard, Karol Horodecki, Łukasz Pawela

Quantum key repeater is the backbone of the future Quantum Internet. It is an open problem to determine, for an arbitrary mixed bipartite state shared between the stations of a quantum key repeater, how much key can be generated between its two end-nodes. We place a novel bound on the quantum key repeater rate, which uses the relative entropy distance from, in general, entangled quantum states. It allows us to generalize bounds on key repeaters of M. Christandl and R. Ferrara [Phys. Rev. Lett. 119, 220506]. As in the latter article, we consider a scenario used for measurement-device-independent quantum cryptography. The derived bound, although not tighter, holds for a more general class of states, thereby avoiding the NP-hard separability problem. We show that the repeated key of the broad class of key correlated states can exceed twice the one-way distillable entanglement by at most the max relative entropy of entanglement of its attacked version. We also provide a non-trivial upper bound on the amount of private randomness of a generic independent bit - a state containing one bit of ideal private randomness.Quantum key repeater is the backbone of the future Quantum Internet. It is an open problem to determine, for an arbitrary mixed bipartite state shared between the stations of a quantum key repeater, how much key can be generated between its two end-nodes. We place a novel bound on the quantum key repeater rate, which uses the relative entropy distance from, in general, entangled quantum states. It allows us to generalize bounds on key repeaters of M. Christandl and R. Ferrara [Phys. Rev. Lett. 119, 220506]. As in the latter article, we consider a scenario used for measurement-device-independent quantum cryptography. The derived bound, although not tighter, holds for a more general class of states, thereby avoiding the NP-hard separability problem. We show that the repeated key of the broad class of key correlated states can exceed twice the one-way distillable entanglement by at most the max relative entropy of entanglement of its attacked version. We also provide a non-trivial upper bound on the amount of private randomness of a generic independent bit - a state containing one bit of ideal private randomness.

arXiv:2206.00993Jun 2, 20220 citations

Limiting one-way distillable secret key via privacy testing of extendible states

Vishal Singh, Karol Horodecki, Aby Philip +1 more

The notions of privacy tests and $k$ -extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary $k$ -extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary $k$ -extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of $k$ -unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the $n$ -shot, one-way distillable key of a state and the $n$ -shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.The notions of privacy tests and $k$-extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary $k$-extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary $k$-extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of $k$-unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the $n$-shot, one-way distillable key of a state and the $n$-shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.

quant-pharXiv:2511.04438Nov 7, 20250 citations

4\otimes 4

are strictly irreducible, that is, after an attack on the key part they become separable states. Providing the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we show a possible way to apply our findings in the context of quantum key repeaters. As a step towards this direction, we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions, yet not large enough to reach the goal of application. However this bound leads to the proof of no-key swapping for some private states in the single copy case without using the fact that they are strictly irreducible.

The notions of privacy tests and

k

-extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary

k

-extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary

k

-extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of

k

-unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the

n

-shot, one-way distillable key of a state and the

n

-shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.