Irreducible private states

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.

Related papers

On distilling secure key from reducible private states and (non)existence of entangled key-undistillable states

Karol Horodecki, Piotr Ćwikliński, A. Rutkowski +1 more

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 the performance of that protocol, given in terms of regularized relative entropy of entanglement instead of relative entropy of entanglement previously known. This implies in particular that the irreducible private states are all strictly irreducible if and only if the entangled but key-undistillable states ('entangled key-undistilable states') exist. In turn, all the irreducible private states of the dimension 4 ⨂ 4 are strictly irreducible, that is, after an attack on the key part they become separable. Provided the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions.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 the performance of that protocol, given in terms of regularized relative entropy of entanglement instead of relative entropy of entanglement previously known. This implies in particular that the irreducible private states are all strictly irreducible if and only if the entangled but key-undistillable states ('entangled key-undistilable states') exist. In turn, all the irreducible private states of the dimension 4 ⨂ 4 are strictly irreducible, that is, after an attack on the key part they become separable. Provided the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions.

arXiv:1612.08938New Journal of PhysicsAug 14, 20184 citations

Private States, Quantum Data Hiding, and the Swapping of Perfect Secrecy

Matthias Christandl, Roberto Ferrara

An important contribution to the understanding of quantum key distribution has been the discovery of entangled states from which secret bits, but no maximally entangled states, can be extracted [Horodecki et al., Phys. Rev. Lett. 94, 200501 (2005)PRLTAO0031-900710.1103/PhysRevLett.94.200501]. The construction of those states was based on an intuition that the quantum mechanical phenomena of data hiding and privacy might be related. In this Letter we firmly connect these two phenomena and highlight three aspects of this result. First, we simplify the definition of the secret key rate. Second, we give a formula for the one-way distillable entanglement of certain private states. Third, we consider the problem of extending the distance of quantum key distribution with help of intermediate stations, a setting called the quantum key repeater. We show that for protocols that first distill private states, it is essentially optimal to use the standard quantum repeater protocol based on entanglement distillation and entanglement swapping.An important contribution to the understanding of quantum key distribution has been the discovery of entangled states from which secret bits, but no maximally entangled states, can be extracted [Horodecki et al., Phys. Rev. Lett. 94, 200501 (2005)PRLTAO0031-900710.1103/PhysRevLett.94.200501]. The construction of those states was based on an intuition that the quantum mechanical phenomena of data hiding and privacy might be related. In this Letter we firmly connect these two phenomena and highlight three aspects of this result. First, we simplify the definition of the secret key rate. Second, we give a formula for the one-way distillable entanglement of certain private states. Third, we consider the problem of extending the distance of quantum key distribution with help of intermediate stations, a setting called the quantum key repeater. We show that for protocols that first distill private states, it is essentially optimal to use the standard quantum repeater protocol based on entanglement distillation and entanglement swapping.

arXiv:1609.04696Physical Review LettersNov 30, 201719 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

Limitations for private randomness repeaters

Karol Horodecki, Ryszard Paweł Kostecki, Roberto Salazar +1 more

Cryptographic protocols are often based on the two main resources: private randomness and private key. In this paper, we develop the relationship between these two resources. First, we show that any state containing perfect, directly accessible, private key (a private state) is a particular case of the state containing perfect, directly accessible, private randomness (an independent state). We then demonstrate a fundamental limitation on the possibility of transferring the privacy of random bits in quantum networks with an intermediate repeater station. More precisely, we provide an upper bound on the rate of repeated randomness in this scenario, similar to the one derived for private key repeaters. This bound holds for states with positive partial transposition. We further demonstrate the power of this upper bound by showing a gap between the localizable and the repeated private randomness for separable Werner states. In the case of restricted class of operations, we provide also a bound on repeated randomness which holds for arbitrary states.Cryptographic protocols are often based on the two main resources: private randomness and private key. In this paper, we develop the relationship between these two resources. First, we show that any state containing perfect, directly accessible, private key (a private state) is a particular case of the state containing perfect, directly accessible, private randomness (an independent state). We then demonstrate a fundamental limitation on the possibility of transferring the privacy of random bits in quantum networks with an intermediate repeater station. More precisely, we provide an upper bound on the rate of repeated randomness in this scenario, similar to the one derived for private key repeaters. This bound holds for states with positive partial transposition. We further demonstrate the power of this upper bound by showing a gap between the localizable and the repeated private randomness for separable Werner states. In the case of restricted class of operations, we provide also a bound on repeated randomness which holds for arbitrary states.

arXiv:2003.07086Physical Review AJul 16, 20201 citation

Bound entangled states with extremal properties

Piotr Badzia ̧g, Karol Horodecki, Michał Horodecki +2 more

Following recent work of Beigi and Shor, we investigate positive partial transpose (PPT) states that are ``heavily entangled.'' We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least $1/4$ . We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of $2\ensuremath{-}\ensuremath{\epsilon}$ from the separable states, we need a dimension of ${2}^{\mathrm{poly}[log(\frac{1}{\ensuremath{\epsilon}})]}$, as opposed to ${2}^{\mathrm{poly}(\frac{1}{\ensuremath{\epsilon}})}$ given by the construction of Beigi and Shor [J. Math. Phys. 51, 042202 (2010)]. We do so by exploiting the so-called private states, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.Following recent work of Beigi and Shor, we investigate positive partial transpose (PPT) states that are ``heavily entangled.'' We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least $1/4$. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of $2\ensuremath{-}\ensuremath{\epsilon}$ from the separable states, we need a dimension of ${2}^{\mathrm{poly}[log(\frac{1}{\ensuremath{\epsilon}})]}$, as opposed to ${2}^{\mathrm{poly}(\frac{1}{\ensuremath{\epsilon}})}$ given by the construction of Beigi and Shor [J. Math. Phys. 51, 042202 (2010)]. We do so by exploiting the so-called private states, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.

arXiv:1309.7992Physical Review AJul 2, 201410 citations

Construction and properties of a class of private states in arbitrary dimensions

A. Rutkowski, Michał Studziński, Piotr Ćwikliński +1 more

We present a construction of quantum states in dimension $d$ that has at least 1 dit of ideal key, called private dits (pdits), which covers most of the known examples of private bits (pbits) $d=2$ . We examine properties of this class of states, focusing mostly on its distance to the set of separable states $\mathcal{S}$ , showing that for a fixed dimension of key part ${d}_{k}$ , the distance increases with ${d}_{s}$ . We provide explicit examples of positive partial transpose states (in $d$ dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of $\mathcal{S}$ is $2\ensuremath{-}\ensuremath{\epsilon}$, where $d$ scales with $\ensuremath{\epsilon}$ as $d\ensuremath{\propto}1/{\ensuremath{\epsilon}}^{3}$, as opposed to $d\ensuremath{\propto}{2}^{{[log(4/\ensuremath{\epsilon})]}^{2}}$ obtained by Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. [Phys. Rev. A 90, 012301 (2014)]. We do not use boosting (taking many copies of pdits to boost the distance) as in the Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. paper.We present a construction of quantum states in dimension $d$ that has at least 1 dit of ideal key, called private dits (pdits), which covers most of the known examples of private bits (pbits) $d=2$. We examine properties of this class of states, focusing mostly on its distance to the set of separable states $\mathcal{S}$, showing that for a fixed dimension of key part ${d}_{k}$, the distance increases with ${d}_{s}$. We provide explicit examples of positive partial transpose states (in $d$ dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of $\mathcal{S}$ is $2\ensuremath{-}\ensuremath{\epsilon}$, where $d$ scales with $\ensuremath{\epsilon}$ as $d\ensuremath{\propto}1/{\ensuremath{\epsilon}}^{3}$, as opposed to $d\ensuremath{\propto}{2}^{{[log(4/\ensuremath{\epsilon})]}^{2}}$ obtained by Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. [Phys. Rev. A 90, 012301 (2014)]. We do not use boosting (taking many copies of pdits to boost the distance) as in the Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. paper.

arXiv:1410.0378Physical Review AJan 27, 20150 citations

4\otimes 4

1/4

. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of $2\ensuremath{-}\ensuremath{\epsilon}$ from the separable states, we need a dimension of ${2}^{\mathrm{poly}[log(\frac{1}{\ensuremath{\epsilon}})]}$, as opposed to ${2}^{\mathrm{poly}(\frac{1}{\ensuremath{\epsilon}})}$ given by the construction of Beigi and Shor [J. Math. Phys. 51, 042202 (2010)]. We do so by exploiting the so-called private states, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.

We present a construction of quantum states in dimension

d

that has at least 1 dit of ideal key, called private dits (pdits), which covers most of the known examples of private bits (pbits)

d=2

. We examine properties of this class of states, focusing mostly on its distance to the set of separable states

\mathcal{S}

, showing that for a fixed dimension of key part

{d}_{k}

, the distance increases with

{d}_{s}

. We provide explicit examples of positive partial transpose states (in

d

dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of

\mathcal{S}

is $2\ensuremath{-}\ensuremath{\epsilon}$, where

d

scales with $\ensuremath{\epsilon}$ as $d\ensuremath{\propto}1/{\ensuremath{\epsilon}}^{3}$, as opposed to $d\ensuremath{\propto}{2}^{{[log(4/\ensuremath{\epsilon})]}^{2}}$ obtained by Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. [Phys. Rev. A 90, 012301 (2014)]. We do not use boosting (taking many copies of pdits to boost the distance) as in the Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. paper.