Jens Eisert:
Certifying continuous-variable quantum systems
Continuous variable quantum systems are key protagonists when it comes to realising quantum information protocols, for good reasons, allowing for accurate and reliable quantum state manipulation. In this context, in many flavours the question arises whether a state preparation or implementation of a protocol has actually been precisely successful. Such questions are specifically important, e.g., in protocols aiming at showing the "quantum supremacy" of quantum over classical machines. In this talk, I will elaborate on this question in two ramifications. In the first part, I will discuss notions of reliable quantum certification for photonic quantum state preparations [1], including ones based on continuous-variables, relying on feasible measurements only, making use of ideas of interacting proof systems. In the second part, I will revisit the geometry of separable Gaussian states [2], based on which optimal witnesses can be constructed from semi-definite problems, now freshly applied to the context of detecting multi-mode entanglement in mechanical solid-state systems [3]. If time allows, I might mention ongoing work on Gaussian channels and superactivation [4].
[1] Reliable quantum certification for photonic quantum technologies, L. Aolita, C. Gogolin, M. Kliesch, J. Eisert, Nature Comm. 6, 8498 (2015).
[2] Optimal entanglement witnesses for continuous-variable systems, P. Hyllus, J. Eisert, New J. Phys. 8, 51 (2006).
[3] In preparation (2016).
[4] In preparation (2016).
Nicolai Friis:
Estimating parameters encoded in Gaussian transformations
We address the issue of precisely estimating small parameters encoded in Bogoliubov transformations, that is, general linear transformations of the modes of a bosonic quantum field. Such Gaussian transformations frequently appear in the context of quantum optics but also in the context of quantum field theory on curved spacetimes. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states for a transformation given in terms of the Bogoliubov coefficients [1]. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires non-classical, but not necessarily entangled states. Our method further allows us to quantify losses in precision arising from being able to control only finitely many modes. We identify a lower bound for these tracing losses and provide a description of the family of “pretty good” states that minimize the losses, while achieving Heisenberg scaling. Finally, we remark on extensions to general mixed states and noisy Gaussian channels [2].
[1] N. Friis, M. Skotiniotis, I. Fuentes, and W. Dür, Phys. Rev. A 92, 022106 (2015) [arXiv:1502.07654].
[2] N. Friis and M. Skotiniotis, in preparation.
Jaromir Fiurasek:
Virtual implementation of noiseless amplification and entanglement distillation and their applications in continuous-variable quantum communication
We consider continuous-variable quantum communication scenarios, where the recipients of the quantum states perform eight-port homodyne detection (i.e. projection onto coherent states) on the received states. We show that in this scenario, several important quantum operations, including noiseless quantum amplification/attenuation and entanglement distillation by iterative Gaussification, do not need to be physically implemented but can simply be emulated in the classical data post-processing stage. This approach enables to circumvent hardware implementation problems with noiseless amplification or iterative multi-copy entanglement distillation. As an illustration of application we present experimental results for iterative entanglement distillation of phase-diffused two-mode squeezed states. Since our procedure necessarily includes the measurement, it is particularly suitable for applications in continuous-variable quantum key distribution, where it could enhance the secure range or tolerable excess noise.
[1] J. Fiurasek and N.J. Cerf, Gaussian postselection and virtual noiseless amplification in continuous-variable quantum key distribution, Phys. Rev. A 86, 060302(R) (2012).
[2] D. Abdelkhalek, M. Syllwasschy, N.J. Cerf, J. Fiurasek, and R. Schnabel, Efficient entanglement distillation without quantum memory, submitted.
Stefano Pirandola:
Fundamental Limits of Repeaterless Quantum Communications
Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks there is a crucial question to answer: What are their optimal rates without quantum repeaters? Our work addresses this basic question for any two parties connected by a quantum channel, without any restriction on their classical communication, which can be unlimited and two-way. We design a method which reduces the most general protocol of quantum communication over a channel to the computation of a simple quantity, that we call entanglement flux. In this way, we bound the ultimate rates that are achievable over the most important bosonic and qubit channels, computing a number of exact formulas for their two-way capacities. In particular, we determine the fundamental rate-loss scaling which affects any optical quantum communication. By setting these limits, our results establish the most general and correct benchmarks for testing the performance of quantum repeaters.
Maciej Lewenstein:
Entangled and non-classical states in qubit-boson systems
We will try to derive sufficient conditions for entanglement based on the analysis of a certain families of positive maps that transform positive operators into separable ones. We will also analyse various criteria characterizing existence of the positive P-representation for separable qubit-boson state. These criteria will be based on the generalized Bochner theorem and on the positive averages of squares fo polynomials
[1] Joint work with Swapan Rana, Manab Bera.
Gerardo Adesso:
Multipartite steering of Gaussian states: monogamy constraints and cryptographical applications
We investigate EPR steerability of multimode Gaussian states by Gaussian measurements. On the one hand, we establish a monogamy--like constraint, preventing joint steerability of a single mode by Gaussian measurements on multiple group of modes; this follows by proving a strong subadditivity inequality for the log-determinant of covariance matrices [1]. On the other hand, we show that a recently introduced quantifier of Gaussian steering [2] obeys a Coffman-Kundu-Wootters--type monogamy inequality for arbitrary multimode Gaussian states. In the case of pure Gaussian states of three modes, we find that the residual steering emerging from such an inequality admits an operational interpretation, related to the key rate of a semi-device-independent implementation of quantum secret sharing, taking into account potential dishonesty of some of the parties [3]. A novel security analysis for the latter protocol is provided [4].
[1] G. Adesso and R. Simon, arXiv:1601.03226 (2016)
[2] I. Kogias, A. R. Lee, S. Ragy, G. Adesso, Phys. Rev. Lett. 114, 060403 (2015)
[3] Y. Xiang et al., in preparation (2016)
[4] I. Kogias et al., in preparation (2016)
Anthony Leverrier:
Security proofs for continuous-variable QKD
Security proofs for quantum key distribution have made tremendous progress in the last few years, and we now have composable security in the finite-size regime for a number of protocols. In the case of BB84 for instance, one even recovers the asymptotic key rate given by the Dewetak-Winter bound for reasonable block sizes.
The situation is more complicated in the case of continuous-variable protocols: composable security in the finite-size setting has only been obtained for a few protocols, and one does not recover the expected asymptotic key rate for reasonable block sizes. In this talk, I will review the state-of-the-art of security proofs for continuous-variable QKD and explain the various challenges that need to be addressed in order to get better security proofs.
[1] Entropy 17, 6072-6092 (2015), arXiv:1506.02888.
Geza Giedke:
Gaussian Local Unitary Equivalence of n-mode Gaussian States and Gaussian LOCC transformations
We investigate which pure multipartite Gaussian states can be transformed into each other by local unitary and non-unitary Gaussian
means. Introducing a easily computed standard form we show that two states are equivalent under Gaussian local unitaries iff their
standard form coincides. We then investigate transformations between non-equivalent states by means of Gaussian local operations assisted by classical communication.
[1] G. Giedke and B. Kraus, Phys. Rev. A 89, 012335 (2014).
Maxim Shirokov:
Conditional mutual information in infinite-dimensional quantum systems and its use
It is shown that the quantum conditional mutual information (defined in a standard way) can be uniquely extended to a lower semicontinuous function on the set of all infinite-dimensional tripartite states which has all basic properties of this quantity valid in the finite-dimensional case. Some corollaries of the lower semicontinuity of the conditional mutual information are discussed. Winter’s type tight continuity bound for the quantum conditional mutual information under energy constraint on one subsystem is obtained and used for analysis of continuity properties of the infinite-dimensional versions of the squashed entanglement and of the entanglement of formation. Some applications to the theory of infinite-dimensional quantum channels and their capacities are also considered. Several open problems (having well known solutions in finite dimensions) are pointed. In particular, the question about extension of any separable state to a short Markov chain is discussed in connection with the basic property of the squashed entanglement.
[1] arXiv:1506.06377.
[2] arXiv:1507.08964.
Raul Garcia-Patron:
Entanglement of Formation for Gaussian states: overview and open questions
Michael Jabbour:
Majorization preservation of Gaussian bosonic channels
It is shown that phase-insensitive Gaussian bosonic channels are majorization-preserving over the set of passive states of the harmonic oscillator. This means that comparable passive states under majorization are transformed into equally comparable passive states. The proof relies on a new preorder relation called Fock-majorization, which coincides with regular majorization for passive states but also induces a mean photon number order, thereby connecting the concepts of energy and disorder of a quantum state. As an application, the consequences of majorization preservation are investigated in the context of the broadcast communication capacity of bosonic Gaussian channels. Most of our results being independent of the bosonic nature of the system under investigation, they could be generalized to other quantum systems and Hamiltonians, providing a general tool that could prove useful in quantum information theory and quantum thermodynamics.
[1] M. G. Jabbour, R. García-Patrón, N. J. Cerf, arXiv:1512.08225 [quant-ph].
Morgan Mitchell:
Quantum enhancement of atomic instruments : learning from LIGO (and growing on GEO-600)
In 2011, the GEO-600 gravitational-wave detector began running with squeezed light to improve its sensitivity, range, and data rate. In 2013, one of the LIGO interferometers was similarly enhanced, becoming the first instrument to . Are there terrestrial applications for these techniques ? I will describe the application of squeezing, optical and atomic, to atomic instruments, e.g. atomic clocks, atomic magnetometers, and atom interferometers. It turns out that atom interferometers offer a few twists that make them an even better platform for quantum enhancement.
Saikat Guha:
On rate-vs.-loss limits in quantum key distribution with and without quantum repeaters
Frédéric Grosshans:
Unidimensional continuous-variable quantum key distribution
We propose the continuous-variable quantum key distribution protocol based on the Gaussian modulation of a single quadrature of the coherent states of light, which is aimed to provide simplified implementation compared to the symmetrically modulated Gaussian coherent-state protocols. The protocol waives the necessity in phase quadrature modulation and the corresponding channel transmittance estimation. The security of the protocol against collective attacks in a generally phase-sensitive Gaussian channels is analysed and is shown achievable upon certain conditions. Robustness of the protocol to channel imperfections is compared to that of the symmetrical coherent-state protocol. The simplified unidimensional protocol is shown possible at a reasonable quantitative cost in terms of key rate and of tolerable channel excess noise.
[1] Joint work with Vladyslav C. Usenko, arXiv:1504.07093, Phys. Rev. A 92, 062337.
Nathan Walk
Channel purification via continuous-variable quantum teleportation with Gaussian postselection
We present a protocol based on continuous-variable quantum teleportation and Gaussian postselection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian operations.
[1] Blandino, R., Walk, N., Lund, A. P., & Ralph, T. C. (2016). Channel purification via continuous-variable quantum teleportation with Gaussian postselection. Physical Review A, 93(1), 012326. http://doi.org/10.1103/PhysRevA.93.012326
Certifying continuous-variable quantum systems
Continuous variable quantum systems are key protagonists when it comes to realising quantum information protocols, for good reasons, allowing for accurate and reliable quantum state manipulation. In this context, in many flavours the question arises whether a state preparation or implementation of a protocol has actually been precisely successful. Such questions are specifically important, e.g., in protocols aiming at showing the "quantum supremacy" of quantum over classical machines. In this talk, I will elaborate on this question in two ramifications. In the first part, I will discuss notions of reliable quantum certification for photonic quantum state preparations [1], including ones based on continuous-variables, relying on feasible measurements only, making use of ideas of interacting proof systems. In the second part, I will revisit the geometry of separable Gaussian states [2], based on which optimal witnesses can be constructed from semi-definite problems, now freshly applied to the context of detecting multi-mode entanglement in mechanical solid-state systems [3]. If time allows, I might mention ongoing work on Gaussian channels and superactivation [4].
[1] Reliable quantum certification for photonic quantum technologies, L. Aolita, C. Gogolin, M. Kliesch, J. Eisert, Nature Comm. 6, 8498 (2015).
[2] Optimal entanglement witnesses for continuous-variable systems, P. Hyllus, J. Eisert, New J. Phys. 8, 51 (2006).
[3] In preparation (2016).
[4] In preparation (2016).
Nicolai Friis:
Estimating parameters encoded in Gaussian transformations
We address the issue of precisely estimating small parameters encoded in Bogoliubov transformations, that is, general linear transformations of the modes of a bosonic quantum field. Such Gaussian transformations frequently appear in the context of quantum optics but also in the context of quantum field theory on curved spacetimes. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states for a transformation given in terms of the Bogoliubov coefficients [1]. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires non-classical, but not necessarily entangled states. Our method further allows us to quantify losses in precision arising from being able to control only finitely many modes. We identify a lower bound for these tracing losses and provide a description of the family of “pretty good” states that minimize the losses, while achieving Heisenberg scaling. Finally, we remark on extensions to general mixed states and noisy Gaussian channels [2].
[1] N. Friis, M. Skotiniotis, I. Fuentes, and W. Dür, Phys. Rev. A 92, 022106 (2015) [arXiv:1502.07654].
[2] N. Friis and M. Skotiniotis, in preparation.
Jaromir Fiurasek:
Virtual implementation of noiseless amplification and entanglement distillation and their applications in continuous-variable quantum communication
We consider continuous-variable quantum communication scenarios, where the recipients of the quantum states perform eight-port homodyne detection (i.e. projection onto coherent states) on the received states. We show that in this scenario, several important quantum operations, including noiseless quantum amplification/attenuation and entanglement distillation by iterative Gaussification, do not need to be physically implemented but can simply be emulated in the classical data post-processing stage. This approach enables to circumvent hardware implementation problems with noiseless amplification or iterative multi-copy entanglement distillation. As an illustration of application we present experimental results for iterative entanglement distillation of phase-diffused two-mode squeezed states. Since our procedure necessarily includes the measurement, it is particularly suitable for applications in continuous-variable quantum key distribution, where it could enhance the secure range or tolerable excess noise.
[1] J. Fiurasek and N.J. Cerf, Gaussian postselection and virtual noiseless amplification in continuous-variable quantum key distribution, Phys. Rev. A 86, 060302(R) (2012).
[2] D. Abdelkhalek, M. Syllwasschy, N.J. Cerf, J. Fiurasek, and R. Schnabel, Efficient entanglement distillation without quantum memory, submitted.
Stefano Pirandola:
Fundamental Limits of Repeaterless Quantum Communications
Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks there is a crucial question to answer: What are their optimal rates without quantum repeaters? Our work addresses this basic question for any two parties connected by a quantum channel, without any restriction on their classical communication, which can be unlimited and two-way. We design a method which reduces the most general protocol of quantum communication over a channel to the computation of a simple quantity, that we call entanglement flux. In this way, we bound the ultimate rates that are achievable over the most important bosonic and qubit channels, computing a number of exact formulas for their two-way capacities. In particular, we determine the fundamental rate-loss scaling which affects any optical quantum communication. By setting these limits, our results establish the most general and correct benchmarks for testing the performance of quantum repeaters.
Maciej Lewenstein:
Entangled and non-classical states in qubit-boson systems
We will try to derive sufficient conditions for entanglement based on the analysis of a certain families of positive maps that transform positive operators into separable ones. We will also analyse various criteria characterizing existence of the positive P-representation for separable qubit-boson state. These criteria will be based on the generalized Bochner theorem and on the positive averages of squares fo polynomials
[1] Joint work with Swapan Rana, Manab Bera.
Gerardo Adesso:
Multipartite steering of Gaussian states: monogamy constraints and cryptographical applications
We investigate EPR steerability of multimode Gaussian states by Gaussian measurements. On the one hand, we establish a monogamy--like constraint, preventing joint steerability of a single mode by Gaussian measurements on multiple group of modes; this follows by proving a strong subadditivity inequality for the log-determinant of covariance matrices [1]. On the other hand, we show that a recently introduced quantifier of Gaussian steering [2] obeys a Coffman-Kundu-Wootters--type monogamy inequality for arbitrary multimode Gaussian states. In the case of pure Gaussian states of three modes, we find that the residual steering emerging from such an inequality admits an operational interpretation, related to the key rate of a semi-device-independent implementation of quantum secret sharing, taking into account potential dishonesty of some of the parties [3]. A novel security analysis for the latter protocol is provided [4].
[1] G. Adesso and R. Simon, arXiv:1601.03226 (2016)
[2] I. Kogias, A. R. Lee, S. Ragy, G. Adesso, Phys. Rev. Lett. 114, 060403 (2015)
[3] Y. Xiang et al., in preparation (2016)
[4] I. Kogias et al., in preparation (2016)
Anthony Leverrier:
Security proofs for continuous-variable QKD
Security proofs for quantum key distribution have made tremendous progress in the last few years, and we now have composable security in the finite-size regime for a number of protocols. In the case of BB84 for instance, one even recovers the asymptotic key rate given by the Dewetak-Winter bound for reasonable block sizes.
The situation is more complicated in the case of continuous-variable protocols: composable security in the finite-size setting has only been obtained for a few protocols, and one does not recover the expected asymptotic key rate for reasonable block sizes. In this talk, I will review the state-of-the-art of security proofs for continuous-variable QKD and explain the various challenges that need to be addressed in order to get better security proofs.
[1] Entropy 17, 6072-6092 (2015), arXiv:1506.02888.
Geza Giedke:
Gaussian Local Unitary Equivalence of n-mode Gaussian States and Gaussian LOCC transformations
We investigate which pure multipartite Gaussian states can be transformed into each other by local unitary and non-unitary Gaussian
means. Introducing a easily computed standard form we show that two states are equivalent under Gaussian local unitaries iff their
standard form coincides. We then investigate transformations between non-equivalent states by means of Gaussian local operations assisted by classical communication.
[1] G. Giedke and B. Kraus, Phys. Rev. A 89, 012335 (2014).
Maxim Shirokov:
Conditional mutual information in infinite-dimensional quantum systems and its use
It is shown that the quantum conditional mutual information (defined in a standard way) can be uniquely extended to a lower semicontinuous function on the set of all infinite-dimensional tripartite states which has all basic properties of this quantity valid in the finite-dimensional case. Some corollaries of the lower semicontinuity of the conditional mutual information are discussed. Winter’s type tight continuity bound for the quantum conditional mutual information under energy constraint on one subsystem is obtained and used for analysis of continuity properties of the infinite-dimensional versions of the squashed entanglement and of the entanglement of formation. Some applications to the theory of infinite-dimensional quantum channels and their capacities are also considered. Several open problems (having well known solutions in finite dimensions) are pointed. In particular, the question about extension of any separable state to a short Markov chain is discussed in connection with the basic property of the squashed entanglement.
[1] arXiv:1506.06377.
[2] arXiv:1507.08964.
Raul Garcia-Patron:
Entanglement of Formation for Gaussian states: overview and open questions
Michael Jabbour:
Majorization preservation of Gaussian bosonic channels
It is shown that phase-insensitive Gaussian bosonic channels are majorization-preserving over the set of passive states of the harmonic oscillator. This means that comparable passive states under majorization are transformed into equally comparable passive states. The proof relies on a new preorder relation called Fock-majorization, which coincides with regular majorization for passive states but also induces a mean photon number order, thereby connecting the concepts of energy and disorder of a quantum state. As an application, the consequences of majorization preservation are investigated in the context of the broadcast communication capacity of bosonic Gaussian channels. Most of our results being independent of the bosonic nature of the system under investigation, they could be generalized to other quantum systems and Hamiltonians, providing a general tool that could prove useful in quantum information theory and quantum thermodynamics.
[1] M. G. Jabbour, R. García-Patrón, N. J. Cerf, arXiv:1512.08225 [quant-ph].
Morgan Mitchell:
Quantum enhancement of atomic instruments : learning from LIGO (and growing on GEO-600)
In 2011, the GEO-600 gravitational-wave detector began running with squeezed light to improve its sensitivity, range, and data rate. In 2013, one of the LIGO interferometers was similarly enhanced, becoming the first instrument to . Are there terrestrial applications for these techniques ? I will describe the application of squeezing, optical and atomic, to atomic instruments, e.g. atomic clocks, atomic magnetometers, and atom interferometers. It turns out that atom interferometers offer a few twists that make them an even better platform for quantum enhancement.
Saikat Guha:
On rate-vs.-loss limits in quantum key distribution with and without quantum repeaters
Frédéric Grosshans:
Unidimensional continuous-variable quantum key distribution
We propose the continuous-variable quantum key distribution protocol based on the Gaussian modulation of a single quadrature of the coherent states of light, which is aimed to provide simplified implementation compared to the symmetrically modulated Gaussian coherent-state protocols. The protocol waives the necessity in phase quadrature modulation and the corresponding channel transmittance estimation. The security of the protocol against collective attacks in a generally phase-sensitive Gaussian channels is analysed and is shown achievable upon certain conditions. Robustness of the protocol to channel imperfections is compared to that of the symmetrical coherent-state protocol. The simplified unidimensional protocol is shown possible at a reasonable quantitative cost in terms of key rate and of tolerable channel excess noise.
[1] Joint work with Vladyslav C. Usenko, arXiv:1504.07093, Phys. Rev. A 92, 062337.
Nathan Walk
Channel purification via continuous-variable quantum teleportation with Gaussian postselection
We present a protocol based on continuous-variable quantum teleportation and Gaussian postselection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian operations.
[1] Blandino, R., Walk, N., Lund, A. P., & Ralph, T. C. (2016). Channel purification via continuous-variable quantum teleportation with Gaussian postselection. Physical Review A, 93(1), 012326. http://doi.org/10.1103/PhysRevA.93.012326