arXiv:1308.6346 · Physical Review A

Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement

Jiang Zhang, Matthias D. Lang, Carlton M. Caves

Published Oct 7, 2013• 51 citations

Abstract

In quantum optics a pure state is considered classical, relative to the statistics of photodetection, if and only if it is a coherent state. A different and newer notion of nonclassicality is based on modal entanglement. One example that relates these two notions is the Hong-Ou-Mandel effect, where modal entanglement is generated by a beamsplitter from the nonclassical photon-number state $|1\ensuremath{\rangle}\ensuremath{\bigotimes}|1\ensuremath{\rangle}$. This suggests that beamsplitters or, more generally, linear-optical networks are mediators of the two notions of nonclassicality. In this Brief Report, we show the following: Given a nonclassical pure-product-state input to an $N$-port linear-optical network, the output is almost always mode entangled; the only exception is a product of squeezed states, all with the same squeezing strength, input to a network that does not mix the squeezed and antisqueezed quadratures. Our work thus gives a necessary and sufficient condition for a linear network to generate modal entanglement from pure-product inputs, a result that is of immediate relevance to the boson-sampling problem.

DOI: 10.1103/physreva.88.044301

Related papers

Enhancing nonclassical properties of quantum states of light using linear optics

E. P. Mattos, A. Vidiella-Barranco

arXiv:2310.04913Optics LettersJun 15, 20232 citations

From squeezing to Gaussian entanglement via beamsplitters

Shuangshuang Fu, Shunlong Luo, Yue Zhang

Journal of Physics B Atomic Molecular and Optical PhysicsFeb 10, 20203 citations

Entanglement and nonclassicality for multimode radiation-field states

J. Solomon Ivan, S. Chaturvedi, Elisa Ercolessi +4 more

arXiv:1009.6104Physical Review AMar 25, 201135 citations

Nonclassical mixed states that generate zero entanglement with a beam splitter

Aaron Z. Goldberg, Daniel F. V. James

arXiv:1712.05425Journal of Physics A Mathematical and TheoreticalAug 3, 20188 citations

Generation of NPT Entanglement from Nonclassical Photon Statistics

J. Solomon Ivan, Narasimhaiengar Mukunda, Simon, R.

arXiv:quant-ph/0603255Mar 28, 20068 citations

Single- and two-mode quantumness at a beam splitter

Matteo Brunelli, Claudia Benedetti, Stefano Olivares +2 more

arXiv:1502.04996Physical Review AJun 11, 201527 citations

arXiv:1308.6346 · Physical Review A

Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement

Jiang Zhang, Matthias D. Lang, Carlton M. Caves

Published Oct 7, 2013• 51 citations

arXiv HTML PDF

Abstract

|1\ensuremath{\rangle}\ensuremath{\bigotimes}|1\ensuremath{\rangle}

. This suggests that beamsplitters or, more generally, linear-optical networks are mediators of the two notions of nonclassicality. In this Brief Report, we show the following: Given a nonclassical pure-product-state input to an

N

-port linear-optical network, the output is almost always mode entangled; the only exception is a product of squeezed states, all with the same squeezing strength, input to a network that does not mix the squeezed and antisqueezed quadratures. Our work thus gives a necessary and sufficient condition for a linear network to generate modal entanglement from pure-product inputs, a result that is of immediate relevance to the boson-sampling problem.

DOI: 10.1103/physreva.88.044301

Related papers

Enhancing nonclassical properties of quantum states of light using linear optics

E. P. Mattos, A. Vidiella-Barranco

In this Letter, we present a simple and versatile scheme for enhancing the nonclassical properties of light states using only linear optics and photodetectors. By combining a coherent state |α〉 and an arbitrary pure state of light |ϕ〉 (excluding coherent states) at two beam splitters, we show that the amplitude α of the coherent state can be tuned to filter out specific Fock components and generate states of light with increased nonclassical features. We provide two examples of input states and demonstrate the effectiveness of our scheme in enhancing the sub-Poissonian statistics or the quadrature squeezing of the output states.In this Letter, we present a simple and versatile scheme for enhancing the nonclassical properties of light states using only linear optics and photodetectors. By combining a coherent state |α〉 and an arbitrary pure state of light |ϕ〉 (excluding coherent states) at two beam splitters, we show that the amplitude α of the coherent state can be tuned to filter out specific Fock components and generate states of light with increased nonclassical features. We provide two examples of input states and demonstrate the effectiveness of our scheme in enhancing the sub-Poissonian statistics or the quadrature squeezing of the output states.

arXiv:2310.04913Optics LettersJun 15, 20232 citations

From squeezing to Gaussian entanglement via beamsplitters

Shuangshuang Fu, Shunlong Luo, Yue Zhang

Abstract We investigate relations between squeezing of separable input states and entanglement of output states generated by a beamsplitter, both of which are valuable nonclassical resources for quantum information processing. Motivated by the seminal work of Asboth´ et al [2005 Phys. Rev. Lett. 94 173602], where a systematic study of nonclassicality (including squeezing) and beamsplitter-generated entanglement is conducted, we consider the situation when the two input states are all Gaussian and identify certain constraints on the squeezing parameters to generate entanglement. Contrary to naive expectation, large squeezing does not necessarily guarantee large entanglement. Depending on the phase relations between the input states and the beamsplitters, the conditions to generate entanglement vary, which may be regarded as a manifestation of interference. In two typical cases, it is the difference or sum of the squeezing amplitudes that plays a key role in generating entanglement. Analytic expressions of entanglement in terms of the difference or sum of the squeezing amplitudes are obtained, which are further shown to be monotonic in the relevant parameters.Abstract We investigate relations between squeezing of separable input states and entanglement of output states generated by a beamsplitter, both of which are valuable nonclassical resources for quantum information processing. Motivated by the seminal work of Asboth´ et al [2005 Phys. Rev. Lett. 94 173602], where a systematic study of nonclassicality (including squeezing) and beamsplitter-generated entanglement is conducted, we consider the situation when the two input states are all Gaussian and identify certain constraints on the squeezing parameters to generate entanglement. Contrary to naive expectation, large squeezing does not necessarily guarantee large entanglement. Depending on the phase relations between the input states and the beamsplitters, the conditions to generate entanglement vary, which may be regarded as a manifestation of interference. In two typical cases, it is the difference or sum of the squeezing amplitudes that plays a key role in generating entanglement. Analytic expressions of entanglement in terms of the difference or sum of the squeezing amplitudes are obtained, which are further shown to be monotonic in the relevant parameters.

Journal of Physics B Atomic Molecular and Optical PhysicsFeb 10, 20203 citations

Entanglement and nonclassicality for multimode radiation-field states

J. Solomon Ivan, S. Chaturvedi, Elisa Ercolessi +4 more

Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multimode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beam splitters, in a transparent manner. For single-mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beam splitter, these conditions turn exactly into signatures of negativity under partial transpose (NPT) entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two-mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beam-splitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three-mode beam-splitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multimode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beam splitters, in a transparent manner. For single-mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beam splitter, these conditions turn exactly into signatures of negativity under partial transpose (NPT) entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two-mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beam-splitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three-mode beam-splitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.

arXiv:1009.6104Physical Review AMar 25, 201135 citations

Nonclassical mixed states that generate zero entanglement with a beam splitter

Aaron Z. Goldberg, Daniel F. V. James

Abstract Beam splitters are routinely used for generating entanglement. Their entangling properties have been studied extensively, with nonclassicality of the input states a prerequisite for entanglement at the output. Here we quantify the amount of entanglement generated by weakly-reflecting beam splitters, and look for nonclassical states that are not entangled by general beam splitters. We extend the known class of results to mixed and non-Gaussian states, finding that inputting highly nonclassical combinations of unpolarized states that are squeezed and displaced onto a beam splitter can still yield separable output states. This is important in light of the challenge of characterizing mixed state entanglement. We further identify a parallel between SU(2) unpolarized states and the two-mode vacuum. Our result is crucial for understanding the generation of modal entanglement by beam splitters.Abstract Beam splitters are routinely used for generating entanglement. Their entangling properties have been studied extensively, with nonclassicality of the input states a prerequisite for entanglement at the output. Here we quantify the amount of entanglement generated by weakly-reflecting beam splitters, and look for nonclassical states that are not entangled by general beam splitters. We extend the known class of results to mixed and non-Gaussian states, finding that inputting highly nonclassical combinations of unpolarized states that are squeezed and displaced onto a beam splitter can still yield separable output states. This is important in light of the challenge of characterizing mixed state entanglement. We further identify a parallel between SU(2) unpolarized states and the two-mode vacuum. Our result is crucial for understanding the generation of modal entanglement by beam splitters.

arXiv:1712.05425Journal of Physics A Mathematical and TheoreticalAug 3, 20188 citations

Generation of NPT Entanglement from Nonclassical Photon Statistics

J. Solomon Ivan, Narasimhaiengar Mukunda, Simon, R.

With a product state of the form $ρ_{in} = ρ_a\otimes|0>_b_b< 0|$ as input, the output two-mode state $ρ_{\rm out}$ , of the beam splitter is shown to be NPT whenever the photon number distribution (PND) statistics $\{p(n_a) \}$ associated with the possibly mixed state $ρ_a$ of the a_mode is antibunched or otherwise nonclassical, i.e., if $\{p(n_a)\}$ fails to respect any one of an infinite of classicality conditions.With a product state of the form $ρ_{in} = ρ_a\otimes|0>_b_b< 0|$ as input, the output two-mode state $ρ_{\rm out}$, of the beam splitter is shown to be NPT whenever the photon number distribution (PND) statistics $\{p(n_a) \}$ associated with the possibly mixed state $ρ_a$ of the a_mode is antibunched or otherwise nonclassical, i.e., if $\{p(n_a)\}$ fails to respect any one of an infinite of classicality conditions.

arXiv:quant-ph/0603255Mar 28, 20068 citations

Single- and two-mode quantumness at a beam splitter

Matteo Brunelli, Claudia Benedetti, Stefano Olivares +2 more

In the context of bipartite bosonic systems, two notions of classicality of correlations can be defined: $P$ classicality, based on the properties of the Glauber-Sudarshan $P$ function; and $C$ classicality, based on the entropic quantum discord. It has been shown that these two notions are maximally inequivalent in a static (metric) sense, as they coincide only on a set of states of zero measure. We extend and reinforce quantitatively this inequivalence by addressing the dynamical relation between these types of nonclassicality in a paradigmatic quantum-optical setting: the linear mixing at a beam splitter of a single-mode Gaussian state with a thermal reference state. Specifically, we show that almost all $P$ -classical input states generate outputs that are not $C$ classical. Indeed, for the case of zero thermal reference photons, the more $P$ -classical resources at the input the less $C$ classicality at the output. In addition, we show that the $P$ classicality at the input---as quantified by the nonclassical depth---does instead determine quantitatively the potential of generating output entanglement. This endows the nonclassical depth with a new operational interpretation: it gives the maximum number of thermal reference photons that can be mixed at a beam splitter without destroying the output entanglement.In the context of bipartite bosonic systems, two notions of classicality of correlations can be defined: $P$ classicality, based on the properties of the Glauber-Sudarshan $P$ function; and $C$ classicality, based on the entropic quantum discord. It has been shown that these two notions are maximally inequivalent in a static (metric) sense, as they coincide only on a set of states of zero measure. We extend and reinforce quantitatively this inequivalence by addressing the dynamical relation between these types of nonclassicality in a paradigmatic quantum-optical setting: the linear mixing at a beam splitter of a single-mode Gaussian state with a thermal reference state. Specifically, we show that almost all $P$-classical input states generate outputs that are not $C$ classical. Indeed, for the case of zero thermal reference photons, the more $P$-classical resources at the input the less $C$ classicality at the output. In addition, we show that the $P$ classicality at the input---as quantified by the nonclassical depth---does instead determine quantitatively the potential of generating output entanglement. This endows the nonclassical depth with a new operational interpretation: it gives the maximum number of thermal reference photons that can be mixed at a beam splitter without destroying the output entanglement.

arXiv:1502.04996Physical Review AJun 11, 201527 citations

|1\ensuremath{\rangle}\ensuremath{\bigotimes}|1\ensuremath{\rangle}

N

With a product state of the form $ρ_{in} = ρ_a\otimes|0>_b_b< 0|$ as input, the output two-mode state

ρ_{\rm out}

, of the beam splitter is shown to be NPT whenever the photon number distribution (PND) statistics

\{p(n_a) \}

associated with the possibly mixed state

ρ_a

of the a_mode is antibunched or otherwise nonclassical, i.e., if

\{p(n_a)\}

fails to respect any one of an infinite of classicality conditions.

In the context of bipartite bosonic systems, two notions of classicality of correlations can be defined:

P

classicality, based on the properties of the Glauber-Sudarshan

P

function; and

C

classicality, based on the entropic quantum discord. It has been shown that these two notions are maximally inequivalent in a static (metric) sense, as they coincide only on a set of states of zero measure. We extend and reinforce quantitatively this inequivalence by addressing the dynamical relation between these types of nonclassicality in a paradigmatic quantum-optical setting: the linear mixing at a beam splitter of a single-mode Gaussian state with a thermal reference state. Specifically, we show that almost all

P

-classical input states generate outputs that are not

C

classical. Indeed, for the case of zero thermal reference photons, the more

P

-classical resources at the input the less

C

classicality at the output. In addition, we show that the

P

classicality at the input---as quantified by the nonclassical depth---does instead determine quantitatively the potential of generating output entanglement. This endows the nonclassical depth with a new operational interpretation: it gives the maximum number of thermal reference photons that can be mixed at a beam splitter without destroying the output entanglement.