Origin of nonclassicality: observed state versus measurement

Contents 1 Introduction 2 2 Quantum Interference and the Construction of Nonclassical States 3 2.1 Nonclassical Light, Quadratures and Phase--Sensitive Observables 3 2.2 Phase--Space and Quasi--Probabilities . . . . . . . . . . . . . . . . 13 2.3 Quantum Noise as a Stochastic Process . . . . . . . . . . . . . . 15 2.4 Marginal distributions and reconstruction of Wigner functions . . 18 2.5 Joint Measurements of p, q in Phase Space . . . . . . . . . . . . . 20 2.6 Phase--Space Sampling Entropies . . . . . . . . . . . . . . . . . . 27 2.7 Mutual Information and Entropic Uncertainty . . . . . . . . . . . 28 2.8 Building Nonclassical States by Quantum Interference . . . . . . 30 3 Dissipation Versus Coherent Evolution: Decoherence 36 4 Unravelling the Master Equation 37 5 Quantum Jumps 43 5.1 Intrinsic or Extrinsic Jumps . . . . . . . . . . . . . . . . . . . . . 43 5.2 Observation of Quantum Jumps . . . . . . . . . . . . . .Contents 1 Introduction 2 2 Quantum Interference and the Construction of Nonclassical States 3 2.1 Nonclassical Light, Quadratures and Phase--Sensitive Observables 3 2.2 Phase--Space and Quasi--Probabilities . . . . . . . . . . . . . . . . 13 2.3 Quantum Noise as a Stochastic Process . . . . . . . . . . . . . . 15 2.4 Marginal distributions and reconstruction of Wigner functions . . 18 2.5 Joint Measurements of p, q in Phase Space . . . . . . . . . . . . . 20 2.6 Phase--Space Sampling Entropies . . . . . . . . . . . . . . . . . . 27 2.7 Mutual Information and Entropic Uncertainty . . . . . . . . . . . 28 2.8 Building Nonclassical States by Quantum Interference . . . . . . 30 3 Dissipation Versus Coherent Evolution: Decoherence 36 4 Unravelling the Master Equation 37 5 Quantum Jumps 43 5.1 Intrinsic or Extrinsic Jumps . . . . . . . . . . . . . . . . . . . . . 43 5.2 Observation of Quantum Jumps . . . . . . . . . . . . . .

Jan 1, 19979 citations

Statistical mixtures of states can be more quantum than their superpositions: Comparison of nonclassicality measures for single-qubit states

Adam Miranowicz, Karol Bartkiewicz, Anirban Pathak +3 more

A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan $P$ function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\ensuremath{\tau}$, related to the minimal amount of Gaussian noise which changes a nonpositive $P$ function into a positive one; (2) the nonclassical distance $D$ , defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance $D$ for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range $[0,1]$ of their values, as well as the NP for a given value of $\ensuremath{\tau}$ such that $\ensuremath{\tau}>0.315\phantom{\rule{0.16em}{0ex}}4$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\ensuremath{\tau}$ for a given $D$ , NP versus $D$ , and $\ensuremath{\tau}$ versus NP. We also show the ``relativity'' of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan $P$ function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\ensuremath{\tau}$, related to the minimal amount of Gaussian noise which changes a nonpositive $P$ function into a positive one; (2) the nonclassical distance $D$, defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance $D$ for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range $[0,1]$ of their values, as well as the NP for a given value of $\ensuremath{\tau}$ such that $\ensuremath{\tau}>0.315\phantom{\rule{0.16em}{0ex}}4$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\ensuremath{\tau}$ for a given $D$, NP versus $D$, and $\ensuremath{\tau}$ versus NP. We also show the ``relativity'' of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.

arXiv:1502.04523Physical Review AApr 8, 201571 citations

Weak measurements and nonClassical correlations

S. Lekshmi, N. Shaji, Anil Shaji

Annals of PhysicsDec 10, 20161 citation

The Essence of Nonclassicality: Non-Vanishing Signal Deficit

S. Aravinda, R. Srikanth

arXiv:1502.05390International Journal of Theoretical PhysicsJul 21, 20153 citations

Direct observation of sub-binomial light

Tim J. Bartley, Gaia Donati, Xian‐Min Jin +3 more

Nonclassical states of light are necessary resources for quantum technologies such as cryptography, computation and the definition of metrological standards. Distinguishing nonclassical states from those that can be described without invoking quantum mechanics is both a necessary practical task and of fundamental interest. In general, signatures of nonclassicality arise from apparently pathological behaviour of the quasi-probability distributions used to describe the states. While tomographic reconstruction of the state is therefore sufficient to show nonclassicality, it is often more convenient to probe photon-counting statistics of on-off detectors directly. This is formalised by the Mandel Q-parameter QM= 〈Δn〉 / 〈n〉 - 1, which is negative for sub-Poissonian statistics arising from nonclassical states. However, the Mandel Q-parameter can fail when using a multiplexed array of on-off detectors to measure states with several photons; splitting in this manner can corrupt its original statistics. A new parameter, QB= N 〈(Δ c) 2 〉 / 〈 c 〉 (N - 〈 c 〉) can be used to witness nonclassicality directly from such click statistics. For QB<; 0, such statistics are thus deemed sub-binomial and therefore arise from a nonclassical state.Nonclassical states of light are necessary resources for quantum technologies such as cryptography, computation and the definition of metrological standards. Distinguishing nonclassical states from those that can be described without invoking quantum mechanics is both a necessary practical task and of fundamental interest. In general, signatures of nonclassicality arise from apparently pathological behaviour of the quasi-probability distributions used to describe the states. While tomographic reconstruction of the state is therefore sufficient to show nonclassicality, it is often more convenient to probe photon-counting statistics of on-off detectors directly. This is formalised by the Mandel Q-parameter QM= 〈Δn〉 / 〈n〉 - 1, which is negative for sub-Poissonian statistics arising from nonclassical states. However, the Mandel Q-parameter can fail when using a multiplexed array of on-off detectors to measure states with several photons; splitting in this manner can corrupt its original statistics. A new parameter, QB= N 〈(Δ c) 2 〉 / 〈 c 〉 (N - 〈 c 〉) can be used to witness nonclassicality directly from such click statistics. For QB<; 0, such statistics are thus deemed sub-binomial and therefore arise from a nonclassical state.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)May 1, 20137 citations

A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan

P

function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\ensuremath{\tau}$, related to the minimal amount of Gaussian noise which changes a nonpositive

P

function into a positive one; (2) the nonclassical distance

D

, defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance

D

for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range

[0,1]

of their values, as well as the NP for a given value of $\ensuremath{\tau}$ such that $\ensuremath{\tau}>0.315\phantom{\rule{0.16em}{0ex}}4$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\ensuremath{\tau}$ for a given

D

, NP versus

D

, and $\ensuremath{\tau}$ versus NP. We also show the ``relativity'' of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.