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Behavior is neither

**classical**, nor an

**accumulated quantum effect**.Think Cooper pairs in a superconductor.

Call a system **macroscopic** if effective size is linear N_{eff} = $\mathcal O$ (N).
I.e. all particles need to be described quantum mechanically.

**Examples:**

GHZ state: N_{eff} = N, *macroscopic*

Cooper pair: N_{eff} = 2, *microscopic*

**Two Measures: (not strictly N _{eff})**

**Extensive difference $\Lambda$**: difference of states vs reference value (e.g. $\Delta \Phi / \Phi_0$)

**Disconnectivity D:** degree of entanglement, possibly reduced entropy

**Examples:**

GHZ State of N particles: D = N

C_{60}: D ≈ 10^{6}

Atomic ensembles: D ≈ 10^{7}

SQUIDs: D ≈ 10^{10}

**Idea:** Compare realistic (e.g. BEC) ↔ GHZ states

$$\ket{\psi} = \frac{1}{\sqrt{K}} \left( \ket{0}^{\otimes N} + \ket{\epsilon}^{\otimes N} \right)$$ where $\left| \braket{0}{\epsilon} \right|^2 = 1 - \epsilon^2$.

Through decoherence rate / max distillation:

$$N_{eff} = N \epsilon^2$$

Can a state improve phase estimation in interferometry?

Let $\ket{\psi(t)} = e^{-i H t / \hbar} \ket{\psi}$ and

$\theta(\ket{\psi}) :=$ time to reach orthogonal state.

Observe that $\theta(\ket{\psi}) \ll \theta(\ket{0}), \theta(\ket{1})$. Think small oscillations in an interference pattern.

Define the effective particle number $$N_\text{eff} = \left[ \frac{\theta(\ket{0}) + \theta(\ket{1})}{\theta{\ket{\psi}}} \right]^2$$

**Examples:**

C_{60}: N ≈ 5

SQUIDs: N ≈ 33

Optomechanical $\ket{0} + \ket{1}$: N = 1 (?)

Divide $\ket{0} + \ket{1}$ into maximal number (n) of subsystems, so that measuring *one* we can distinguish the states with precision $\delta$.

Define effective size N_{eff} = n.

Here, macroscopic = locally distinguishable

How many **single-particle** operations do we need to apply to get from $\ket{0}$ to $\ket{1}$?

- Start with $\mathcal H_0 = \operatorname{span} \{ \ket{0} \}$
- Define $\mathcal H_i$ by applying all possible single particle operations on $\mathcal H_{i-1}$
- Write $\ket{1} = \sum_i P_i \ket{1} = \sum_i \lambda_i \ket{v_i}$
- Effective size $\avg{N} = \avg{\sum_i i \ket{v_i} \bra{v_i}}$

**Examples:**

GHZ: N_{eff} = N

Superconducting qubit: N_{eff} ≈ 1

Many photon states: N_{eff} = 1

**Fisher Information:** expected value of the observed information.

Inverse of **Cramer-Rao-Bound**, "key resource" for metrology.

Implicit form: $$(ds)_\text{Bures} = \frac{1}{2\hbar} \sqrt{\mathcal F(\rho, H)} dt$$

For pure states: $$\mathcal F(\psi, H) = 4 \left[ \langle H^2 \rangle - \langle H \rangle^2 \right]$$

Let $A = \sum_i A_i$ be a locally acting operator with $||A_i|| = 1$. $$N_\text{eff}^F = \operatorname{max}_A \mathcal F(\rho, A) / (4 N)$$

Better measure: **relative** Fisher information $$N_\text{eff} = \frac{2 N_\text{eff}^F(\ket{\psi})}{N_\text{eff}^F(\ket{0}) + N_\text{eff}^F(\ket{1})}$$

- Dependence on set of basis statesMany states look trivial in the right basis.
- Particle number ≠ Macroscopicitye.g. 10 photon NOON state is visible = macroscopic.
- What is the elementary particle?Molecules, Atoms, Protons & Neutrons, …

… call a quantum system the more macroscopic the better its experimental demostration rules out a minimal modification of quantum mechanics …

All extensions *classicalize* the system:

- Invariant under Galilean trafo
- Leave exchange symmetry unaffected
- Innocent bystander condition Adding an uncorrelated system leaves reduced state unchanged.
- Scale invariant with respect to CM

For a single particle (using Holevo 1993):

$$\mathcal L_1 = \frac{1}{\tau} \left[ \int d^3 s d^3 q g(s, q) W(s, q) \rho W^\dagger (s, q) - \rho \right ]$$where $W(s, q) = \exp [ i/\hbar (P s - q X) ]$.

$g(s, q)$ a positive, isotropic, normalized phase space distribution with standard deviations $\sigma_s$, $\sigma_q$.

Matrix elements of $\rho$ that are larger than $\sigma_s$, $\sigma_q$ decay with $\tau$.

Using the other conditions we get, for a massive object (M, mass density $\varrho$) approximated by a single particle:

$$\frac{1}{\tau} = \frac{1}{\tau_e} \frac{1}{m_e^2} \int d^3 s d^3 q g_e(s, q) |\tilde \varrho(q)|^2$$ $$g(s, q) = \frac{\tau M^3}{\tau_e m_e^5} g_e(s M/m_e, q) |\tilde \varrho(q)|^2$$ Note that for an elementary particle $\tau = \tau_e (m_e/m)^2$Different experimental bounds:

Define macroscopicity as

$$\mu = \log_{10} \left( \frac{\tau_e}{1~\text{s}} \right)$$ For double-slit-like experiments: $$\mu = \log_{10} \left[ \frac{1}{|\ln f|} \left(\frac{M}{m_e}\right) \frac{t}{1~\text{s}} \right]$$ where $f$ is the visibility.Experiment | μ |
---|---|

Oscillating micromembrane | 11.5 |

Hypothetical large SQUID | 14.5 |

Talbot-Lau interference at 10^{5} amu |
14.5 |

Satellite atom (Cs) interferometer | 14.5 |

Oscillating micromirror | 19.0 |

Nanosphere interference | 20.5 |

Talbot-Lau interference at 10^{8} amu |
23.3 |

Schrödinger gedanken experiment | ~57 |

My gut reaction is that while the idea in this paper is a clever one, it is facing in an irrelevant direction. Rather than referring to quantum mechanics in its formulation, macroscopicity should instead reflect our ‘common-sense’ intuition of the difference between an electron being in an indefinite state and a cat being in an indefinite state.

- Tony Leggett

Subjective stuff is hard!

Approaches: particle number / experimental effects

What insight did we gain?