[Paper Review] Remark on multi-particle observables and entangled states with constant complexity

Dominik Janzing, Thomas Beth|ArXiv.org|Mar 24, 2000

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper demonstrates that quantum networks of constant depth cannot prepare or measure highly entangled multi-particle states that exhibit macroscopic quantum uncertainty, as quantified by a parameter $ e_{ ho} $. It proves that such states and observables require network depth on the order of $ \log n $, establishing a fundamental complexity threshold for macroscopic quantum behavior.

ABSTRACT

We show that every density matrix of an n-particle system prepared by a quantum network of constant depth is asymptotically commuting with the mean-field observables. We introduce certain pairs of hypersurfaces in the space of density matrices and give lower bounds for the depth of a network which prepares states lying outside those pairs. The measurement of an observable which is not asymptotically commuting with the mean-field observables requires a network of depth in the order of log n, if one demands the measurement to project the state into the eigenspace of the measured observable.

Motivation & Objective

• To investigate the limitations of constant-depth quantum networks in preparing or measuring highly entangled many-body states that exhibit macroscopic quantum behavior.
• To formalize the connection between quantum network depth and the ability to access states violating macrorealism, as captured by the parameter $ e_{ ho} $.
• To establish lower bounds on the depth required for preparing or measuring observables incompatible with mean-field approximations, which are central to classical macroscopic behavior.
• To analyze the role of network depth in the context of Schrödinger’s Cat-type paradoxes, where quantum superpositions of macroscopically distinct states challenge classical intuition.

Proposed method

• Defined a parameter $ e_{ ho} $ to quantify the degree of macroscopic quantum uncertainty in a many-particle density matrix $ \rho $, with large $ e_{ ho} $ indicating strong non-classical behavior.
• Introduced the concept of mean-field observables $ \overline{a} = \frac{1}{n}\sum_{i=1}^n a_i $, representing collective properties of the system.
• Proved that any observable $ c $ measurable via a constant-depth network must asymptotically commute with $ \overline{a} $, with the commutator norm bounded by $ \| [\overline{a}, A^*(P)] \| \leq \frac{2^k}{\sqrt{2n}} $, where $ k $ is the network depth.
• Used spectral projections $ P $ of observables to analyze the compatibility of measurement outcomes with mean-field behavior, showing that non-commuting observables require deeper circuits.
• Leveraged the structure of tensor product Hilbert spaces and support overlap between local operators to bound the number of interacting terms in the commutator expression.
• Established that the number of operator pairs with overlapping support is bounded by $ 4^k $ per local operator, leading to the $ \sqrt{4^k / n} $ scaling in the commutator bound.

Experimental results

Research questions

• RQ1 Can a quantum network of constant depth prepare a many-particle state with large $ e_{\rho} $, indicating macroscopic quantum superposition?
• RQ2 What is the minimum depth of a quantum network required to prepare or measure an observable incompatible with mean-field observables?
• RQ3 How does the asymptotic commutativity between network-implemented observables and mean-field observables constrain the set of accessible entangled states?
• RQ4 To what extent do resource constraints in quantum circuits—specifically depth—impose fundamental limits on macroscopic quantum phenomena like those in Schrödinger’s Cat thought experiment?

Key findings

• Any observable that can be measured via a quantum network of depth $ k $ must have its spectral projections asymptotically commuting with mean-field observables, with the commutator norm bounded by $ \frac{2^k}{\sqrt{2n}} $.
• States with large $ e_{\rho} $, indicating macroscopic quantum uncertainty, cannot be prepared by networks of constant depth, as they require depth $ \Theta(\log n) $.
• The depth required to measure an observable incompatible with mean-field behavior is also $ \Theta(\log n) $, matching the depth required for state preparation.
• The bound on the commutator norm implies that constant-depth networks cannot access the full set of entangled states relevant to macrorealism or Schrödinger’s Cat paradox.
• The number of operator pairs with overlapping support in a depth-$ k $ network is at most $ 4^k $ per local operator, which limits the growth of non-commutativity.
• The results imply that 'very-low-complexity entanglement'—such as that from depth-2 networks—cannot generate the kind of highly non-local, macroscopic entanglement associated with $ e_{\rho} $-large states.