Schr odinger cats and quantum complementarity

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Schr¨odinger cats and quantum complementarity

Lorenzo Maccone

Dip. Fisica, Univ. of Pavia, INFN Sez. Pavia, via Bassi 6, I-27100 Pavia, Italy

Complementarity tells us we cannot know precisely the values of all the properties of a quantum

object at the same time: the precise determination of one property implies that the value of some

other (complementary) property is undeﬁned. E.g. the precise knowledge of the position of a particle

implies that its momentum is undeﬁned. Here we show that a Schr¨odinger cat has a well deﬁned

value of a property that is complementary to its “being dead or alive” property. Then, thanks to

complementarity, it has an undeﬁned value of the property “being dead or alive”.

In other words, the cat paradox is explained through quantum complementarity: of its many

complementary properties, any quantum system, such as a cat, can have a well deﬁned value only

of one at a time. Schr¨odinger’s cat has a deﬁnite value of a property which is complementary to

“being dead or alive”, so it is neither dead nor alive. Figuratively one can say it is both dead

and alive. While this interpretation only uses textbook concepts (the Copenhagen interpretation),

apparently it has never explicitly appeared in the literature. We detail how to build an Arduino

based simulation of Schr¨odinger’s experiment based on these concepts for science outreach events.

The Schr¨odinger’s cat argument was published by

Schr¨odinger’s in [1] and devised during a discussion with

Einstein. It was a provocation to show that quantum ef-

fects cannot be naively hidden in the microscopic realm,

but through linearity (and entanglement) they can af-

fect also macroscopic systems such as cats, giving rise, in

Schr¨odinger’s words, to “ridiculous cases”. While quan-

tum mechanics is by now well established, the debate on

its interpretation is still very much open: a huge num-

ber of competing interpretations vie for an explanation of

the quantum formulas. The Schr¨odinger cat is a perfect

testbed for such explanations [2].

In this paper, we give a very simple interpretation to

the cat, the “complementarity interpretation”, based on

textbook concepts, namely on the Copenhagen interpre-

tation. We then show how to use it to create a simulated

experiment which is eﬀective to communicate to the gen-

eral public concepts such as the superposition principle

and quantum complementarity. Even though our expla-

nation could have been proposed by Bohr himself (who

famously championed quantum complementarity), Bohr

never explicitly addressed the cat [3]. In his discussions

with Schr¨odinger [4], he did address classicality, but only

limiting himself to the measurement apparatuses, which

is not what Schr¨odinger’s argument embodies. Surpris-

ingly, to our knowledge the proposal presented in this

paper (Schr¨odinger’s cat through complementarity) has

not appeared in the English literature previously [5].

The outline of the paper follows. We start by explain-

ing quantum complementarity and how it is embodied in

quantum mechanics. We then recall Schr¨odinger’s ar-

gument and show how quantum complementarity can

be used to make sense of the superposition. We con-

clude detailing how one can construct a simulation of

Schr¨odinger’s experiment in a cardboard box controlled

through an Arduino microcontroller with a simulated cat.

Quantum complementarity:— All systems are de-

scribed by a collection of their properties. For exam-

ple, the state of a featureless particle is described by two

properties, its position and its momentum. A spin 1/2

is described by the components Jx, Jy, Jzof its angular

momentum. A cat is described by a myriad of compli-

cated properties (color, weight, furriness, position, etc.).

One of these is its “being dead or alive”: it can have two

distinct values ‘dead’ and ‘alive’ (we will neglect inter-

mediate ‘moribund’ values).

Properties in quantum mechanics are described by ob-

servables, such as the position observable X. The states

of quantum systems are described by vectors |ψ⟩, and the

state of systems which have a deﬁnite value of a property

are eigenstates |x⟩of the observable. The corresponding

eigenvalue xis the value of the property in such state.

States are vectors because of the superposition principle:

any state is a linear combination of eigenvectors of some

observable. Also any eigenstate of an observable is a lin-

ear combination of eigenstates of other observables. The

Born rule tells us that, if we prepare a system in a (non-

trivial) superposition of eigenstates of the observable we

are measuring, we will get a probabilistic answer: that

observable property is not well deﬁned in that system

state. (The probabilities are given by the square moduli

of the linear combination coeﬃcients. We discuss below

whether such probabilities are due to ignorance of some-

thing, or to the fact that this something is undeﬁned.)

Complementarity, hence, follows by joining superposi-

tion with the Born rule. Consider two properties: ob-

servable ˆ

Hand observable ˆ

S. Suppose that the ˆ

Heigen-

states are all nontrivial superpositions of ˆ

Seigenstates

and that the state of the system is in an eigenvector of

the ˆ

Sobservable. Then property ˆ

Swill have a deﬁnite

value, but ˆ

Hwill not. Complementarity [6, 7] can be

loosely stated as: “if the value of one property is deter-

mined exactly, the values of some other properties will be

undeﬁned”. E.g. if the position of the particle is exactly

arXiv:2210.01083v2 [quant-ph] 29 Aug 2023

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摘要：

Schr¨odingercatsandquantumcomplementarityLorenzoMacconeDip.Fisica,Univ.ofPavia,INFNSez.Pavia,viaBassi6,I-27100Pavia,ItalyComplementaritytellsuswecannotknowpreciselythevaluesofallthepropertiesofaquantumobjectatthesametime:theprecisedeterminationofonepropertyimpliesthatthevalueofsomeother(complementar...

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