物理學 1935

能認為量子力學對物理實在的描述是完備的嗎？

阿爾伯特·愛因斯坦、鮑里斯·波多爾斯基、內森·羅森

若測量一個粒子便鎖定其遠方夥伴，量子理論必不完備。

Choose your version

In depth · the introduction

在這邊測量一個粒子，它遠方的夥伴彷彿在同一瞬間就打定了主意——這條聯繫，愛因斯坦無法忍受，並以此論證：量子理論一定漏掉了什麼。

核心想法

量子力學說，一個粒子在你測量之前，並沒有確定的位置或動量——只有機率。愛因斯坦，連同波多爾斯基與羅森，認為這不可能是故事的全部。他們設想：兩個粒子一同誕生、再被送往遠方，命運彼此鎖死——你為其中一個測出什麼，立刻就知道另一個相匹配的答案。

他們布下的陷阱是這樣的。假設你透過測量粒子 A，就能完全確定地預測粒子 B 的結果，而根本不去碰 B。那麼 B 想必早已具有那項性質——它是實在的，早在你去看之前，就好端端待在那裡。可量子力學偏偏拒絕預先給 B 賦予那個確定值。於是 EPR 斷言：這套理論必定不完備——其中一定有它略去未提的隱藏細節。愛因斯坦希望實在是定域的（沒有瞬時的超距影響），而且是確定的。量子力學，這兩樣似乎都給不了。

它是如何誕生的

到 1935 年，愛因斯坦幫著建起了量子理論，卻已對它那幅世界圖景生出不信任。在普林斯頓高等研究院安頓下來的他，與兩位年輕些的同事——鮑里斯·波多爾斯基和內森·羅森——合作，要把這份不安，化成一個誰也揮之不去的論證。文章由波多爾斯基執筆，結果，他在正式發表前把它透給了報界——《紐約時報》登出了標題，而愛因斯坦對那種說法頗為惱火。

據說，量子力學的偉大旗手尼爾斯·波耳，為此投入了數週的緊張工作，又以一模一樣的標題發文回擊。他論證道：EPR 那種「粒子獨立於你如何選擇測量、自身便擁有種種性質」的想法，對糾纏系統根本不適用。這場爭論就這樣擱了三十年：對同一組方程的兩種精彩解讀，卻沒有一個實驗能在它們之間作出裁決。

它為何重要

EPR 做成了一件罕見的事：他們把一個關於實在的哲學問題，變成了一個精確、可回答的問題。他們本想揭露量子力學的一個破綻。結果，1964 年，物理學家約翰·貝爾找到了一條路，把他們的思想實驗變成一個真實的檢驗——而其後數十年的實驗，交出了一個愛因斯坦會深惡痛絕的判決：自然界確實如方程所言那般離奇；EPR 所盼望挽救的那個安穩的、定域而確定的世界，並不存在。

他們的誠實，正是關鍵所在。在他們的假設之下，EPR 的邏輯完美無缺；落空的，是其中一條假設——影響不能跑得比分離的速度更快——而自然界不肯遵守它。少有哪篇旨在贏得爭論的論文，錯得如此富有成果。

一個可以想像的畫面

想像一雙手套，被分裝進兩個密封的盒子，運往地球的兩端。打開其中一個盒子，看到一隻左手套，你立刻就知道：遠處那個盒子裡裝的是右手套——沒有任何信號傳過去；答案早在出發時就被封了進去。愛因斯坦希望糾纏粒子正是這樣：答案在源頭就已決定，只是被藏了起來。

可量子粒子偏偏不符合這幅圖景。對手套，你只能查「左還是右」，如此而已。對糾纏粒子，你卻可以在最後一刻，挑選去問哪一個問題——而無論你怎麼挑，遠方夥伴的答案，都吻合得太過完美。任何一雙事先封好的手套，都變不出這個戲法。正是這道落差，被貝爾變成可檢驗的，最終了結了爭論。

它的位置

這篇論文，是量子力學的奠基——普朗克、波耳、海森堡、薛丁格——與量子資訊時代之間的樞紐。它把糾纏凝結為一個概念，催生了貝爾 1964 年的定理，並引出克勞澤、阿斯佩與蔡林格那些贏得 2022 年諾貝爾獎的實驗。愛因斯坦所拒斥的那份離奇，如今正驅動著量子密碼學與量子電腦。在本館中，它緊挨著它所質疑的工作，以及它無意間播下的種種技術。

The original document

Original source text

A. Einstein, B. Podolsky, N. Rosen · Physical Review 47 (1935): 777–780 · Received March 25, 1935

Abstract

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system.

In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

§1 — The condition of completeness and the criterion of reality

Whatever the meaning assigned to the term complete, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counterpart in the physical theory. We shall call this the condition of completeness. The second question is thus easily answered, as soon as we are able to decide what are the elements of the physical reality.

If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.

[ … ]

§2 — The two-particle thought experiment

Suppose now that we have two systems, I and II, which we permit to interact … after which the systems no longer interact. We assume that the states of the two systems … are known. We can then calculate … the state of the combined system. … By measuring either A or B we are in a position to predict with certainty, and without in any way disturbing the second system, either the value of the quantity P (that is p_k) or the value of the quantity Q (that is q_r). In accordance with our criterion of reality, in the first case we must consider the quantity P as being an element of reality, in the second case the quantity Q is an element of reality. But, as we have seen, both wave functions … belong to the same reality.

Previously we proved that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2).

We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.

One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. … This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

Institute for Advanced Study · Princeton, New Jersey · received March 25, 1935