Why is it impossible to measure position and momentum at the same time with arbitrary precision?

Question

I'm aware of the uncertainty principle that doesn't allow $\Delta x$ and $\Delta p$ to be both arbitrarily close to zero. I understand this by looking at the wave function and seeing that if one is sharply peeked its fourier transform will be wide.

But how does this stop one from measuring both position and momentum at the same time? I've googled this question, but all I found were explantions using the 'Observer effect'. I'm not sure, but I think this effect is very different from the intrinsic uncertainty principle.

So what stops us from measuring both position and momentum with arbitrairy precision? Does a quantum system always have to change when observerd? Or does it have to do with the uncertainty principle?

Thank you in advance.

This post imported from StackExchange Physics at 2025-03-08 12:38 (UTC), posted by SE-user catmousedog

Comments

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This post imported from StackExchange Physics at 2025-03-08 12:38 (UTC), posted by SE-user rob

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Since it was deleted, here's 3blue1brown's explanation of the general uncertainty principle. The tl;dw is that the "uncertainty principle" is actually a property of all waves.

This post imported from StackExchange Physics at 2025-03-08 12:38 (UTC), posted by SE-user BlueRaja - Danny Pflughoeft

Answer 1

When someone asks "Is it really impossible to simultaneously measure position and momentum with arbitrary precision in quantum theory?", the best preliminary answer is another question: "what do you exactly mean by measurement, by precision, and by position and momentum?". Those words have several meanings each in quantum theory, reflected in literature and experimental practice. There is a sense in which a simultaneous and arbitrarily precise measurement of position and momentum is not only possible, but also routinely made in many quantum labs, for example quantum-optics labs. Such measurement is indeed at the core of modern quantum applications such as quantum-key distribution.

I think it's best first to make clear what the different meanings of measurement, position, momentum are in actual applications and in the literature, and then to give examples of the different experimental procedures that are called "measurement of position" etc. What's important is to understand what's being done; the rest is just semantics.

Let me get there step by step. The answer below summarizes what you can find in current articles published in scientific journals and current textbooks, works and results which I have experienced myself as a researcher in quantum optics. All references are given throughout the answer, and some additional ones at the end. I strongly recommend that you go and read them. Also, this answer is meant to discuss the uncertainty principle and simultaneous measurement within quantum theory. Maybe in the future we'll all use an alternative theory in which the same experimental facts are given a different meaning; there are such alternative theories proposed at present, and many researchers indeed are working on alternatives. Finally, this answer tries to avoid terminological debates, explaining the experimental, laboratory side of the matter. Warnings about terminology will be given throughout. (I don't mean that terminology isn't important, though: different terminologies can inspire different research directions.)

We must be careful, because our understanding of the uncertainty principle today is very different from how people saw it in the 1930–50s. The modern understanding is also borne out in modern experimental practice. There are two main points to clarify.

1. What do we exactly mean by "measurement" and by "precision" or "$\Delta x$"?

The general picture is this:

1. We can prepare one copy of a physical system according to some specific protocol. We say that the system has been prepared in a specific state (generally represented by a density matrix $\pmb{\rho}$). Then we perform a specific operation that yields an outcome. We say that we have performed one instance of a measurement on the system (generally represented by a so-called positive-operator-valued measure $\{\pmb{O}_i\}$, where $i$ labels the possible outcomes).

2. We can repeat the procedure above anew – new copy of the system – as many times as we please, according to the same specific protocols. We are thus making many instances of the same kind of measurement, on copies of the system prepared in the same state. We thus obtain a collection of measurement results, from which we can build a frequency distribution and statistics. Throughout this answer, when I say "repetition of a measurement" I mean it in this specific sense.

There's also the question of what happens when we make two or more measurements in succession, on the same system. But I'm not going to discuss that here; see the references at the end.

This is why the general empirical statements of quantum theory have this form: "If we prepare the system in state $\pmb{\rho}$, and perform the measurement $\{\pmb{O}_i\}$, we have a probability $p_1$ of observing outcome $i=1$, a probability $p_2$ of observing outcome $i=2$, ..." and so on (with appropriate continuous limits for continuous outcomes).

Now, there's a measurement precision/error associated with each single instance of the measurement, and also a variability of the outcomes across repetitions of the measurement. The first kind of error can be made as small as we please. The variability across repetitions, however, generally appears not to be reducible below some nonzero amount which depends on the specific state and the specific measurement. This latter variability is what the "$\Delta x$" in the Heisenberg formula refers to.

So when we say "cannot be measured with arbitrary precision", what we mean more exactly is that "its variability across measurement repetitions cannot be made arbitrarily low". The fundamental mystery of quantum mechanics is the lack – in a systematic way – of reproducibility across measurement instances. But the error in the outcome of each single instance has no theoretical lower bound.

Of course this situation affects our predictive abilities, because whenever we repeat the same kind of measurement on a system prepared on the same kind of state, we don't really know what to expect, within $\Delta x$.

This important distinction between single and multiple measurement instances was first pointed out by Ballentine in 1970:

• Ballentine: The Statistical Interpretation of Quantum Mechanics (1970).

see especially the very explanatory Fig. 2 there. And it's not a matter of "interpretation", as the title might today suggest. It's an experimental fact. Clear experimental examples of this distinction are given for example in

• Leonhardt: Measuring the Quantum State of Light (1997)

see for example Fig. 2.1 there and its explanation. Also the more advanced

• Mandel, Wolf: Optical Coherence and Quantum Optics (2008).

See also the textbooks given below.

The distinction between error of one measurement instance and variability across measurement instances is also evident if you think about a Stern-Gerlach experiment. Suppose we prepare a spin in the state $x+$ and we measure it in the direction $y$. The measurement yields only one of two clearly distinct spots, corresponding to either the outcome $+\hbar/2$ or $-\hbar/2$ in the $y$ direction. This outcome may have some error in practice, but we can in principle clearly distinguish whether it is $+\hbar/2$ or $-\hbar/2$. However, if we prepare a new spin in the state $x+$ and measure $y$ again, we can very well find the opposite outcome – again very precisely measured. Over many measurements we observe these $+$ and $-$ outcomes roughly 50% each. The standard deviation is $\hbar/2$, and that's indeed the "$\Delta S_y$" given by the quantum formulae: they refer to measurement repetitions, not to one single instance in which you send a single electron through the apparatus.

It must be stressed that some authors (for example Leonhardt above) use the term "measurement result" to mean, not the result of a single experiment, but the average value $\bar{x}$ found in several repetitions of an experiment. Of course this average value has uncertainty $\Delta x$. There's no contradiction here, just a different terminology. You can call "measurement" what you please – just be precise in explaining what your experimental protocol is. Some authors use the term "one-shot measurement" to make the distinction clear; as an example, check these titles:

• Pyshkin et al: Ground-state cooling of quantum systems via a one-shot measurement (2016).
• Yung et al: One-shot detection limits of quantum illumination with discrete signals (2020).

The fact that, even though the predictive uncertainty $\Delta x$ is finite, we can have infinite precision in a single (one-shot) measurement, is not worthless, but very important in applications such as [quantum key distribution](https://en.wikipedia.org/wiki/Quantum_key_distribution). In many key-distribution protocols the two key-sharing parties compare the precise values $x$ they obtained in single-instance measurements of their entangled states. These values will be correlated to within their single-instance measurement error, which is much smaller than the predictive uncertainty $\Delta x$. The presence of an eavesdropper would destroy this correlation. The two parties can therefore know that there's an eavesdropper if they see that their measured values only agree to within $\Delta x$, rather than to within the much smaller single-instance measurement error. This scheme wouldn't work if the single-instance measurement error were $\Delta x$. See for example

• Reid: Quantum cryptography with a predetermined key, using continuous-variable Einstein-Podolsky-Rosen correlations (2000).
• Grosshans et al: Quantum key distribution using gaussian-modulated coherent states (2003). In Fig. 2 one can see very well the difference between single-instance measurement error and the variability $\Delta x$ across measurements.
• Madsen et al: Continuous variable quantum key distribution with modulated entangled states (2012). See especially Fig. 4 and its explanation.

2. What is exactly a "measurement of position" or of "momentum"?

In classical mechanics there's only one measurement (even if it can be realized by different technological means) of any specific quantity $Q$, such as position or spin or momentum. And classical mechanics says that the error in one measurement instance and the variability across instances can both be made as low as we please.

In quantum theory there are many different experimental protocols that we can interpret, for different reasons, as "measurements" of that quantity $Q$. Usually they all yield the same mean value across repetitions (for a given state), but differ in other statistical properties such as variance. Because of this, and of the variability explained above, Bell (of the famous Bell's theorem) protested that we actually shouldn't call these experimental procedures "measurements":

• Bell: Against "measurement" (1990).

In particular, in classical physics there's one joint, simultaneous measurement of position and momentum. In quantum theory there are several measurement protocols that can be interpreted as joint, simultaneous measurements of position and momentum, in the sense that each instance of such measurement yields two values, the one is position, the other is momentum. In the classical limit they become the classical simultaneous measurement of $x$ and $p$. This possibility was first pointed out by Arthurs & Kelly in 1965:

• Arthurs, Kelly: On the simultaneous measurement of a pair of conjugate observables (1965).

and further discussed, for example, in

• Stenholm: Simultaneous measurement of conjugate variables (1992).

This simultaneous measurement is not represented by $\hat{x}$ and $\hat{p}$, but by a pair of commuting operators $(\hat{X}, \hat{P})$ satisfying $\hat{X}+\hat{x}=\hat{a}$, $\hat{P}+\hat{p}=\hat{b}$, for specially chosen $\hat{a}, \hat{b}$. The point is that the joint operator $(\hat{X}, \hat{P})$ can rightfully be called a simultaneous measurement of position and momentum, because it reduces to that measurement in the classical limit (and obviously we have $\bar{X}=\bar{x}, \bar{P}=\bar{p}$). In fact, from the equations above we could very well say that $\hat{x},\hat{p}$ are defined in terms of $\hat{X},\hat{P}$, rather than vice versa.

This kind of simultaneous measurement – which is possible for any pairs of conjugate variables, not just position and momentum – is not a theoretical quirk, but is a daily routine measurement in quantum-optics labs for example. It is used to do quantum tomography, among other applications. As far as I know one of the first experimental realizations was made in 1984:

• Walker, Carroll: Simultaneous phase and amplitude measurements on optical signals using a multiport junction (1984).

You can find detailed theoretical and experimental descriptions of it  in Leonhardt's book above, chapter 6, tellingly entitled "Simultaneous measurement of position and momentum".

But as I said, there are several different protocols that may be said to be a simultaneous measurement of conjugate observables, corresponding to different choices of $\hat{a},\hat{b}$. What's interesting is the way in which these measurements differ. They can be seen as forming a continuum between two extremes (see references above):

• At one extreme, the variability across measurement repetitions of $X$ has a lower bound (which depends on the state of the system), while the variability of $P$ is infinite. Basically it's as if we were measuring $X$ without measuring $P$. This corresponds to the traditional $\hat{x}$.
• At the other extreme, the variability across measurement repetitions of $P$ has a lower bound, while the variability for $X$ is infinite. So it's as if we were measuring $P$ without measuring $X$. This corresponds to the traditional $\hat{p}$.
• In between, there are measurement protocols which have more and more variability for $X$ across measurement instances, and less and less variability for $P$. This "continuum" of measurement protocols interpolates between the two extremes above. There is a "sweet spot" in between in which we have a simultaneous measurement of both quantities with a finite variability for each. The product of their variabilities, $\Delta X\ \Delta P$, for this "sweet-spot measurement protocol" satisfies an inequality similar to the well-known one for conjugate variables, but with an upper bound slightly larger than the traditional $\hbar/2$ (just twice as much, see eqn (12) in Arthurs & Kelly). So there's a price to pay for the ability to measure them simultaneously.

This kind of "continuum" of simultaneous measurements is also possible for the famous double-slit experiment. It's realized by using "noisy" detectors at the slits. There are setups in which we can observe a weak interference beyond the two-slit screen, and at the same time have some certainty about the slit at which a photon could be detected. See for example:

• Wootters, Zurek: Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle (1979).
• Banaszek et al: Quantum mechanical which-way experiment with an internal degree of freedom (2013).
• Chiao et al: Quantum non-locality in two-photon experiments at Berkeley (1995), for variations of this experiment.

We might be tempted to ask "OK but what's the real measurement of position an momentum, among all these?". But within quantum theory this is a meaningless question, similar to asking "In which frame of reference are these two events really simultaneous?" within relativity theory. The classical notions and quantities of position and momentum simply don't exist in quantum theory. We have several other notions and quantities that have some similarities to the classical ones. Which to consider? it depends, on the context and application. The situation indeed has some similarities with that for "simultaneity" in relativity: there are "different simultaneities" dependent on the frame of reference; which we choose depends on the problem and application.

In quantum theory we can't really say "the system has these values", or "these are the actual values". All we can say is that when we do such-and-such to the system, then so-and-so happens. For this reason many quantum physicists (check eg Busch et al. below) prefer to speak of "intervention on a system" rather than "measurement of a system" (I personally avoid the term "measurement" too).

Summing up: we can also say that a simultaneous and arbitrarily precise measurement of position and momentum is possible – and in fact a routine.

So the answer to your question is that in a single measurement instance we actually can (and do!) measure position and simultaneously and both with arbitrary precision. This fact is important in applications such as quantum-key distribution, mentioned above.

But we also observe an unavoidable variability upon identical repetitions of such measurement. This variability makes the arbitrary single-measurement precision unimportant in other applications, where consistency through repetitions is required instead.

Moreover, we must specify which of the simultaneous measurements of momentum and position we're performing: there isn't just one, as in classical physics.

Even if the answer to your question is positive, we must stress that:
(1) Heisenberg's principle is not violated, because it refers to the variability across measurement repetitions, not the the error in a single measurement. (2) It's still true that the operators $\hat{x}$ and $\hat{p}$ cannot be measured simultaneously. What we're measuring is a slightly different operator; but this operator can be rightfully called a joint measurement of position and momentum, because it reduces to that measurement in the classical limit.

Old-fashioned statements about the uncertainty principle must therefore be taken with a grain of salt.

Comments

Here are several good books discussing these matters with clarity, precision, and experimental evidence:

• de Muynck: Foundations of Quantum Mechanics, an Empiricist Approach (2004).
• Peres: Quantum Theory: Concepts and Methods (2002).
• Holevo: Probabilistic and Statistical Aspects of Quantum Theory (2011).
• Busch, Grabowski, Lahti: Operational Quantum Physics (1995)
• Nielsen, Chuang: Quantum Computation and Quantum Information (2010).
• Bengtsson, Życzkowski: Geometry of Quantum States: An Introduction to Quantum Entanglement (2017).