# Tomaytos, Tomahtos and Non-local Measurements

In the interest of keeping this blog active, i’m recycling one of my old IQC blog posts

One of my discoveries as a physicist was that, despite all attempts at clarity, we still have different meanings for the same words and use different words to refer the the same thing. When Alice says measurement, Bob hears a `quantum to classical channel’, but Alice, a hard-core Everettian, does not even believe such channels exist. When Charlie says non-local, he means Bell non-local, but string theorist Dan starts lecturing him about non-local Lagrangian terms and violations of causality. And when I say non-local measurements, you hear #\$%^ ?e#&*?.  Let me give you a hint, I do not mean ‘Bell non-local quantum to classical channels’, to be honest, I am not even sure what that would mean.

So what do I mean when I say measurement? A measurement is a quantum operation that takes a quantum state as its input and spits out a quantum state and a classical result as an output (no, I am not an Everettian). For simplicity I will concentrate of a special case of this operation, a projective measurement of an observable A. The classical result of a projective measurement is an eigenvalue of A, but what is the outgoing state?

A textbook (projective) measurement.  A Quantum state $|\psi\rangle$  goes in and a classical outcome “r” comes out together with a corresponding  quantum state $|\psi_r\rangle$.

### The Lüders measurement

Even the term projective measurement can lead to confusion, and indeed in the early days of quantum mechanics it did. When von Neumann wrote down the mathematical formalism for quantum measurements, he missed an important detail about degenerate observables (i.e Hermitian operators with a degenerate eigenvalue spectrum). In the usual projective measurement, the state of the system after the measurement is uniquely determined by the classical result (an eigenvalue of the observable). Consequently,  if we don’t look at the classical result, the quantum channel is a standard dephasing channel. In the case of a degenerate observable, the same eigenvalue corresponds to two or more orthogonal eigenstates. Seemingly the state of the system should correspond to one of those eigenstates, and the channel is a standard dephasing channel. But a degenerate spectrum means that the set of orthogonal eigenvectors is not unique, instead each eigenvalue has a corresponding subspace of eigenvectors. What Lüders suggested is that the dephasing channel does nothing within these subspaces.

### Example

Consider the two qubit observable $A=|00\rangle\langle 00 |$. It has eigenvalues $1,0,0,0$. A $1$ result in this measurement corresponds to “The system is in the state $|00\rangle$“.  Following a measurement with outcome $1$, the outgoing state will be $|00\rangle$. Similarly, a $0$ result corresponds to “The system is not in the state $|00\rangle$ “. But here is where the Lüders rule kicks in. Given a generic input state $\alpha|00\rangle+\beta|01\rangle+\gamma|{10}\rangle+\delta|{11}\rangle$ and a Lüders measurement of $A$ with outcome 0, the outgoing state will be $\frac{1}{\sqrt{|\alpha|^2+|\beta|^2+|\gamma|^2}}\left[\beta|{01}\rangle+\gamma|{10}\rangle+\delta|{11}\rangle\right]$.

### Non-local measurements

The relation to non-locality may already be apparent from the example, but let me start with some definitions. A system can be called non-local if it has parts in different locations, e.g. one part on Earth and the other on the moon. A measurement is non-local if it reveals something about a non-local system as a whole. In principle these definitions apply to classical and quantum systems. Classically a non-local measurement is trivial, there is no conceptual reason why we can’t just measure at each location. For a quantum system the situation is different. Let us use the example above, but now consider the situation where the two qubits are in separate locations. Local measurements of $\sigma_z$ will produce the desired measurement statistics (after coarse graining) but reveal too much information and dephase the state completely, while a Lüders measurement should not. What is quite neat about this example is that the Lüders measurement of $|{00}\rangle$ cannot be implemented without entanglement (or quantum communication) resources and two-way classical communication. To prove that entanglement is necessary, it is enough to give an example where entanglement is created during the measurement. To show that communication is necessary, it is enough to show that the measurement (even if the outcome is unknown) can be used to transmit information. The detailed proof is left as an exercise to the reader. The lazy reader can find it here (see appendix A).

This is a slighly modified version of a  Feb 2016 IQC blog post.

# Three papers published

When it rains it pours. I had three papers published in the last week. One experimental paper and two papers about entanglement.

1. Experimental violation of the Leggett–Garg inequality in a three-level system. A cool experimental project with IQC’s liquid state NMR group.    Check out the outreach article  about this experiment.
2. Extrapolated quantum states, void states and a huge novel class of distillable entangled states. My first collaboration with Tal Mor and Michel Boyer and my first paper to appear in a bona fide CS journal (although the content is really mathematical physics). It took about 18 months to get the first referee reports.
3. Entanglement and deterministic quantum computing with one qubit. This is a follow up to the paper above, although it appeared on arXiv  a few months earlier.

# Entangled cats and quantum discord

It turns out that few people appreciate the relation between Schrödinger’s cat and entanglement.  When we hear entanglement, the first paper that comes to mind is Einstein Podolsky and Rosen’s “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”  EPR were the first to point out a strange prediction of quantum mechanics (which we now call entanglement),  but the   term entanglement (or its German equivalent) was coined by Schrödinger in a  paper  inspired by EPR. In the same paper Schrödinger describes an experiment involving a cat interacting with a  “small flask of hydrocyanic acid” (and a Geiger counter etc.) such that, at some point the best quantum description of the cat-flask system is an entangled state. If one then ignores the state of the flask, the cat is in a mixture of being dead and alive (and not in a superposition as some wold have you believe). Schrödinger noted that this is a peculiar situation,  the cat’s state has large uncertainty (it is maximally mixed) while the cat+flask etc. are in a well defined state, that is, the uncertainty is at the minimum allowed by the theory. We call such a state a pure state.

Schrödinger coined the term entanglement in the context of pure quantum states. A pure quantum state describing two subsystems is entangled  if (and only if) the state of each subsystem is mixed, i.e (within the context of the relevant operators) there is no (rank 1) measurement that yields a definite outcome1.  But in reality the states we encounter are mixed and Schrödinger’s definition cannot be applied in a straightforward way.

A mixed quantum state is similar to a composite color such as pink, brown or white which have no specific wavelength.  Any composite color can be made by mixing elements from a set of primary colors such as red green and blue (RBG) but one can choose different conventions to produce the same color2. Similarly, a mixed quantum state does not have a unique decomposition in terms of pure quantum states.  The cat in the box is in a mixture of being dead and alive, yet it is also in a mixture of being in various superpositions of dead and alive.  It turns out that this creates a serious problem when we try to define mixed state entanglement.

The standard way to define entanglement is to look for a decomposition into pure non-entangled states. So, if we can find some way to describe the mixed quantum state as a mixture of non-entangled (i.e separable) states, then the state is separable.  This is a convenient mathematical definition 3 but it is not consistent with the physical manifestation of entanglement.

What is the physical manifestation of entanglement?

One way  to think of entanglement is as a resource for some physical tasks such as teleportation or quantum communication.  Ideally one would want to make a claim such as “If you gave me enough copies of an entangled state I could perform perfect teleportation”.  Indeed this would be the case if the states were pure, but in the case of mixed states there are counterexamples to this statement for practically any physical task (except channel discrimination).

Another way to think about entanglement is as a way to quantify complexity.  The intuition comes from the fact that a good enough description4  of an entangled state usually requires a very large memory.  If a system is in a pure quantum state and it is not entangled, we can fully describe it by specifying each part.  If it is entangled, we must also specify some global properties. Roughly speaking, these global properties describe the relations between the subsystems, and the number of parameters we need to keep track of grows exponentially with the number of subsystems.  However, as it turns out, some highly entangled states can be described in a very concise way.  When it comes to mixed states, the situation is different and it is unclear if we can give a concise description of a separable system.

The bottom line is,  the physical manifestation of entanglement is not trivial, especially when we consider mixed states. As a result, there is no obvious one-size-fits-all way to extend various ideas about entanglement to mixed states.

Quantum correlations and discord

So, while there is no unique way to generalize entanglement to mixed states,  one particular method (entangled = not separable) has become canonical. Other ways of generalizing entanglement from pure states must be given a different name. Many of these fall into the broad category of quantum correlations (or discord).  These quantities are equivalent to entanglement in pure states, but don’t correspond to non-separable in the case of mixed states.

Ok, but why should we care?

Entanglement is one of the central features of quantum theory, and there is good reason to suspect that it plays a crucial role in many physical scenarios, from many body physics to black holes and of course quantum information processing.  Unfortunately, it is not trivial to extend our mathematical treatment of entanglement beyond the two party, pure state case.   There are many examples  where separable mixed states or ensembles of separable pure states,  behave in a way that resembles pure entangled states.  Apart from the obvious joy of playing around with the mathematical structure of quantum states, there are many things we can learn by trying to understand this rich structure beyond the usual separable vs non-separable states.  Discord is one, and there are others, most notably Bell non-locality.

And if you want to know more, check out my paper with Danny Terno , arXiv:1608.01920

Footnotes

1.  The caveats here are simply to ensure that the measurement is not trivial in some sense. For example if the states are entangled in spin, asking about their position is not relevant, similarly making a trivial measurement (one that has outcome 1 if the spin is up and the same outcome 1 if the spin is down) is not interesting.
2. Actually,  the situation with colors is far more complex than I described, but as far as the human eye is concerned the statement is more or less correct.  Spectroscopy would reveal a unique decomposition to any color.  Quantum states on the other hand have no unique decomposition, in fact, if they did we would be in big trouble with relativistic causality (i.e we would be able to send information faster than the speed of light). As a side note: Schrödinger was interested in our perception of colors and made some interesting contributions to the field.
3. Given the complete description of a (mixed) quantum state, it can be very difficult (computationally) to decide if it is entangled or separable.
4. Think of trying to keep a description of the state in the memory of a computer for the purpose of simulating the evolution and finally reproducing some measurement statistics.

These last four and a half months have been exciting in may ways. Three papers submitted to arXiv: The first on Entanglement in DQC1, the second, a Leggett Garg experiment in liquid state NMR; and the third, a book chapter titled Why should we care about quantum discord? I also had two papers published one on quantum money and the second on  sequential measurements.

In August I organized a workshop on Semi-quantum computing and recently wrote about it on the IQC blog.  I also attended a workshop on Entanglement and quantumnes  in Montréal.

Earlier this month I got sucked into a discussion about publishing.

# Nonlocal Measurements

My paper Nonlocal Measurements Via Quantum Erasure has  finally been published in PRL.  There is a short news story on the work on the IQC website. I also recently wrote a related blog post on nonlocal measurements for the IQC blog.

#### IQC blog post: Tomaytos, Tomahtos and Non-local Measurements

Nonlocal measurements via quantum erasure, A. Brodutch and E. Cohen  Phys. Rev. Lett 116 (2016)

# Misconceptions about weak measurements: 2. Weak, not noisy

It’s about time that I continue writing about misunderstandings surrounding weak measurements and weak values. This time I will try to explain the difference between weak measurements and noisy measurements.

## Why weak?

One of the first things we learn about quantum mechanics is that  the measurement process causes an unavoidable back-action on the measured system. As a consequence, some measurements are incompatible, i.e. the result of a measurement on observable  A  can change significantly if a different observable,  B, is measured before A.  A well known example is the measurements of position and momentum where the back-action leads to the Heisenberg uncertainty relation.

The measurement back-action can create  some seemingly paradoxical situations when we make counterfactual arguments such as

We measured A and got the result a, but had we measured B we would have go the result b which is incompatible with a.

These situations appear very often  when we consider  systems both past and future boundary conditions. In these cases they are known as pre and post selection (PPS) paradoxes. In PPS paradoxex the measurement back-action is important even when A and B commute.  An example is the three box paradox that I explain without mathematical detail:

A single particle is placed in one of three boxes A,B,C (actually in a superposition) at time t0 and is later found to be in some other superposition state at t1.  At time t0< t m < t1 one box is opened. The initial t0 and final t1 states of the particle are chosen in such a way that the following happens:

If box A is opened, the particle will be discovered with certainty. If box B is opened, the particle will also be found with certainty. If box C is opened the particle will be found with some probability. The situation seems paradoxical:

If the ball is found with certainty in box A, then it must have been in box A to begin with. But if it is also found with certainty in box B, so it must have been there …

One way to solve this apparent paradox is to note that the measurements are incompatible. i.e opening box A and not B,C is incompatible with opening box B and not A,C etc.

These are the types of questions that Aharonov Albert and Vaidman were investigating  1980s1 . Weak measurements were studied as a way to minimize the measurement back-action. These measurements then  provided a picture that arguably gives a solid (if somewhat strange) foundation to statements like the one above.

The motivation of weak measurements is therefore an attempt to derive a consistent picture where all observables are mutually compatible in a way which is similar to classical physics. In quantum mechanics this comes at a cost. The classical information gained by reading out the result of a single weak measurement is usually indistinguishable from noise. In other words  weak measurements are noisy measurements.

## Weak, not noisy

Part of the confusion around weak measurements lies in the fact that the statement above is not a sufficient condition for a weak measurement. One may argue noise is not even a necessary requirement, it is rather, a consequence of quantum mechanics. Weak measurements may be noisy, but noisy measurements are, in most cases, not weak. To understand this fact it is good to examine  both a classical and quantum scenario.

#### The classical scenario

Walking on the beach you see a person drowning. Being  a good swimmer you go in and try to save this person. As you get back to the beach you see that he is not responsive and decide to to find if he is alive. You are now faced with the choice of how to perform the measurement.

A weak measurement – You try to get a pulse – The measurement is somewhat noisy since the pulse may be too weak to notice. It is also a weak measurement since it is unlikely to change this person’s state.

A noisy measurement – You start screaming for help. There is some small chance that the guy will wake up and tell you to shut up.

A noisy, strong measurement – You start kicking the guy in the head, hoping that he regains conciseness. This is a strong measurement, but it is also noisy. The person might be alive and you still won’t notice after kicking his head, moreover the kick in the head might kill him.

#### The quantum scenario

You want to find the $\sigma_z$ component of a spin 1/2 particle.

A weak measurement – Perform the usual von Neumann measurement with weak coupling. There is still some back-action but if the coupling is sufficiently weak you can ignore it. The down side is that you will get very little information.

A noisy measurement – Perform the weak measurement as above, but follow it with a unitary rotation and some dephasing.

A noisy, strong measurement – Perform a standard projective measurement, but then add extra noise at the readout stage. This could, for example,  be the result of a defective amplifier.

While all of the measurements above are noisy, only the weak measurements follow the original motivation of making a measurement with a weak back-action.

## An extreme example

One neat example of a measurement which is noisy but not weak involves a wave function with a probability distribution that has no tails.

Take the measurement of a Pauli observable that has results $\pm1$ and imagine that after the readout we get the following probability distributions: If the system was initially in the state corresponding to +1 we get a flat distribution between -9 and 11, if the result is -1 we get a flat distribution between -11 and 9. The measurement is noisy, in fact any result between -9 and +9 will give us no information about the system. However it is not weak since any result outside this range will cause the state of the system to collapse into an eigenstate.

A pointer with no tails: The probability density function for the result of a  dichotomic measurement. A +1 state will produce the blue distribution while a   -1 state will produce the orange one. Although a result between -9 and +9 will provide no information, the measurement is still not weak.

It is not surprising that this type of measurement will not produce a weak value as the expectation value of a given set of measurements on a pre and post selected system.  While this is is obviously an extreme case,  any situation where the probability density function for the readout probabilities has no tails will not be weak for the same reason. The same is usually true in cases where the derivatives of the probability density function are very large. In less technical terms – noise is not a sufficient condition for a weak measurement.

1. To get a partial historic account of what AAV were thinking see David Albert’s remarks in Howard Wiseman’s QTWOIII talk on weak measurements (around minute 25-29)