Are there measuring standards (and units) for the identification of qubits?

The representation of bits in different technological areas:

1. Normal digital bits are mere abstractions of the underlying electric current through wires. Different standards, like CMOS or TTL, assign different thresholds to such signals: "if the voltage goes above this level, then the bit is 1; if it goes below this level, then the bit is 0; discard in any other case".

2. In genetics, we usually consider a signal as a 1 if it is "enough" to trigger the target response; 0 otherwise. In this scenario, the thresholding is qualitative.

From the point of view of quantum information, qubits also abstractions, but in practice measurements will need standards to be comparable.

Question: From the point of view of quantum engineering, is there any standard technique/method to identify their value e.g. based on detection thresholds or fidelity verification like Bell inequalities violation? Are there units for that hidden signals?

The best possible answer would probably contain specific details for different architectures (e.g. superconductors vs photons) or contexts (e.g. quantum computing vs quantum communications).

2 Answers

It seems to me that, from the point of view of quantum engineering, we are a few years away from being at the point of fixing standards.

Standards are a good way to ensure the reproducibility of the behaviour of a piece of information technology, and the interoperability of the functionality of multiple pieces of information technology. It is clear that at some point we will require such standards. The question is: how would one begin to formulate those standards?

• Before we ensure the reproducibility of a piece of quantum technology, we should build a piece of quantum technology whose behaviour we want to reproduce, rather than immediately set out to improve upon. With perhaps one exception, I expect that almost everyone in the quantum technologies game are more interested in bettering their own designs — possibly involving significant revisions to any design parameters which could play the role of standards — rather than setting down parameters which they expect that everyone will be happy to use.

Conceivably D-WAVE is at this stage — obviously they would also like to improve upon their existing technology (as for instance do conventional chip manufacturers), but my understanding is that they are in the business of making $\,N>1\,$ computers of a given model whose behaviour is intended to achieve a certain, well, 'standard'. Whatever the computational power of their machines, it seems that they are in the business of engineering complex systems with multiple components, and doing so in a reproducible manner. So it seems very likely that they have standards for their qubits: but it is not clear that anyone apart from them will be interested in conforming to those standards (rather than building more versatile quantum computers for instance), or to what extent D-WAVE's standards are publicly available.

Another incipient exception is in the field of optics, where the inclination is very strong to use the existing materials technology of fibre optic cables: while there is no formal standard likely exists, a practical standard of using wavelengths of light which have very low attenuation in modern-day optical fibre is one that could be comfortably predicted to continue for the foreseeable future.

• Given this situation for individual approaches to quantum technologies, the question of interoperation is even more premature. No-one knows what their equipment is going to be interoperating with — again, with the probable exception of fibre-optic cable, and perhaps the mains frequency of your electrical grid if this is somehow important to take into account.

But ask the question again in five or ten years (more likely ten), and you may get a more interesting answer.

This answer is divided into three parts: Standards, what you're measuring, and how to derive and specify the accuracy of the measurement.

Question: Are there measuring standards (and units) for the identification of qubits?

There isn't an ISO Standard for "measuring qubits", there are ISO Standards called "17.020 - Metrology and measurement in general" which set forth standards for: 'preferred numbers', 'expression of uncertainty in measurement', 'Propagation of distributions using a Monte Carlo method', 'uncertainty', 'vocabulary', 'Statistical interpretation of data and methods of uncertainty evaluation', 'accuracy', 'Measurement management systems', 'calibration', 'Capability of detection', 'repeatability, reproducibility and trueness estimates in measurement uncertainty evaluation', etc.

When using equipment calibrated and used in accordance with those Standards you can claim ISO Compliance. Those Standards are a couple of hundred dollars each, of interest might be the BIPM "GUM: Guide to the Expression of Uncertainty in Measurement" - they write standards that might be incorporated into the ISO Standards (so they are free to review).

Question: From the point of view of quantum engineering, is there any standard technique/method to identify their value e.g. based on detection thresholds or fidelity verification like Bell inequalities violation? Are there units for that hidden signals?

Once you set forth a Standard for your laboratory to comply with, purchase and calibrate your equipment, and both use it correctly and audit its use, you come to your first big hurdle: uncertainty.

"In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position $x$ and momentum $p$, can be known.".

"... the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology. It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.".

So: It's not how good you are, how much you spend, or what you do; there will always be something about which it is uncertain in quantum systems. Also important: Measurement affects the result.

Question: The best possible answer would probably contain specific details for different architectures (e.g. superconductors vs photons) or contexts (e.g. quantum computing vs quantum communications).

[Exceedingly brief explanation of quantum mechanics and computing - lots omitted. The question calls for: "The best possible answer would probably contain specific details for different architectures ..." - this answer is not about measuring the physical properties of a qubit, it's about measuring the information contained within any qubit - architecture neutral.]

(Well known and respected) Scott Aaronson's webpage: "PHYS771 Lecture 9: Quantum" explains:

"... nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex. ... Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the numbers we used to call "probabilities" can be negative numbers. ...

[Lots omitted]

Let's consider a single bit. In probability theory, we can describe a bit as having a probability $p$ of being $0$, and a probability $1-p$ of being $1$. But if we switch from the 1-norm to the 2-norm, now we no longer want two numbers that sum to $1$, we want two numbers whose squares sum to $1$. (I'm assuming we're still talking about real numbers.) In other words, we now want a vector $(α,β)$ where $α^2 + β^2 = 1$. Of course, the set of all such vectors forms a circle:

The theory we're inventing will somehow have to connect to observation. So, suppose we have a bit that's described by this vector $(α,β)$. Then we'll need to specify what happens if we look at the bit. Well, since it is a bit, we should see either $0$ or $1$! Furthermore, the probability of seeing $0$ and the probability of seeing $1$ had better add up to $1$. Now, starting from the vector $(α,β)$, how can we get two numbers that add up to $1$? Simple: we can let $α^2$ be the probability of a $0$ outcome, and let $β^2$ be the probability of a $1$ outcome.

But in that case, why not forget about $α$ and $β$, and just describe the bit directly in terms of probabilities? Ahhhhh. The difference comes in how the vector changes when we apply an operation to it. In probability theory, if we have a bit that's represented by the vector $(p,1-p)$, then we can represent any operation on the bit by a stochastic matrix: that is, a matrix of nonnegative real numbers where every column adds up to $1$. So for example, the "bit flip" operation -- which changes the probability of a $1$ outcome from $p$ to $1-p$ -- can be represented as follows: $$\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix} \begin{pmatrix} p\\ 1-p\end{pmatrix}=\begin{pmatrix} 1-p\\ p\end{pmatrix}$$ Indeed, it turns out that a stochastic matrix is the most general sort of matrix that always maps a probability vector to another probability vector.

[A little omitted.]

This "2-norm bit" that we've defined has a name, which as you know is qubit. Physicists like to represent qubits using what they call "Dirac ket notation", in which the vector $(α,β)$ becomes $α\left| 0 \right>+β\left| 1 \right>$. Here $α$ is the amplitude of outcome $\left| 0 \right>$, and $β$ is the amplitude of outcome $\left| 1 \right>$.

This notation usually drives computer scientists up a wall when they first see it -- especially because of the asymmetric brackets! But if you stick with it, you see that it's really not so bad. As an example, instead of writing out a vector like $(0,0,3/5,0,0,0,4/5,0,0)$, you can simply write $\frac{3}{5}\left|3\right>+\frac{4}{5}\left|7\right>$, omitting all of the $0$ entries.

So given a qubit, we can transform it by applying any 2-by-2 unitary matrix -- and that leads already to the famous effect of quantum interference. For example, consider the unitary matrix

$$\begin{pmatrix} \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \quad \frac{1}{\sqrt{2}}\end{pmatrix}$$

which takes a vector in the plane and rotates it by 45 degrees counterclockwise. Now consider the state $\left| 0 \right>$. If we apply U once to this state, we'll get $\frac{1}{\sqrt{2}}\left(\left| 0 \right>+\left| 1 \right>\right)$ -- it's like taking a coin and flipping it. But then, if we apply the same operation U a second time, we'll get $\left| 1 \right>$:

$$\begin{pmatrix} \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \quad \frac{1}{\sqrt{2}}\end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \end{pmatrix}= \begin{pmatrix} 0\\ 1 \end{pmatrix}$$

So in other words, applying a "randomizing" operation to a "random" state produces a deterministic outcome! Intuitively, even though there are two "paths" that lead to the outcome $\left| 0 \right>$, one of those paths has a positive amplitude and the other has a negative amplitude. As a result, the two paths interfere destructively and cancel each other out. By contrast, the two paths leading to the outcome $\left| 1 \right>$ both have positive amplitude and therefore interfere constructively.

$$\require{enclose} \begin{array}{} & & & \left| 0 \right> \\ \quad & \quad & \swarrow & \quad \; & \quad \searrow \\ & \; \left| 0 \right> & & & & \left| 1 \right> \\ \quad \swarrow & & \searrow & \quad & \quad \swarrow & & \searrow \\ \enclose{downdiagonalstrike,updiagonalstrike}[mathcolor="red"]{\color{black}{\left|0\right>}} & & \quad \left| 1 \right> & & \enclose{downdiagonalstrike,updiagonalstrike}[mathcolor="red"]{\color{black}{-\left|0\right>}} & & \quad \left| 1 \right> \\ \end{array}$$

The reason you never see this sort of interference in the classical world is that probabilities can't be negative. So, cancellation between positive and negative amplitudes can be seen as the source of all "quantum weirdness" -- the one thing that makes quantum mechanics different from classical probability theory. How I wish someone had told me that when I first heard the word "quantum"!". [End of quote from Scott Aaronson's webpage.]

[Read the remainder of his webpage linked above to learn about: Mixed States, The Squaring Rule, Real vs. Complex Numbers, and Linearity.]

He concludes by recommending reading: "Unknown Quantum States: The Quantum de Finetti Representation", but that paper is from 18 Apr 2001; a lot has changed since then, and the article draws attention to the shortcoming of the explanation it offers:

"To prove the quantum version of the de Finetti theorem, we rely on the classical theorem as much as possible. ... With this in mind, the proof is expedited by making use of the theory of generalized quantum measurements or positive operator-valued measures (POVMs). We give a brief introduction to that theory. ... That there is no quantum de Finetti theorem in real Hilbert space means that there are fundamental differences between real and complex Hilbert spaces with respect to learning from measurement results. The ultimate reason for this is that in ordinary, complex-Hilbert-space quantum mechanics, exchangeability implies separability, i.e., the absence of entanglement. This follows directly from the quantum de Finetti theorem because of states of the form Eq. (3.15) are not entangled. This implication does not carry over to real Hilbert spaces.".

Now to a partial explanation of modern tomography measurement techniques, that recover the data (not physical properties) from qubits - though physical properties can also be determined (and in some types of qubits it is a more easily noticeable physical property, size, rather than a less noticeable one, spin, that determines the data).

Further information on Quantum Computer Science and Measurement can be found on this webpage, along with numerous links: "Chem/CS/Phys191: Qubits, Quantum Mechanics, and Computers". In particular see Chapters 1 and 2: "Qubits and Quantum Measurement" and "Entanglement".

Possibly the takeaway, relevant to your question, is:

".6 The Measurement Principle

This linear superposition $\left| \psi \right> = \sum_{j=0}^{k-1} \alpha_j \left|j\right>$ is part of the private world of the electron. Access to the information describing this state is severely limited — in particular, we cannot actually measure the complex amplitudes $\alpha_j$ This is not just a practical limitation; it is enshrined in the measurement postulate of quantum physics.

A measurement on this k state system yields one of at most $k$ possible outcomes: i.e. an integer between $0$ and $k − 1$. Measuring $\left| \psi \right>$ in the standard basis yields $j$ with probability $\left| \alpha_j \right|^2$.

One important aspect of the measurement process is that it alters the state of the quantum system: the effect of the measurement is that the new state is exactly the outcome of the measurement. I.e., if the outcome of the measurement is $j$, then following the measurement, the qubit is in state $\left| j \right>$. This implies that you cannot collect any additional information about the amplitudes $\alpha_j$ by repeating the measurement.

Intuitively, a measurement provides the only way of reaching into the Hilbert space to probe the quantum state vector. In general, this is done by selecting an orthonormal basis $\left|e_0\right>,\ldots, \left|e_{k−1}\right>$. The outcome of the measurement is $\left|e_j\right>$ with probability equal to the square of the length of the projection of the state vector $\psi$ on $\left|e_j\right>$. A consequence of performing the measurement is that the new state vector is $\left|e_j\right>$. This measurement may be regarded as a probabilistic rule for projecting the state vector onto one of the vectors of the orthonormal measurement basis.

Accurate measurement and error specification:

See how measurements were performed in 2010, in one particular paper: "Optimal, reliable estimation of quantum states", 20 Apr 2010, by Robin Blume-Kohout:

"Abstract

Accurately inferring the state of a quantum device from the results of measurements is a crucial task in building quantum information processing hardware. The predominant state estimation procedure, maximum likelihood estimation (MLE), generally reports an estimate with zero eigenvalues. These cannot be justified. Furthermore, the MLE estimate is incompatible with error bars, so conclusions drawn from it are suspect. I propose an alternative procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues, its eigenvalues provide a bound on their own uncertainties, and under certain circumstances, it is provably the most accurate procedure possible. I show how to implement BME numerically, and how to obtain natural error bars that are compatible with the estimate. Finally, I briefly discuss the differences between Bayesian and frequentist estimation techniques.".

But a later paper, "Optimal error regions for quantum state estimation" (13 Dec 2013), by Jiangwei Shang, Et al, supports an extension to MLE:

"Abstract

An estimator is a state that represents one's best guess of the actual state of the quantum system for the given data. Such estimators are points in the state space. To be statistically meaningful, they have to be endowed with error regions, the generalization of error bars beyond one dimension. As opposed to standard ad hoc constructions of error regions, we introduce the maximum-likelihood region — the region of largest likelihood among all regions of the same size — as the natural counterpart of the popular maximum-likelihood estimator. Here, the size of a region is its prior probability. A related concept is the smallest credible region — the smallest region with pre-chosen posterior probability. In both cases, the optimal error region has a constant likelihood on its boundary. This surprisingly simple characterization permits concise reporting of the error regions, even in high-dimensional problems. For illustration, we identify optimal error regions for single-qubit and two-qubit states from computer-generated data that simulate incomplete tomography with few measured copies.".

"Determining which quantum measurement performs better for state estimation" (13 July 2015), by Jaroslav Řeháček, Yong Siah Teo, and Zdeněk Hradil, Phys. Rev. A 92, 012108:

"Abstract

We introduce an operational and statistically meaningful measure, the quantum tomographic transfer function, that possesses important physical invariance properties for judging whether a given informationally complete quantum measurement performs better tomographically in quantum-state estimation relative to other informationally complete measurements. This function is independent of the unknown true state of the quantum source and is directly related to the average optimal tomographic accuracy of an unbiased state estimator for the measurement in the limit of many sampling events. For the experimentally appealing minimally complete measurements, the transfer function is an extremely simple formula. We also give an explicit expression for this transfer function in terms of an ordered expansion that is readily computable and illustrate its usage with numerical simulations and its consistency with some known results.".

Compare with: "Practical and Reliable Error Bars in Quantum Tomography" (1 July 2016), by Philippe Faist and Renato Renner, Phys. Rev. Lett. 117, 010404:

"Abstract

Precise characterization of quantum devices is usually achieved with quantum tomography. However, most methods which are currently widely used in experiments, such as maximum likelihood estimation, lack a well-justified error analysis. Promising recent methods based on confidence regions are difficult to apply in practice or yield error bars which are unnecessarily large. Here, we propose a practical yet robust method for obtaining error bars. We do so by introducing a novel representation of the output of the tomography procedure, the quantum error bars. This representation is (i) concise, being given in terms of few parameters, (ii) intuitive, providing a fair idea of the “spread” of the error, and (iii) useful, containing the necessary information for constructing confidence regions. The statements resulting from our method are formulated in terms of a figure of merit, such as the fidelity to a reference state. We present an algorithm for computing this representation and provide ready-to-use software. Our procedure is applied to actual experimental data obtained from two superconducting qubits in an entangled state, demonstrating the applicability of our method.".

"Fast universal performance certification of measurement schemes for quantum tomography" (15 Aug 2016), by Dominik Koutný, Yong Siah Teo, Zdeněk Hradil, and Jaroslav Řeháček, Phys. Rev. A 94, 022113:

"Abstract

Prior to a measurement in a quantum-state tomography experiment, it is important to evaluate the performance of this measurement with respect to the average accuracy in state estimation. We propose a fast and reliable numerical certification of measurement performance that is applicable to any known quantum measurement. This numerical method is based on the statistical theory of unbiased estimation that is valid for any physically accessible quantum state that is necessarily full rank in the limit of a large number of measurement copies, and the Hoeffding inequality that applies to bounded statistical quantities in the quantum state space. We present the use of this straightforward certification procedure by illustrating the convergence to optimal pure-state tomography with an increasing number of overcomplete measurement outcomes. Furthermore, we demonstrate that the performances of symmetric informationally complete measurements and mutually unbiased bases, which are commonly regarded as optimal measurements, can be easily beaten in tomographic performance with randomly generated measurements that are only slightly more informationally overcomplete. Two important classes of random measurements are also discussed with the help of our numerical machinery.".

The design of the measurement protocol will change over the years, but there are Standards under which one can operate while calibrating equipment, documenting procedures, and making measurements.

If results were not statistically reproducible you would have a random number generator, not a quantum computer.