What Is Quantum Information?

Quantum information studies what physical systems can communicate, compute, preserve, or reveal when their states and transformations obey quantum mechanics. Its results concern tasks: distinguishing preparations, sending a message, estimating a field, or protecting a logical state against noise. The same state can be useful for one task and unhelpful for another.

The subject uses ordinary quantum mechanics. It does not introduce a new substance called information, or one number that measures the usefulness of every quantum system. A channel capacity, a discrimination probability, and a sensor’s uncertainty answer different questions.

Required background. Quantum states and the Born rule connect preparations to probabilities; tensor products describe combined systems.

States, distinguishability, and relative phase

Quantum information uses ordinary quantum mechanics, but several ordinary information-processing assumptions no longer hold.

A state is not a readable classical label

A classical register can encode a symbol in mutually distinguishable states. An ideal measurement can read the symbol and leave a copy in another register. A quantum preparation may instead be chosen from nonorthogonal states. No measurement can identify arbitrary nonorthogonal alternatives perfectly in one shot.

For pure states $|\psi\rangle$ and $|\phi\rangle$, the overlap

$$\bigl|\langle\psi|\phi\rangle\bigr|$$

therefore has operational meaning. Orthogonal states can be perfectly distinguished; identical rays carry no distinguishable alternative; intermediate overlap implies an unavoidable error or inconclusive outcome. Trace Distance gives the canonical distinguishability metric, while the landing page’s worked state-discrimination anchor shows the optimal equal-prior success probability.

This limitation is not merely detector imperfection. It follows from the geometry of quantum states. The related No-Cloning and No-Signaling theorem page rules out a universal operation that makes perfect copies of arbitrary unknown states. Neither result forbids copying a known state or a set of mutually orthogonal codewords.

Relative phase becomes an operational variable

Consider the qubit family

$$|\psi_\varphi\rangle = \frac{|0\rangle+e^{i\varphi}|1\rangle}{\sqrt2}.$$

A measurement in the computational basis gives

$$p(0|\varphi)=p(1|\varphi)=\frac12,$$

so that measurement contains no information about $\varphi$. In the $X$ basis,

$$|\pm\rangle = \frac{|0\rangle\pm|1\rangle}{\sqrt2},$$

the probabilities are

$$\begin{aligned} p(+|\varphi) &= \bigl|\langle+|\psi_\varphi\rangle\bigr|^2 = \frac{1+\cos\varphi}{2}, \\ p(-|\varphi) &= \frac{1-\cos\varphi}{2}. \end{aligned}$$

The phase is neither a classical label hidden inside the qubit nor directly visible in every measurement. It becomes accessible through an interference experiment with a suitable reference and measurement basis. This small example already contains the logic of coherent control and quantum sensing:

1. Encode a parameter.
2. Preserve relative phase.
3. Choose an informative measurement.
4. Infer the parameter from repeated data.

One shot produces one random outcome, not the numerical value of $\varphi$. Estimation requires a statistical model, repeated preparations or an equivalent ensemble, and an uncertainty statement.

Composition permits nonclassical correlations

Two systems combine through a tensor product,

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B.$$

Some joint states cannot be written as product states or classical mixtures of product states. Their correlations are entangled. Entanglement can support teleportation, superdense coding, distributed protocols, metrology, and error correction, but it does not by itself transmit a controllable message faster than light.

Composite Systems and Entanglement develops the structure of joint states; entanglement in quantum information connects it to these tasks.

Physical evolution becomes an information channel

Closed-system evolution is unitary, but an information-processing component may include an environment, loss, discarded subsystems, random control errors, or measurement. With a fixed preparation of an initially uncorrelated environment, its unconditional input-output action is represented by a quantum channel

$$\mathcal E: \mathcal L(\mathcal H_A) \longrightarrow \mathcal L(\mathcal H_B),$$

a completely positive, trace-preserving linear map. A Kraus representation has the form

$$\mathcal E(\rho) = \sum_a K_a\rho K_a^\dagger, \qquad \sum_a K_a^\dagger K_a=I.$$

These equations specify an input-output transformation without uniquely identifying the microscopic cause of noise. Different Hamiltonians or environments can induce the same channel. Preparation-dependent correlations or memory can require a larger system description; separately estimated one-step channels need not compose correctly when the environment carries information between steps. Quantum channels and noise develops these qualifications.

Measurement is part of the task

A measurement with outcomes $y$ is represented at the probability level by positive operators $\{M_y\}$ satisfying

$$M_y\geq0, \qquad \sum_y M_y=I, \qquad p(y|\rho)=\operatorname{Tr}(M_y\rho).$$

The chosen measurement determines which distinctions become observable. If the post-measurement state matters, probabilities alone are insufficient; one needs a quantum instrument. The canonical treatment is in Generalized Measurements and Instruments.

Measurement in Circuits specializes this outcome-versus-poststate distinction to circuit diagrams, declared records, terminal or reusable outputs, and finite-shot postprocessing.

The lesson is not that observation mysteriously creates all information. It is that an operational claim must specify the preparation, process, measurement, and inference rule that connect a physical system to recorded data.

Resources and information measures

A density operator $\rho\geq0$, $\operatorname{Tr}\rho=1$ specifies probabilities for measurements on the modeled system. Different ensembles with the same density operator cannot be distinguished on that system alone, although a correlated reference may distinguish how they were prepared. A channel can describe a gate, a memory, a communication line, or discarding a subsystem; it need not represent spatial transmission.

A resource theory specifies which operations are available without consuming the resource. For a deterministic free operation $\Lambda$, a monotone obeys

$$R(\Lambda(\rho))\leq R(\rho).$$

For example, local operations and classical communication cannot create entanglement from separable states. The restriction allows arbitrary classical messages; it excludes quantum communication and shared entanglement supplied for free. Coherence relative to a basis, asymmetry relative to a symmetry, and magic relative to stabilizer operations use different free operations. See resource theories and Chitambar–Gour for the general framework.

Information quantities answer different questions

Quantity	Operational question	Important qualification
von Neumann entropy $S(\rho)$	How mixed is a state, or what asymptotic compression rate arises in a source model?	Not a universal score of usefulness
Trace distance $D(\rho,\sigma)$	How distinguishable are two states under the best measurement?	Requires specified priors for a decision probability
Fidelity $F(\rho,\sigma)$	How close are two states under a chosen convention?	State fidelity and average gate fidelity are distinct quantities
Mutual information $I(A:B)$	How much total correlation is present?	Includes classical and quantum correlation
Entanglement monotone	How much of a declared entanglement resource is present?	Depends on state class and allowed operations
Fisher information	How sensitive is a data model to a parameter?	A local sensitivity measure; an attainable precision also depends on statistical assumptions
Channel capacity	At what asymptotic rate can information be transmitted?	Classical, quantum, private, and assisted capacities differ

Use the Quantum Information Formula Reference for compact identities and Math Needed for Quantum Information for prerequisites.

Four kinds of task

Computation. A circuit prepares an output distribution, for example

$$p(z\mid C)=|\langle z|U_C|0^n\rangle|^2.$$

A useful algorithm makes a desired answer recoverable from its samples. The dimension $2^n$ is not a promise that all amplitudes can be read or that a classical input can be loaded cheaply. Algorithms and complexity compares explicit tasks and access assumptions. Coherent controlled operations preserve superpositions of alternatives; measurement and feedforward use classical records to select subsequent actions.

Communication. Classical messages, unknown quantum states, shared entanglement, and secret keys are different outputs. Teleportation transmits an unknown qubit using shared entanglement and two classical bits. Quantum key distribution produces a classical secret key under an explicit adversary model and requires authenticated communication. Measurement disturbance alone is not a security proof.

Sensing. A parameter $\theta$ changes a state or channel, and observed data determine an estimate $\hat\theta$. For independent, identically distributed repetitions, a regular likelihood with finite positive Fisher information and a locally unbiased estimator gives

$$\operatorname{Var}_\theta(\hat\theta) \geq\frac{1}{\nu F(\theta)}.$$

This is a local lower bound, not a general guarantee of an estimator attaining it or resolving a periodic ambiguity. Cramér–Rao bounds treats those assumptions, and quantum Fisher information optimizes the measurement sensitivity. Physical comparisons must also fix probe number, interrogation time, loss, preparation, and readout costs.

Simulation. A controllable quantum system represents a target model and estimates chosen properties. For a time-independent Hamiltonian, the target dynamics is $U(t)=e^{-iHt/\hbar}$. A digital implementation approximates it with gates; an analog implementation engineers suitable interactions. Neither verifies itself: model mapping, initial preparation, observable measurements, and approximation errors all matter. What is quantum simulation? develops these connections.

Physical models and technological conclusions

Quantum information changes which questions we ask of quantum mechanics; it does not replace its dynamics or probabilities. An experimentally successful protocol can agree with several interpretations of the same theory. It is not, for that reason alone, a test distinguishing those interpretations.

Similarly, observing a quantum effect differs from building a useful information-processing device. A long coherence time does not determine gate accuracy, and a large number of physical qubits does not determine logical computation size. To compare devices, specify the task, acceptable error, allowed operations, and total resources. Reading quantum-technology claims explains how to interpret experimental and theoretical conclusions.

The volume map gives chapter choices. The information foundations chapter develops the carriers and measures introduced here, while the learning roadmap connects them to the prerequisite physics and mathematics.

Exercises

1. Measurement basis and phase information

For

$$|\psi_\varphi\rangle = \frac{|0\rangle+e^{i\varphi}|1\rangle}{\sqrt2},$$

show that measuring in the $Y$ basis,

$$|{+i}\rangle = \frac{|0\rangle+i|1\rangle}{\sqrt2}, \qquad |{-i}\rangle = \frac{|0\rangle-i|1\rangle}{\sqrt2},$$

gives probabilities that depend on $\sin\varphi$. Which pair of bases would let repeated experiments estimate both $\cos\varphi$ and $\sin\varphi$?

Solution

The overlaps are

$$\begin{aligned} \langle {+i}|\psi_\varphi\rangle &= \frac{1-i e^{i\varphi}}{2}, \\ \langle {-i}|\psi_\varphi\rangle &= \frac{1+i e^{i\varphi}}{2}. \end{aligned}$$

Therefore

$$\begin{aligned} p(+i|\varphi) &= \frac{1+\sin\varphi}{2}, \\ p(-i|\varphi) &= \frac{1-\sin\varphi}{2}. \end{aligned}$$

Repeated $X$-basis measurements estimate $\cos\varphi$, while repeated $Y$-basis measurements estimate $\sin\varphi$. Together they determine $\varphi$ modulo $2\pi$, subject to sampling uncertainty. A $Z$-basis measurement gives neither component for this equatorial family.

2. Perfect discrimination implies orthogonality

Suppose a two-outcome POVM $\{M_\psi,M_\phi\}$ perfectly identifies whether a system was prepared in $|\psi\rangle$ or $|\phi\rangle$. Show that the two states must be orthogonal.

Solution

Perfect discrimination requires

$$\begin{aligned} \langle\psi|M_\psi|\psi\rangle&=1, & \langle\psi|M_\phi|\psi\rangle&=0, \\ \langle\phi|M_\phi|\phi\rangle&=1, & \langle\phi|M_\psi|\phi\rangle&=0. \end{aligned}$$

Because $M_\phi\geq0$, the zero expectation value implies

$$M_\phi|\psi\rangle=0.$$

Similarly, $M_\psi|\phi\rangle=0$. Using $M_\psi+M_\phi=I$,

$$\langle\psi|\phi\rangle = \langle\psi| (M_\psi+M_\phi) |\phi\rangle =0.$$

Thus distinct pure states can be identified with certainty in one shot only if they are orthogonal.

3. Dephasing removes the accessible phase

Apply complete dephasing in the computational basis to $|\psi_\varphi\rangle\langle\psi_\varphi|$:

$$\Delta(\rho) = |0\rangle\langle0|\rho|0\rangle\langle0| + |1\rangle\langle1|\rho|1\rangle\langle1|.$$

What is the output state, and what happens to the $X$-basis outcome probabilities?

Solution

Before dephasing,

$$\rho_\varphi = \frac12 \begin{pmatrix} 1 & e^{-i\varphi}\\ e^{i\varphi} & 1 \end{pmatrix}.$$

Complete dephasing removes the off-diagonal terms:

$$\Delta(\rho_\varphi) = \frac12 \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} = \frac I2.$$

The $X$-basis probabilities become

$$p(+)=p(-)=\frac12,$$

independent of $\varphi$. The channel has erased the coherence that carried phase information for this measurement task.

4. When high accuracy means no information

A source emits label 0 with probability 0.99 and label 1 with probability 0.01, but prepares the same quantum state for both labels. What is the best minimum-error success probability? Does that success mean the receiver learned the label from the system?

Solution

Every measurement has the same distribution conditional on either label. The system therefore supplies no information about which label occurred. Always guessing 0 nevertheless succeeds with probability 0.99. High accuracy relative to an imbalanced prior need not demonstrate distinguishability.

5. Resource monotonicity

Suppose $R$ is an entanglement monotone under local operations and classical communication. A proposed deterministic protocol claims

$$R(\Lambda(\rho))>R(\rho)$$

for an allowed local protocol $\Lambda$. Give three logically distinct explanations for the apparent violation.

Solution

Possible explanations include:

1. $\Lambda$ was not actually free; it consumed shared entanglement, quantum communication, or another resource excluded from the chosen free operations.
2. The protocol was probabilistic and reported only a successful postselected branch. For a monotone that is also nonincreasing on average over outcomes, the selected branch can have larger $R$ while the outcome-weighted average does not.
3. $R$ is not a valid monotone for the state class or free-operation set being used.

A fourth possibility is a calculation or calibration error. The exercise illustrates why a resource statement must include the free operations, consumed auxiliaries, and conditioning rule.

References

• E. Chitambar and G. Gour, “Quantum resource theories,” Reviews of Modern Physics 91, 025001, 2019, doi:10.1103/RevModPhys.91.025001.
• R. P. Feynman, “Simulating physics with computers,” International Journal of Theoretical Physics 21, 467–488, 1982, doi:10.1007/BF02650179.
• V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nature Photonics 5, 222–229, 2011, doi:10.1038/nphoton.2011.35.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, doi:10.1017/CBO9780511976667.
• J. Preskill, Lecture Notes for Physics 219/Computer Science 219: Quantum Computation, California Institute of Technology, course materials.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, doi:10.1017/9781316848142.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017, doi:10.1017/9781316809976.
• W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299, 802–803, 1982, doi:10.1038/299802a0.