Information-Theoretic Foundations

Information-theoretic foundations turn a formal quantum model into an operational question. Given a problem or claim, the goal is to identify the carrier and encoding, state or labeled ensemble, allowed transformation, measurement, classical output, figure of merit, and resource restrictions before choosing a calculation. This guide supplies that routing discipline and two small numerical audits. It does not replace the canonical derivations of state space, measurement theory, entropy inequalities, Schmidt decomposition, or no-go theorems.

Required background. Quantum States supplies the distinction between a state and one of its representations, while the Born Rule supplies the probability audit used throughout.

Helpful background. Tensor Products prepares the subsystem language, and Entropy prepares the classical information conventions.

Information Foundations as an Operational Layer

A useful information-processing description follows a chain rather than starting with a favorite formula:

source or preparation
  -> state or labeled ensemble
  -> channel or instrument
  -> measurement
  -> classical record
  -> declared loss, payoff, rate, or resource claim

A classical source uses symbols $x$ with probabilities $p_x$. A quantum source may attach a state $\rho_x$ to each label, producing the ensemble $\{p_x,\rho_x\}$ and the receiver state

$$\bar\rho=\sum_x p_x\rho_x.$$

The average state is not the whole source specification when correlations with the label $x$ matter. Two different ensembles can have the same $\bar\rho$ and therefore be indistinguishable to a receiver who lacks the label, while supporting different preparation stories or later side information.

If a channel $\mathcal N$ is followed by an instrument $\{\mathcal I_y\}$, then the probability of record $y$ conditional on source label $x$ is

$$\begin{aligned} p(y\mid x) &=\operatorname{Tr}\!\left[\mathcal I_y\!\left(\mathcal N(\rho_x)\right)\right]\\ &=\operatorname{Tr}\!\left[M_y\mathcal N(\rho_x)\right], \qquad M_y=\mathcal I_y^\dagger(I). \end{aligned}$$

The effects $M_y$ determine outcome probabilities, but they do not by themselves determine the conditional output states. That distinction is why an operational question must name the instrument, not merely a POVM, whenever backaction or future use of the state matters. The full theory belongs to Generalized Measurements and Instruments.

Readiness and Entry Points

The chapter can be entered from physics, computer science, or information theory. Use the smallest repair that makes the next operational statement unambiguous.

Capability to test	Substantive repair or next step
Normalize a state, distinguish a ray from a coordinate vector, and state a basis	Quantum States, Rays and Global Phase, and Finite-Dimensional Hilbert Spaces
Compute probabilities for a declared measurement	Born Rule
Keep subsystem order and local versus global operators straight	Tensor Products
Compute $H(X)$, conditional entropy, and classical relative entropy with a declared logarithm base and support	Entropy and Relative Entropy
Distinguish a physical device from an abstract or encoded carrier	Bits, Qubits, Qudits, and Modes
Work with noisy preparations, discarded subsystems, or measurement branches	Density Operators for Quantum Information

These are capability checks, not a demand to reread an entire prerequisite tree. A physics reader may need the source-channel-decision vocabulary; a computer-science reader may need state, measurement, and subsystem discipline; an information theorist may need the distinction between a classical channel and a completely positive map. The Quantum Information Roadmap supplies a longer sequence, while Math Needed for Quantum Information supplies targeted mathematical repairs.

Carriers, Encodings, and State-Space Choices

An information carrier is an abstract state space together with a declared encoding and an allowed set of operations and measurements. The same abstract qubit can be realized by many devices, and one device can support several inequivalent encodings.

Term	Operational question
Classical bit	Which two distinguishable symbols are used, and what stochastic errors act on them?
Qubit	Which two-dimensional Hilbert space or code subspace carries the state, and which basis defines the labels $0$ and $1$?
Qudit	What is the dimension $d$, which levels form the computational basis, and which operations preserve the chosen subspace?
Mode	Which field or oscillator degree of freedom is used, and is the encoding continuous-variable, finite-energy, or restricted to a code subspace?
Physical qubit	Which controlled device degrees of freedom implement preparation, gates, storage, and readout?
Logical qubit	Which code, syndrome process, decoder, and logical operators define protected information?

The canonical carrier and encoding taxonomy is Bits, Qubits, Qudits, and Modes. General facts about $\mathbb C^d$, bases, rays, and spectra remain at their Mathematical Toolkit and Core Formalism homes.

For $n$ qubits, the register dimension is $2^n$, but this does not mean that one run reveals $2^n$ classical numbers. A measurement returns a classical record drawn from a declared outcome set. State-description size, accessible information, channel capacity, sample complexity, and computational cost are different quantities and must not be exchanged without an argument.

Before using a state vector or matrix, record the computational basis and subsystem order. For example, the column $(0,1,0,0)^T$ is not operationally identified until one says whether the basis is $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ or follows another convention.

Qubit Geometry and Calibrated Measurements

Every one-qubit density operator has the exact representation

$$\rho=\frac12\left(I+\mathbf r\cdot\boldsymbol\sigma\right), \qquad \lVert\mathbf r\rVert\le 1.$$

The vector $\mathbf r$ is operational: its components are the Pauli expectation values. For a projective measurement along unit vector $\mathbf n$,

$$p(+\mathbf n)=\frac{1+\mathbf n\cdot\mathbf r}{2}.$$

The Bloch Sphere for Quantum Information owns the calibrated connection among states, measurement axes, tomography, gate rotations, and affine noise maps. The picture is exact for one qubit, not a generic ball representation for qudits, registers, or continuous-variable modes.

Numerical routing audit: a noisy qubit. Consider

$$\rho= \begin{pmatrix} 0.8 & 0.3\\ 0.3 & 0.2 \end{pmatrix}.$$

The trace is one, and comparison with the Bloch form gives

$$\mathbf r=(0.6,0,0.6), \qquad \lVert\mathbf r\rVert=\sqrt{0.72}=0.848528.$$

Its eigenvalues, purity, and entropy are

$$\begin{aligned} \lambda_\pm &=\frac{1\pm\sqrt{0.72}}{2} =0.924264,\ 0.075736,\\ \operatorname{Tr}(\rho^2)&=0.86,\\ S(\rho)&=h_2(0.075736)=0.386973\ \text{bits}. \end{aligned}$$

The two immediate Born checks are $p(+x)=0.8$ and $p(0z)=0.8$. Positivity, purity, and possible ensemble interpretations belong to Density Operators for Quantum Information; axes and one-qubit readout geometry belong to the Bloch-sphere page; entropy interpretation belongs to Quantum Entropy. A physical channel or backaction question exits to Measurement and Open Systems. The number $S(\rho)$ is not, by itself, the accessible information of an unspecified source ensemble.

Mixed States, Reductions, and Purifications

A finite-dimensional density operator satisfies

$$\rho=\rho^\dagger, \qquad \rho\ge 0, \qquad \operatorname{Tr}\rho=1.$$

Those conditions describe the operational state available to the observer. They do not select a unique ensemble decomposition. If

$$\rho=\sum_x p_x|\psi_x\rangle\langle\psi_x|,$$

the label $x$ may represent accessible preparation information, an inaccessible record, or merely one mathematical decomposition among many. Keeping an accessible label requires the classical-quantum state

$$\rho_{XQ}=\sum_x p_x|x\rangle\langle x|_X\otimes\rho_x,$$

not only the marginal $\bar\rho_Q$.

A subsystem can also be mixed because it is entangled with something discarded. For a joint state $\rho_{AB}$, the reduced state $\rho_A=\operatorname{Tr}_B\rho_{AB}$ predicts every local measurement on $A$, but it does not retain the correlations with $B$. Conversely, a purification represents a mixed state as the marginal of a larger pure state; the purifying system and basis are not unique.

Pure versus Mixed States owns the general criteria. Density Operators for Quantum Information owns the protocol workflow involving source ensembles, classical labels, reduced states, purification, channels, instruments, trace distance, and fidelity. Composite Systems owns the full partial-trace and purification constructions.

An instrument branch $\mathcal I_y(\rho)$ can be subnormalized: its trace is the probability of outcome $y$. Only after division by that probability, when nonzero, is the conditional output a normalized state. Confusing these two objects is a common source of incorrect probabilities and nonlinear update rules.

Entropy, Correlation, and Distinguishability

Use base-$2$ logarithms here, so information is reported in bits. The principal quantum entropy is

$$S(\rho)=-\operatorname{Tr}(\rho\log_2\rho).$$

It depends on the spectrum of $\rho$, not on the outcome distribution of an arbitrarily chosen measurement. Several related quantities answer different questions:

Quantity	Question and canonical scope
$H(X)$	Uncertainty of a classical label; see Mathematical Toolkit Entropy.
$S(\rho)$	Spectral mixedness of a quantum state; see Quantum Entropy.
$I(A{:}B)=S(A)+S(B)-S(AB)$	Total correlation between declared subsystems; it is not generally an entanglement measure.
$D(\rho\|\sigma)$	Directed distinguishability with a support condition; its monotonicity under channels is data processing.
$\chi=S(\bar\rho)-\sum_xp_xS(\rho_x)$	The Holevo quantity of a labeled ensemble; it bounds accessible classical information but need not be attained.

Quantum conditional entropy $S(A|B)=S(AB)-S(B)$ can be negative. That is not a failed probability calculation; it is a quantum correlation statement whose operational meaning depends on the task. Likewise, $I(A{:}B)>0$ detects total correlation, including classical correlation in separable mixed states.

Quantum Entropy owns the QI definitions, Rényi variants, conditional and mutual information, quantum relative entropy, data processing, strong subadditivity, Holevo quantities, and finite-dimensional cautions. Mutual Information owns structural bipartite examples, while Mathematical Toolkit Relative Entropy owns classical Kullback–Leibler divergence, support conditions, and statistical interpretation.

Schmidt Spectra and Entanglement Diagnostics

For a finite-dimensional pure bipartite state, the Schmidt decomposition has the form

$$|\psi\rangle_{AB} =\sum_i\sqrt{\lambda_i}\,|i_A\rangle|i_B\rangle, \qquad \lambda_i\ge0, \qquad \sum_i\lambda_i=1.$$

The nonzero $\lambda_i$ are the eigenvalues of both reduced states. Schmidt Decomposition owns the theorem, singular-value proof, reduced spectra, rank, and basis freedom. This chapter uses the spectrum to select an operational quantity.

For a pure bipartite state, the entropy of entanglement is

$$E(|\psi\rangle_{AB})=S(\rho_A)=S(\rho_B).$$

That equality is special to pure bipartite states. Mixed-state entanglement requires a declared measure, and multipartite states require a declared partition and notion of resource.

Numerical routing audit: a partially entangled pair. Let

$$|\psi\rangle=\sqrt{0.9}|00\rangle+\sqrt{0.1}|11\rangle.$$

Then

$$\rho_A=\operatorname{diag}(0.9,0.1), \qquad E(|\psi\rangle)=h_2(0.1)=0.468996\ \text{ebits}.$$

The Schmidt rank is two, but the state is not maximally entangled and does not contain one ebit. Route the theorem and reduced spectrum to Composite Systems, selection among entropy, concurrence, negativity, formation, or distillation quantities to Entanglement Measures, and conversion under local operations to the substantive LOCC Preview.

Resources, Free Operations, and Conversion Tasks

A resource theory begins by declaring more than a desirable state. At minimum it specifies:

1. the object type, such as states, channels, measurements, or processes;
2. free objects;
3. free operations;
4. a target task or conversion;
5. exact or approximate error criteria;
6. a one-shot, finite-copy, or asymptotic regime;
7. catalysts, classical communication, postselection, and other side resources.

Only after that contract is fixed can a monotone $R$ be required to satisfy

$$R(\Lambda(\rho))\le R(\rho) \qquad \text{for every declared free operation }\Lambda.$$

Different choices of free operations produce different orderings and conversion laws. There is no basis-independent scalar called simply “quantumness” that simultaneously measures entanglement, coherence, magic, asymmetry, and thermodynamic value.

Resource setting	Current substantive entry point
Bipartite entanglement under local processing	Entanglement Measures and LOCC Preview
Magic relative to stabilizer operations	Magic State Distillation
Athermality under thermal operations	Thermal Operations Preview
Entanglement as a protocol resource	Entanglement in Quantum Information

Resource Theories owns this comparison rule, while the specialist pages retain their detailed conversion theories. A claim about resource advantage is incomplete until it names the free-operation class, target, accuracy, success probability, copy regime, and all consumed side resources.

Operational Limits and Disturbance

No-go results rule out precisely specified tasks under precisely specified assumptions. Their slogans are not interchangeable.

• Universal exact deterministic cloning of arbitrary unknown pure states is impossible, but known orthogonal states can be copied in their distinguishing basis.
• No-broadcasting extends the issue to mixed-state families: exact broadcasting is possible for commuting families.
• No-signaling constrains remote outcome statistics; it is not the same assumption as Bell locality and is not the proof of no-cloning.
• Bell and Kochen–Specker results concern particular hidden-variable or value-assignment assumptions, not the impossibility of every realist model.

No-Cloning and No-Signaling owns the QI operational proofs and communication consequences. Information and Foundations Results supplies compact assumption-aware theorem cards.

Information–disturbance claims require equal precision. State the input ensemble, instrument, information criterion, disturbance criterion, and whether the claim is average-case, worst-case, or state-dependent. A POVM fixes $p(y|x)$ but not the postmeasurement channel, so the same POVM can be realized by instruments with different disturbance. There is no universal positive scalar disturbance attached to every informative measurement on every possible input.

Measurement Backaction and Quantum Instruments own the measurement dynamics. Why Generalized Measurements Are Needed owns discrimination-driven motivation. BB84 owns the protocol-specific connection between Eve’s intervention and observed statistics, while Why Quantum Error Correction Is Possible owns the distinction between learnable syndrome information and protected logical information.

Choose the Canonical Owner

Use this owner firewall before expanding a derivation. The local QI page owns the operational use named below; the linked upstream page retains the general theorem or formalism.

If the next question asks about …	Use this canonical owner and boundary
A classical source, stochastic channel, code and decoder, error or secrecy criterion, access model, or fair quantum comparator	Classical Information Review owns the operational baseline ledger; Mathematical Toolkit retains the Shannon and Kullback–Leibler derivations, while specialist QI pages retain capacities, security, complexity, and advantage evidence.
Carrier dimension, physical versus logical encoding, leakage, or modes	Bits, Qubits, Qudits, and Modes owns the QI taxonomy; Mathematical Toolkit owns finite-dimensional Hilbert-space theory.
One-qubit states, measurement axes, tomography, gates, or affine noise geometry	Bloch Sphere for Quantum Information owns the operational geometry; spin and linear-algebra pages own the geometric derivations.
Ensembles, classical labels, reduced states, purification, instruments, fidelity, or trace distance in a protocol	Density Operators for Quantum Information owns the workflow; Core and Composite Systems own the general formalism.
Von Neumann or Rényi entropy, conditional information, mutual information, relative entropy, data processing, or Holevo quantities	Quantum Entropy owns the QI use; Math and Composite Systems own classical and structural foundations.
Schmidt theorem, spectrum, rank, or basis freedom	Schmidt Decomposition owns the result and proof.
Which entanglement quantity answers a specified task	Entanglement Measures owns task-aware selection; measure-specific derivations remain in Composite Systems.
Free objects, free operations, monotones, or conversion rates	Resource Theories owns the general conversion ledger; then use the precise specialist owner, such as LOCC Preview, Magic State Distillation, or Thermal Operations Preview.
POVMs, instruments, channels, or backaction	Generalized Measurements and Instruments and Measurement/Open Systems own the formalism.
No-cloning, no-broadcasting, no-signaling, Bell, or Kochen–Specker assumptions	Use No-Cloning and No-Signaling or the Reference theorem gateway rather than creating a second proof.
A compact formula with conventions	Use Quantum Information Formulas and then follow its canonical explanation link.

The firewall prevents a local convenience calculation from silently becoming a second canonical home. Cross-links should translate the object and task between pages, not repeat a proof with slightly different notation.

A Working Sequence Through the Chapter

Choose a route by the task you need to perform.

Represent and control one carrier. Start with Bits, Qubits, Qudits, and Modes, continue to the Bloch Sphere for calibrated one-qubit work, and exit to Gates, Circuits, and Computation Models when the state and measurement conventions are fixed.

Handle uncertainty, discarded systems, or noise. Use Density Operators for Quantum Information first. Continue to Quantum Entropy when the question is spectral uncertainty, correlation, distinguishability, or a data-processing bound.

Diagnose entanglement. Learn the theorem at Schmidt Decomposition, then use Entanglement Measures to match the state class and operational task to a quantity. Use Entanglement in Quantum Information for protocol context.

Audit a measurement or security claim. Move from Generalized Measurements and Instruments to No-Cloning and No-Signaling or BB84, depending on whether the claim is a formal prohibition or a protocol security statement.

At the end of the sequence, a reader should be able to write the source–state–process–measurement–record chain, name all conventions, calculate a small validity or information check, and identify one canonical owner for the next nontrivial step.

Common Routing and Interpretation Mistakes

Treating Hilbert-space dimension as accessible information. Dimension counts independent state-space directions. Accessible information depends on an ensemble, a measurement, and an operational optimization; one measurement shot does not reveal a generic amplitude list.

Assigning a density operator a unique ignorance ensemble. A density operator admits many decompositions. Preserve an accessible preparation label explicitly as a classical-quantum register when its correlations matter.

Using a POVM as a state-update rule. POVM effects determine probabilities. An instrument is required to determine conditional output states and disturbance.

Extending the Bloch ball to arbitrary systems. The three-component Bloch ball is exact for one qubit. Qudits, multiqubit registers, and modes require different state descriptions.

Calling mutual information an entanglement measure. Mutual information counts total correlation and can be nonzero for separable states. Entanglement claims require a declared measure, partition, and task.

Equating Schmidt rank with amount of entanglement. Rank distinguishes product from entangled pure states and counts nonzero Schmidt terms. It does not determine their weights or imply maximal entanglement.

Defining a resource theory only by free states. Free operations, object type, target, error, regime, catalysts, and side resources are also part of the theory. Changing them can change both monotones and possible conversions.

Rejecting negative quantum conditional entropy as an error. Classical conditional entropy is nonnegative, but quantum conditional entropy can be negative. Interpret it through the declared quantum task rather than a classical probability slogan.

Deriving no-cloning from no-signaling. No-cloning follows directly from linearity or inner-product preservation. No-signaling is a distinct constraint on remote observable statistics.

Claiming a universal information–disturbance number. Disturbance depends on the ensemble, instrument, metric, and averaging convention. Even a fixed POVM can have multiple instrument realizations.

Applying finite-dimensional formulas without a domain check. Infinite-dimensional systems require trace-class, support, domain, and often energy-constraint assumptions. A finite truncation is a model choice, not an automatic theorem transfer.

Exercises

1. Carrier and encoding triage

A report says, “This transmon is a qubit that stores a logical qubit.” Separate the physical device, selected code subspace, abstract qubit state, and error-corrected logical encoding. Identify the canonical owner for each layer.

Solution

A transmon is a physical anharmonic circuit with more than two energy levels. Choosing two controlled levels, commonly denoted $|0\rangle$ and $|1\rangle$, defines a two-dimensional computational or code subspace. An abstract qubit state is a normalized ray in that subspace,

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle, \qquad |\alpha|^2+|\beta|^2=1,$$

independent of whether the carrier is a transmon, ion, spin, or photon. Population outside the selected subspace is leakage.

A logical qubit requires an error-correcting encoding, logical operators, syndrome extraction, and a decoder. It is generally distributed across several physical degrees of freedom; calling one unencoded transmon a logical qubit collapses two architecture layers.

Bits, Qubits, Qudits, and Modes owns the carrier and encoding taxonomy. Hardware pages own the circuit, control, coherence, and readout physics. Quantum Error Correction owns the code, syndrome, decoder, and logical-error contract.

2. Qubit-state validity and owner choice

For

$$\rho= \begin{pmatrix} 0.8 & 0.3\\ 0.3 & 0.2 \end{pmatrix},$$

check physical validity, find the Bloch vector, purity, and the probabilities $p(+x)$ and $p(0z)$. Explain which retained page owns each part of the analysis.

Solution

The matrix is Hermitian and has trace one. In the computational basis,

$$\rho=\frac12 \begin{pmatrix} 1+r_z & r_x-ir_y\\ r_x+ir_y & 1-r_z \end{pmatrix},$$

so $\mathbf r=(0.6,0,0.6)$ and $\lVert\mathbf r\rVert=\sqrt{0.72}<1$. The eigenvalues are

$$\lambda_\pm=\frac{1\pm\sqrt{0.72}}2 =0.924264,\ 0.075736,$$

which are nonnegative, so $\rho$ is positive semidefinite and therefore a valid density operator.

The purity is

$$\operatorname{Tr}(\rho^2) =\frac{1+\lVert\mathbf r\rVert^2}{2} =0.86.$$

For the $x$-axis and computational $z$-axis measurements,

$$p(+x)=\frac{1+r_x}{2}=0.8, \qquad p(0z)=\frac{1+r_z}{2}=0.8.$$

Density Operators for Quantum Information owns positivity, normalization, purity, and ensemble interpretation. Bloch Sphere for Quantum Information owns the vector and measurement-axis geometry. Quantum Entropy owns the spectral entropy if that is the next question.

3. Bell reduction and improper mixture

For $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$, compute the reduced state of subsystem $A$. Explain why the result does not select a unique ignorance ensemble.

Solution

The joint density operator is

$$\rho_{AB}=\frac12\left( |00\rangle\langle00| +|00\rangle\langle11| +|11\rangle\langle00| +|11\rangle\langle11| \right).$$

Taking the partial trace over $B$ removes the cross terms because $\langle1|0\rangle=\langle0|1\rangle=0$:

$$\rho_A=\operatorname{Tr}_B\rho_{AB} =\frac12\left(|0\rangle\langle0|+|1\rangle\langle1|\right) =\frac I2.$$

The global state remains pure because $\rho_{AB}^2=\rho_{AB}$, while the local state has purity $\operatorname{Tr}(\rho_A^2)=1/2$. The same $I/2$ can be decomposed as an equal mixture of $z$-basis states, $x$-basis states, or infinitely many other ensembles. Without an accessible classical label, none is the unique physical ignorance story.

Composite Systems owns partial trace and entanglement structure. Density Operators for Quantum Information owns the protocol-level distinction among labeled preparations, discarded subsystems, and operationally equivalent local states.

4. Biased-bit entropy bridge

Let a classical bit have distribution $p=(0.9,0.1)$ and encode its values in orthogonal states, giving $\rho=\operatorname{diag}(0.9,0.1)$. Compute $H(X)$ and $S(\rho)$, and state the scope of their equality.

Solution

With base-$2$ logarithms,

$$\begin{aligned} H(X) &=-0.9\log_2(0.9)-0.1\log_2(0.1)\\ &=h_2(0.1)=0.468996\ \text{bits}. \end{aligned}$$

The eigenvalues of the diagonal density operator are also $0.9$ and $0.1$, so

$$S(\rho)=h_2(0.1)=0.468996\ \text{bits}.$$

The equality holds because the classical probabilities are the eigenvalues of an orthogonal encoding. For a nonorthogonal ensemble, the label entropy $H(X)$ and the entropy of the average state $S(\bar\rho)$ need not agree; moreover, accessible information depends on an optimized measurement and is bounded by the Holevo quantity. Mathematical Toolkit Entropy owns the classical calculation, while Quantum Entropy owns the spectral and operational quantum distinctions.

5. Product-versus-Bell mutual information

Compute $I(A{:}B)$ for the pure product state $|00\rangle$ and for $|\Phi^+\rangle$. Compare each result with the entropy of entanglement.

Solution

For $|00\rangle$, the global and reduced states are pure, so

$$S(A)=S(B)=S(AB)=0, \qquad I(A{:}B)=0.$$

Its entropy of entanglement is also zero.

For $|\Phi^+\rangle$, the global state is pure but both reduced states are maximally mixed:

$$S(AB)=0, \qquad S(A)=S(B)=1\ \text{bit}.$$

Therefore

$$I(A{:}B)=S(A)+S(B)-S(AB)=2\ \text{bits}.$$

The entropy of entanglement is $S(A)=1$ ebit, not two ebits. Mutual information measures total correlation and counts both subsystem entropy contributions in this identity; it is not numerically the same object as the pure-state entanglement resource.

Quantum Entropy owns the identity, Mutual Information owns structural interpretation, and Entanglement Measures owns measure selection.

6. Schmidt spectrum to measure selection

For $|\psi\rangle=\sqrt{0.9}|00\rangle+\sqrt{0.1}|11\rangle$, find the Schmidt rank and entropy of entanglement. Explain why rank, entropy, concurrence, negativity, and conversion rate are different answers.

Solution

The state is already in Schmidt form, with nonzero Schmidt probabilities $0.9$ and $0.1$. Hence its Schmidt rank is two and

$$E(|\psi\rangle)=h_2(0.1)=0.468996\ \text{ebits}.$$

Rank records only how many Schmidt coefficients are nonzero. Entropy also uses their weights. For this two-qubit pure state, concurrence is

$$C=2\sqrt{0.9\times0.1}=0.6,$$

while negativity is $\sqrt{0.9\times0.1}=0.3$. These numbers use different normalizations and answer different mathematical or operational questions. In the asymptotic pure bipartite LOCC setting, the entropy gives the reversible conversion rate in ebits per copy; one-shot, approximate, mixed-state, or restricted-operation tasks require other quantities and error conventions.

Schmidt Decomposition owns the theorem and spectrum. Entanglement Measures owns the decision about which quantity matches the state class and task.

7. Resource-theory tuple

Audit the statement “a $T$ state contains quantumness.” List the information required to turn it into a meaningful stabilizer-resource claim.

Solution

First declare the object: for example, a single-qubit state intended for non-Clifford gate injection. Next choose the free states, commonly the stabilizer-state polytope, and the free operations, such as a specified class of stabilizer operations with Clifford gates, Pauli measurements, classical control, and explicitly declared postselection rules.

Then declare the target task: preparing a higher-fidelity magic state, injecting a $T$ gate, or supplying a fault-tolerant factory. Choose a monotone appropriate to that free-operation class, such as robustness-type magic or mana where applicable. State whether conversion is exact or approximate, deterministic or probabilistic, one-shot or asymptotic, and whether catalysts are returned unchanged.

Finally count consumed stabilizer ancillas, Clifford operations, measurements, classical randomness, discarded failure branches, and any correlated-input assumptions. Only with that tuple can “magic” be a resource claim. A task- and basis-independent scalar “quantumness” is not defined by the sentence.

Magic State Distillation owns the concrete protocol and factory implications. The general lesson is that free objects alone do not define a resource theory.

8. BB84 cloning, disturbance, and no-signaling audit

Evaluate the claim: “BB84 is secure because every informative measurement causes a fixed disturbance, and cloning is impossible only because it would signal.” Identify the correct owners of each corrected statement.

Solution

BB84 uses four nonorthogonal signal states in two conjugate bases. Eve must be modeled by an instrument acting on that ensemble, together with any retained quantum system and classical record. The observable disturbance is inferred from declared parameter-estimation statistics. It is not a universal constant attached to every informative measurement.

For the intercept–resend example in which Eve chooses a BB84 basis uniformly, she chooses the wrong basis half the time. Conditional on that event and on Alice and Bob retaining a same-basis round, Bob receives the wrong bit with probability one half. The sifted-key error rate is therefore $\tfrac12\times\tfrac12=\tfrac14$. Other attacks produce different information–disturbance relations, and a complete security proof also requires authentication, finite statistics, error correction, privacy amplification, and device assumptions.

No-cloning does not require no-signaling. If a universal unitary cloner satisfied

$$U|\psi\rangle|0\rangle=|\psi\rangle|\psi\rangle$$

for every input, inner-product preservation for $|\psi\rangle$ and $|\phi\rangle$ would require

$$\langle\phi|\psi\rangle =\langle\phi|\psi\rangle^2,$$

which fails for distinct nonorthogonal states. No-signaling is instead a constraint that one party’s local choice cannot change another party’s unconditional local statistics.

BB84 and its QKD parent own the protocol and security conditions. Measurement Backaction and Quantum Instruments own disturbance modeling. No-Cloning and No-Signaling owns the operational proofs, and the Reference theorem cards own compact assumption checks.

References

• E. Chitambar and G. Gour, “Quantum Resource Theories,” Reviews of Modern Physics 91, 025001 (2019), doi:10.1103/RevModPhys.91.025001.
• T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed., Wiley (2006).
• R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum Entanglement,” Reviews of Modern Physics 81, 865–942 (2009), doi:10.1103/RevModPhys.81.865.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press (2010), doi:10.1017/CBO9780511976667.
• C. E. Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal 27, 379–423 and 623–656 (1948), doi:10.1002/j.1538-7305.1948.tb01338.x.
• 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.