Quantum Discord and Non-Classical Correlations

Entanglement is usually sold as the signature of quantumness — the thing that makes Bell inequalities fail and quantum computers powerful. But entanglement is not the whole story of quantum correlation. A shared state $\rho_{AB}$ can be perfectly separable — a mere probabilistic mixture of product states, with zero entanglement by any measure — and still hold correlations that no classical joint probability distribution can reproduce. Quantum discord is the quantity built to catch exactly that residue. The idea starts from a small classical fact with large consequences. For two random variables the mutual information can be written two algebraically identical ways: once as marginal entropies minus the joint entropy, and once through the conditional entropy you get after learning one variable. Classically these two formulas are the same number. Quantum-mechanically they split apart, because to "learn about $A$ " you must measure it, and measurement disturbs the joint state and depends on the basis you chose. The measurement-free formula $I(\rho_{AB})$ counts the total correlation; the measured formula $J_A(\rho_{AB})$ , optimized over all local measurements, counts only what a classical readout of $A$ can recover. Their gap is the discord $D_A = I - J_A$ . That gap has teeth. For pure states discord equals the entanglement entropy, so nothing new. But for mixed states the two part ways: most states carry nonzero discord, and the zero-discord states form a vanishingly thin set. The payoff of this article is a single concrete state — the Werner state below its separability threshold — where entanglement is provably zero and discord is provably positive. That state is the proof that "unentangled" and "classically correlated" are not the same thing.

Key takeaways

Discord is the gap between two classically-equivalent formulas for mutual information: the total correlation $I(\rho_{AB})$ and the classically-accessible correlation $J_A(\rho_{AB})$ extracted by measuring $A$ .
The formulas split because a quantum measurement on $A$ disturbs the state and depends on the basis chosen — so $J_A \le I$ , and the difference $D_A = I - J_A \ge 0$ .
Unlike entanglement, discord can be nonzero for separable (unentangled) mixed states — non-classical correlation survives with no entanglement present.
Zero discord on $A$ characterizes exactly the classical-quantum states $\sum_i p_i\,|i\rangle\langle i|_A\otimes\rho_B^{(i)}$ : some complete local measurement leaves the state untouched.
Discord is generally asymmetric ( $D_A\neq D_B$ ) and hard to compute (a minimization over all local measurements), but closed-form for Bell-diagonal two-qubit states.
Key idea: two classically equivalent formulas for mutual information split under quantum measurement; their minimized gap is the discord — non-classical correlation that survives in separable mixed states where entanglement is exactly zero.

The core insight

Two classically equivalent formulas for the mutual information split in quantum mechanics because measurement disturbs the state; their gap, minimized over local measurements, is the quantum discord — non-classical correlation that persists in separable mixed states where entanglement is zero.

Where this lives today

Discord matters most where entanglement is scarce but quantumness still helps. The flagship is the DQC1 "power of one qubit" model: a single clean control qubit steering a maximally mixed register estimates a normalized unitary trace exponentially faster than known classical methods, yet the state built along the way has negligible or zero entanglement. Datta, Shaji and Caves argued the operative resource is discord, and that reframing launched a decade of work on "quantumness without entanglement." Discord also proves robust: under many noise channels entanglement suffers "sudden death" — vanishing in finite time — while discord decays only asymptotically or even freezes at a constant value over a finite interval, making it a more durable carrier of quantum correlation for memories and metrology. There is a clean link to the emergence of classicality. A state loses its discord with respect to $A$ exactly when $A$ has a stable readable basis — precisely the pointer states that "The Pointer Basis and Environment-Induced Superselection" identifies as the ones the environment protects. Decoherence, viewed this way, is the continual erasure of discord along the environment-selected basis: what survives is a classical-quantum correlation, which is why the world looks classical even though the correlations underneath were quantum all along.

The physical picture

Picture reading subsystem $A$ to learn about $B$ . Classically you can inspect $A$ as gently as you like — the underlying joint distribution $p(a,b)$ is just a table, and looking up a row does not rewrite it. So "how much $A$ and $B$ are correlated" and "how much you learn about $B$ by reading $A$ " are the same number. Quantum-mechanically the table is a density operator, and reading $A$ means measuring it: you pick a basis, superpositions collapse, coherences vanish. The most you can classically extract is generally less than the total correlation actually present. Discord is precisely that shortfall — the correlation that hides in the coherences a local measurement destroys.

Figure

Two formulas that agree classically and split quantumly

There is an even sharper way to see it. Take a separable state written as a classical mixture of products, $\sum_i p_i\,|\psi_i\rangle\langle\psi_i|_A\otimes|\phi_i\rangle\langle\phi_i|_B$ , but let the local states $|\psi_i\rangle$ on $A$ be non-orthogonal. There is no entanglement — the state factorizes term by term. Yet because the $|\psi_i\rangle$ overlap, no single measurement can perfectly tell them apart, and that indistinguishability leaves a fingerprint of quantum correlation. Discord measures this "memory of non-orthogonality" that entanglement is blind to. The slogan: entanglement is about non-separability of the state; discord is about the disturbance caused by local measurement. When some local basis exists in which $A$ reads out cleanly like a classical pointer, discord is zero — the state is a classical-quantum state — and otherwise it is not.

Figure

The correlation hierarchy: product, classical-quantum, separable, entangled

The core equations

Everything is built from the von Neumann entropy, the quantum stand-in for Shannon entropy that "Entanglement Entropy and Measures of Entanglement" introduces as $S(\rho)=-\mathrm{Tr}\,\rho\ln\rho$ (zero for a product state, $\ln 2$ for a Bell pair). Here we work in bits, using the base-2 logarithm, so a maximally mixed qubit carries exactly one bit. The entropy is a function of the eigenvalues alone.

S(\rho) = -\,\mathrm{Tr}\,(\rho \log_2 \rho) = -\sum_k \lambda_k \log_2 \lambda_k \qquad (1)

where: $\rho$ — density operator; $\lambda_k$ — its eigenvalues; $\log_2$ — base-2 logarithm (entropy in bits). Read it as the average surprise of the outcome distribution $\{\lambda_k\}$ you would see in $\rho$ 's own eigenbasis; it is basis-independent because it depends only on the spectrum. The quantum mutual information copies the classical formula $H(A)+H(B)-H(A,B)$ verbatim, replacing Shannon by von Neumann entropies and using the reduced states supplied by "The Reduced Density Matrix and Partial Trace" ( $\rho_A=\mathrm{Tr}_B\,\rho_{AB}$ , the operator obtained by tracing out $B$ ).

I(\rho_{AB}) = S(\rho_A) + S(\rho_B) - S(\rho_{AB}) \qquad (2)

where: $\rho_A,\rho_B$ — reduced states; $S(\rho_{AB})$ — joint entropy. This is the total-correlation side: it needs no measurement, is symmetric under $A\leftrightarrow B$ , and vanishes exactly when $\rho_{AB}=\rho_A\otimes\rho_B$ . The other side asks a physical question — measure $A$ , then how uncertain is $B$ ? — and so requires a complete local measurement $\{\Pi_j^A\}$ . Each outcome $j$ leaves $B$ in a conditional state, and averaging their entropies gives the measured conditional entropy.

S(\rho_{AB} \,|\, \{\Pi_j^A\}) = \sum_j p_j\, S(\rho_{B|j}),\qquad \rho_{B|j} = \frac{\mathrm{Tr}_A(\Pi_j^A\,\rho_{AB}\,\Pi_j^A)}{p_j} \qquad (3)

where: $\{\Pi_j^A\}$ — complete set of orthogonal projectors, $\sum_j\Pi_j^A=\mathbb{1}$ ; $p_j=\mathrm{Tr}(\Pi_j^A\,\rho_{AB})$ — outcome probability; $\rho_{B|j}$ — post-measurement conditional state of $B$ . This is the one quantity with no classical analogue: its value depends on which basis you measure, which is the whole source of the split. Maximizing the information $A$ 's measurement reveals about $B$ — equivalently minimizing the residual conditional entropy — defines the classical correlation of Henderson and Vedral.

J_A(\rho_{AB}) = S(\rho_B) - \min_{\{\Pi_j^A\}} \sum_j p_j\, S(\rho_{B|j}) \qquad (4)

where: $J_A$ — the largest correlation about $B$ recoverable by measuring $A$ ; the minimization runs over all local projective measurements on $A$ . It is the classically-accessible part of the correlation, and by construction $J_A \le I$ . Their difference is the quantum discord. Subtracting (4) from (2), the $S(\rho_B)$ terms cancel and only a minimization-dependent object survives.

D_A(\rho_{AB}) = I(\rho_{AB}) - J_A(\rho_{AB}) = S(\rho_A) - S(\rho_{AB}) + \min_{\{\Pi_j^A\}}\sum_j p_j\, S(\rho_{B|j}) \qquad (5)

where: $D_A$ — discord with respect to a measurement on $A$ . It is non-negative (a consequence of strong subadditivity), it is generally asymmetric ( $D_A\neq D_B$ because which party you read matters), and it is zero exactly for the classical-quantum states.

D_A(\rho_{AB}) \ge 0,\qquad D_A = 0 \iff \rho_{AB} = \sum_i p_i\,|i\rangle\langle i|_A \otimes \rho_B^{(i)} \qquad (6)

where: $\{|i\rangle\}$ — an orthonormal basis of $A$ ; $\rho_B^{(i)}$ — arbitrary states of $B$ . Read the right side as " $A$ is a classical register": each pointer value $|i\rangle$ is perfectly readable and correlated with some quantum state of $B$ .

Derivation AWhy the two mutual-information formulas split under measurement

This derivation pins down the origin of discord: the two classical expressions for mutual information are provably the same number for any probability table, but their quantum translations are two different quantities, and their gap is the discord. The strategy is to first prove the classical identity from Bayes' rule, then quantize each side and watch exactly where the equality breaks — it breaks at the conditional entropy, the only piece that demands a measurement. Watch for the asymmetry it introduces: $I$ stays symmetric, $J_A$ does not.

Step 1. Start from a classical joint distribution $p(a,b)$ . The conditional entropy obeys $H(B|A)=\sum_a p(a)\,H(B|A{=}a) = H(A,B)-H(A)$ , a direct consequence of $p(b|a)=p(a,b)/p(a)$ . Substituting gives $H(B)-H(B|A) = H(A)+H(B)-H(A,B)$ , so the "marginals-minus-joint" form and the "conditional" form are literally equal. Why this matters — classically the choice of formula is cosmetic, so nothing forces us to prefer one. Quantum mechanics will remove that freedom.

Step 2. Quantize the first form directly: replace each Shannon entropy by a von Neumann entropy to get $I=S(\rho_A)+S(\rho_B)-S(\rho_{AB})$ . Nothing was measured; the object is symmetric and basis-independent.

Step 3. Quantize the second form. " $H(B|A)$ " has no measurement-free meaning: to condition on $A$ you must choose a measurement $\{\Pi_j^A\}$ , obtain outcome $j$ with probability $p_j$ , and leave $B$ in $\rho_{B|j}$ . The best you can do is optimize the basis, giving $J_A = S(\rho_B)-\min_{\{\Pi_j^A\}}\sum_j p_j S(\rho_{B|j})$ . Why the gap is one-signed — measurement can only add uncertainty about $B$ relative to the ideal joint description, so $J_A\le I$ always, and the deficit is non-negative.

Step 4. Their difference $D_A = I - J_A$ is the discord. For any classical distribution both quantizations return the same number, so $D_A=0$ : discord is a purely quantum object, born from the fact that reading $A$ is an intervention, not a lookup. Verified: reduces to $0$ for every classical-quantum state and equals $S(\rho_A)$ for pure states; symbolic check of the classical two-form identity and the non-negativity gap in verify_aq-qm-quantum-discord.py.

Go deeper · Derivation B · 2 minThe Werner state: entanglement zero, discord positive

Derivation BThe Werner state: entanglement zero, discord positive

This is the article's owned result — a single family of two-qubit states in which entanglement provably vanishes over a whole interval while discord stays strictly positive, so the two correlation measures genuinely disagree. The route is: write the Werner state in Bell-diagonal form, read off its (maximally mixed) marginals and its four eigenvalues, use the closed-form Bell-diagonal classical correlation, compare against the concurrence, and locate the window $0<p\le 1/3$ where concurrence is exactly zero but discord is not. Every number below is recomputed numerically.

Step 1. The Werner state is a Bell state mixed with white noise, $\rho_W = p\,|\Phi^+\rangle\langle\Phi^+| + \tfrac{1-p}{4}\,\mathbb{1}\otimes\mathbb{1}$ with $0\le p\le 1$ . Expanding $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$ in Pauli operators gives the Bell-diagonal form $\rho_W=\tfrac14(\mathbb 1 + p\,\sigma_x{\otimes}\sigma_x - p\,\sigma_y{\otimes}\sigma_y + p\,\sigma_z{\otimes}\sigma_z)$ , i.e. correlation coefficients $(c_1,c_2,c_3)=(p,-p,p)$ , so $c\equiv\max_k|c_k|=p$ .

Step 2. The single-qubit marginals are $\rho_A=\rho_B=\tfrac12\mathbb 1$ , hence $S(\rho_A)=S(\rho_B)=1$ bit. All the information lives in the correlations; none is locally accessible without measurement. Why this is the interesting regime — with maximally mixed marginals, $I$ and $J$ differ only through the joint structure, so any gap is purely correlational.

Step 3. Diagonalizing $\rho_W$ gives eigenvalues $\{\tfrac{1+3p}{4},\ \tfrac{1-p}{4},\ \tfrac{1-p}{4},\ \tfrac{1-p}{4}\}$ — one enhanced, three degenerate — so $S(\rho_{AB})=-\sum_k\lambda_k\log_2\lambda_k$ and $I = 2 - S(\rho_{AB})$ .

Step 4. For a Bell-diagonal state the optimal local measurement is along the Pauli axis with the largest $|c_k|$ , and the classical correlation collapses to the closed form $J = \tfrac{1+c}{2}\log_2(1+c) + \tfrac{1-c}{2}\log_2(1-c)$ with $c=p$ . Discord is then $D=I-J$ .

Step 5. Entanglement is measured by the concurrence, which for the Werner state has the closed form $C(\rho_W)=\max\!\big(0,\tfrac{3p-1}{2}\big)$ . It is exactly zero for $p\le 1/3$ — the Peres–Horodecki separability threshold — and positive above it. Why the thresholds differ — concurrence hits zero at $p=1/3$ , but the discord's classical part $J$ stays strictly below the total $I$ for every $p>0$ , so the two measures cross at different places.

Step 6. Evaluate the window. At $p=1/3$ : $I=0.2075$ bits, $J=0.0817$ bits, so $D=0.1258$ bits while $C=0$ . Sweeping $p\in(0,\tfrac13]$ keeps $C=0$ and $D>0$ throughout; discord vanishes only at $p=0$ . Entanglement-free non-classical correlation is real. Verified: eigenvalues, $S(\rho_A)=S(\rho_B)=1$ , the closed-form $J$ , the concurrence against Wootters' formula, and the entire $C{=}0<D$ window recomputed in verify_aq-qm-quantum-discord.py (anchors $I{=}0.2075$ , $J{=}0.0817$ , $D{=}0.1258$ bits at $p{=}1/3$ ).

Go deeper · Derivation C · 1 minZero discord means A is a classical register

Derivation CZero discord means A is a classical register

Here we characterize the exceptional states that carry no discord, proving $D_A=0$ if and only if $\rho_{AB}$ is block-diagonal in some orthonormal local basis of $A$ — the classical-quantum states. The strategy is to recognize the optimal measurement as a local dephasing channel and read discord as the entropy it produces; the states it leaves untouched are exactly the ones that were already "classical from $A$ 's side."

Step 1. For the optimal rank-one measurement $\{\Pi_i^A=|i\rangle\langle i|\}$ , define the local dephasing channel $\Phi_A(\rho_{AB}) = \sum_i (\Pi_i^A\otimes\mathbb 1)\,\rho_{AB}\,(\Pi_i^A\otimes\mathbb 1)$ , which deletes the off-diagonal $A$ -coherences in the $\{|i\rangle\}$ basis.

Step 2. Dephasing never lowers entropy: $S(\rho_{AB})\le S(\Phi_A(\rho_{AB}))$ , with equality exactly when $\rho_{AB}=\Phi_A(\rho_{AB})$ — i.e. when the state was already block-diagonal in that basis.

Step 3. With the optimal measurement, the minimized conditional entropy assembles with the marginals so that $D_A = S(\Phi_A(\rho_{AB})) - S(\rho_{AB})$ — discord is the entropy the local dephasing injects. Why the "if and only if" is tight — the entropy gap is zero precisely when dephasing changes nothing, which is precisely the block-diagonal form.

Step 4. Therefore $D_A=0 \iff \rho_{AB}=\sum_i p_i\,|i\rangle\langle i|_A\otimes\rho_B^{(i)}$ : some complete local readout of $A$ leaves the global state invariant, and $A$ behaves as a classical register correlated with quantum states $\rho_B^{(i)}$ of $B$ . Such states form a measure-zero set, so almost every state is discordant. Verified: a classical-quantum state built from an orthonormal $A$ -basis is invariant under $A$ -dephasing (gap $=0$ ), while a coherent-in- $A$ state has a strictly positive $1$ -bit gap — checked in verify_aq-qm-quantum-discord.py.

Limiting cases

Case	Result	Meaning
Pure bipartite state	$D_A = E = S(\rho_A)$	Discord equals the entanglement entropy and is symmetric ( $D_A=D_B$ ); the two notions of correlation coincide.
Product state	$\rho_{AB}=\rho_A\otimes\rho_B \Rightarrow D_A=D_B=0$	No correlation of any kind; both $I$ and $J$ vanish.
Classical-quantum state	$\rho_{AB}=\sum_i p_i\,\vert i\rangle\langle i\vert_A\otimes\rho_B^{(i)}$	$D_A=0$ : an orthonormal $A$ -measurement leaves the state unchanged. May still have $D_B>0$ (asymmetry).
Separable but discordant	$D_A>0$ with $\rho_{AB}$ separable	Entanglement zero, discord positive — the generic separable mixed state (Werner state, $0<p\le\tfrac13$ ).
Werner at $p=1/3$	$C=0$ , $D=0.1258$ bits	The threshold state: exactly unentangled yet visibly discordant.

Worked examples

Example 1 — The Werner state at its separability threshold ( $p=1/3$ ). Show explicitly that this state has zero entanglement but positive discord, and quote the numbers.

Step 1. Write the state as Bell-diagonal: $\rho_W=\tfrac14(\mathbb 1 + \tfrac13\sigma_x{\otimes}\sigma_x - \tfrac13\sigma_y{\otimes}\sigma_y + \tfrac13\sigma_z{\otimes}\sigma_z)$ , so $c=\max_k|c_k|=1/3$ .

Step 2. Eigenvalues are $\{\tfrac{1+3p}{4},(\tfrac{1-p}{4})^{\times 3}\}=\{\tfrac12,\tfrac16,\tfrac16,\tfrac16\}$ . The joint entropy is $S(\rho_{AB})=-\tfrac12\log_2\tfrac12 - 3\cdot\tfrac16\log_2\tfrac16 = \tfrac12 + \tfrac12\log_2 6 = 0.5 + 1.2925 = 1.7925$ bits.

Step 3. Marginals are maximally mixed, $S(\rho_A)=S(\rho_B)=1$ bit, so the total correlation is $I = 1+1-1.7925 = \mathbf{0.2075}$ bits.

Step 4. The classical part uses $c=1/3$ : $J=\tfrac{1+1/3}{2}\log_2(1+\tfrac13)+\tfrac{1-1/3}{2}\log_2(1-\tfrac13)=\tfrac23\log_2\tfrac43+\tfrac13\log_2\tfrac23=\mathbf{0.0817}$ bits.

Step 5. Discord is the gap, $D = I - J = 0.2075 - 0.0817 = \mathbf{0.1258}$ bits. The concurrence is $C=\max(0,\tfrac{3(1/3)-1}{2})=\mathbf{0}$ . Zero entanglement, $0.126$ bits of discord — the state is separable yet non-classical.

Example 2 — A classical-quantum state has exactly zero discord. Take $\rho_{AB}=\tfrac12|0\rangle\langle 0|_A\otimes|+\rangle\langle+|_B + \tfrac12|1\rangle\langle 1|_A\otimes|-\rangle\langle-|_B$ , where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2$ .

Step 1. The $A$ -basis $\{|0\rangle,|1\rangle\}$ is orthonormal and the state is already block-diagonal in it, so it fits the classical-quantum form $\sum_i p_i|i\rangle\langle i|_A\otimes\rho_B^{(i)}$ with $p_0=p_1=\tfrac12$ .

Step 2. Apply the local dephasing $\Phi_A$ in the $\{|0\rangle,|1\rangle\}$ basis: it removes off-diagonal $A$ -blocks, but there are none, so $\Phi_A(\rho_{AB})=\rho_{AB}$ .

Step 3. Discord is the entropy gap $D_A=S(\Phi_A(\rho_{AB}))-S(\rho_{AB})=0$ . The state carries $D_A=0$ : subsystem $A$ is a perfect classical pointer. (Measuring $B$ instead can give $D_B>0$ , since $|+\rangle,|-\rangle$ are the ones that overlap under a $B$ -measurement — the promised asymmetry.)

Origins · 1 minharold ollivier and how this was discovered

harold ollivier

b. 1972 · French physicist and engineer; co-defined quantum discord with Wojciech Zurek in 2001, giving the first operational separation between entanglement and non-classical correlation

Ollivier is the "O" of the Ollivier–Zurek definition of discord (Phys. Rev. Lett. 88, 017901, 2001), written while at Los Alamos; the same quantity was defined independently and simultaneously by Leah Henderson and Vlatko Vedral (J. Phys. A 34, 6899, 2001) from the classical-correlation side. He later worked on quantum error correction and, in a very different career turn, on modern machine-learning systems.

History

1932

— von Neumann introduces the density operator and its entropy

S(\rho)=-\mathrm{Tr}\,\rho\log\rho

1935

— EPR and Schrödinger frame entanglement as the characteristic trait of quantum mechanics

2001

— Ollivier and Zurek define quantum discord; Henderson and Vedral independently define the classical correlation

J

2008

— Datta, Shaji and Caves argue discord, not entanglement, powers the DQC1 mixed-state speed-up

2010

— Ferraro and coworkers show the zero-discord states form a set of measure zero

2012

— Modi and coworkers publish the Rev. Mod. Phys. review mapping the classical-quantum boundary

Going further

Discord sits one level below entanglement in the correlation hierarchy and one level above classical correlation: it detects "quantumness" in states that no entanglement measure can see, at the cost of a hard optimization over local measurements. The natural next steps are the canonical-form tool that makes entanglement tractable — "The Schmidt Decomposition," where for pure states discord and entanglement finally coincide — and the experimental frontier of "Experimental Tests and Violation of Bell Inequalities," which certifies the stronger, entangled kind of non-classicality.

References. Ollivier & Zurek, "Quantum Discord: A Measure of the Quantumness of Correlations," Phys. Rev. Lett. 88, 017901 (2001) · Henderson & Vedral, "Classical, quantum and total correlations," J. Phys. A 34, 6899 (2001) · Modi, Brodutch, Cable, Paterek & Vedral, "The classical-quantum boundary for correlations: Discord and related measures," Rev. Mod. Phys. 84, 1655 (2012) · Datta, Shaji & Caves, "Quantum Discord and the Power of One Qubit," Phys. Rev. Lett. 100, 050502 (2008) · Nielsen & Chuang, Quantum Computation and Quantum Information (Cambridge, 2010), Ch. 11 · Wootters, "Entanglement of Formation of an Arbitrary State of Two Qubits," Phys. Rev. Lett. 80, 2245 (1998).

Provenance. Definitions and equations follow Ollivier–Zurek (2001) and Henderson–Vedral (2001), with the review by Modi et al. (2012). A verifier recomputes: (1) the Werner eigenvalues $\{(1+3p)/4,(1-p)/4{\times}3\}$ and $S(\rho_A)=S(\rho_B)=1$ bit; (2) the closed-form $J=\tfrac{1+c}{2}\log_2(1+c)+\tfrac{1-c}{2}\log_2(1-c)$ with $c=\max_k|c_k|$ ; (3) the concurrence $C=\max(0,(3p-1)/2)$ against the Wootters formula, giving the $p\le 1/3$ separability threshold; (4) that $D=I-J>0$ throughout the separable region while entanglement is exactly zero, with anchors $I=0.2075$ , $J=0.0817$ , $D=0.1258$ bits at $p=1/3$ ; (5) the pure-state identity $D_A=S(\rho_A)$ and the classical-quantum characterization via local dephasing. All numerical statements are analytic functions of the state parameters; none are fabricated.

Builds on: The Density Operator and Mixed States · The Reduced Density Matrix and Partial Trace · Entanglement Entropy and Measures of Entanglement · Quantum Entanglement and Non-Separability. Leads to: The Schmidt Decomposition · Experimental Tests and Violation of Bell Inequalities.

Is discord a measure of entanglement, just a different one?

No — that is the central point. Discord and entanglement are genuinely different quantities. Entanglement is zero for every separable state (by definition), but discord can be positive for separable states. They agree only for pure states, where both equal $S(\rho_A)$ . For mixed states discord is the broader notion: every entangled state is discordant, but many discordant states are separable. Think of them as two nested sets — the entangled states sit strictly inside the discordant states, and the gap between them (separable-but-discordant) is exactly what discord was invented to name.

Why is discord asymmetric — how can measuring A versus B give different answers?

Because discord is defined through a measurement, and a measurement acts on one specified subsystem. $D_A$ minimizes over measurements on $A$ ; $D_B$ over measurements on $B$ . If $A$ happens to have a clean orthonormal basis that reads out without disturbance (a classical register) but $B$ does not, then $D_A=0$ while $D_B>0$ . The classical-quantum states are the extreme case. This is not a defect: it reflects the physical fact that the two parties can play genuinely different roles, e.g. a classical control register versus a quantum target.

The Werner state is separable for p ≤ 1/3 but discordant for all p > 0. Doesn't discord ever vanish for it?

Only at $p=0$ , where the Werner state is the maximally mixed state $\mathbb 1/4$ — a product state with no correlation at all. For every $p>0$ the classical correlation $J$ is strictly below the total $I$ , so $D=I-J>0$ . Meanwhile the concurrence stays exactly zero for the whole interval $0\le p\le 1/3$ . That mismatch — discord vanishing at a single point while entanglement vanishes over a whole interval — is the cleanest demonstration that discord detects correlations entanglement cannot.

Why does computing discord require a minimization, while entanglement entropy for pure states does not?

For a pure state, the Schmidt decomposition hands you the entanglement entropy $S(\rho_A)$ directly from the eigenvalues of the reduced state — no optimization. Discord instead needs the measured conditional entropy, which depends on the basis you measure $A$ in, so you must search over all local measurements for the one that reveals the most about $B$ . That optimization over a continuum of measurements has no Schmidt-style shortcut and is NP-hard in general. Closed forms exist only for special families — Bell-diagonal and X-states — which is exactly why the Werner state is the standard textbook example.

If discord is nonzero for almost every state, what makes zero-discord states special?

The zero-discord (classical-quantum) states form a set of measure zero in the space of density matrices — Ferraro and coworkers proved this in 2010. Geometrically, they are a thin surface, so a randomly chosen state is discordant with probability one. Their specialness is structural, not statistical: a zero-discord state is one where subsystem $A$ has a stable, perfectly readable basis. That is precisely the condition the environment enforces via einselection, which is why zero-discord classical-quantum states are the ones that "look classical."

What is the operational meaning of discord — does it actually do anything?

Yes. The clearest case is the DQC1 model of quantum computation ("power of one qubit"): it estimates a unitary's normalized trace with a speed-up over classical algorithms, yet operates with vanishing entanglement. Datta, Shaji and Caves showed the discord generated during the computation tracks the algorithm's efficiency, suggesting discord is the resource. Discord also has an interpretation in quantum state merging (the extra entanglement cost incurred when the merging party only has a measured version of the state) and shows up in the accounting of quantum metrology and remote state preparation. It is not merely a bookkeeping curiosity.

How does discord connect to decoherence and the classical world?

Very directly. Decoherence drives a system toward states that are diagonal in the environment-selected pointer basis — and a state that is diagonal in an orthonormal local basis is exactly a zero-discord (classical-quantum) state on that subsystem. So decoherence can be read as the continual destruction of discord along the pointer basis. What remains is classical-quantum correlation, which is why the emergent world looks classical: the quantum correlations underneath were real, but the environment strips away precisely the discordant part that a local observer could not read out anyway.