Entanglement and Bell Tests
The tensor-product structure of composite systems was written down in The Postulates of Quantum Mechanics without comment. Its consequences are the subject of this chapter, and they are the least classical thing in physics. Almost every state of a composite system fails to factorize; Schrödinger called such states entangled and said of them, correctly, that they are not one but the characteristic trait of quantum mechanics [Schroedinger:1935a]. Einstein, Podolsky and Rosen had just argued from the same states that the quantum description must be incomplete [Einstein:1935]. For thirty years the dispute was regarded as philosophical. It stopped being philosophical in 1964, when Bell showed that any theory in which the outcomes are fixed by local variables carried by the particles obeys an inequality that quantum mechanics violates [Bell:1964]. The question became experimental, and it has been answered.
This chapter develops the theory: what entanglement is, what Bell's theorem does and does not prove, what quantum mechanics may not do with entanglement (signal, clone, share it freely), and how much of it a state contains. The experimental record — Freedman and Clauser, Aspect, and the loophole-free tests of 2015 — lives in Experiment: Bell Tests, and the reader who wants the evidence before the formalism should read that chapter first. What this chapter insists on is the logic connecting the two: a measured violation excludes a precisely specified class of theories and nothing wider, and the treatise's position on the wider claims is stated in Interpretations (Evidence-Anchored). Standard references are [Nielsen:2010] [Peres:1993] [Bell:1987]; the modern review is [Brunner:2014] [Horodecki:2009].
Entanglement and Bell Tests: all derivations of this chapter are pending.
Entangled states
Tensor products and separability
[Reserved: the composite Hilbert space \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\) and the dimension count that makes product states a measure-zero subset; the definition of a pure entangled state as one that is not a product; the extension to mixed states, where separable means a convex combination of products, due to Werner [Werner:1989], together with his example of a mixed state that is entangled yet admits a local hidden-variable model for projective measurements — so that entanglement and Bell violation are not the same property; the two-qubit Bell basis as the working example throughout.]
The Schmidt decomposition
[Reserved: every pure bipartite state can be written \(\sum_{i}\sqrt{p_{i}}\ket{a_{i}}\otimes\ket{b_{i}}\) with orthonormal local bases, the coefficients being the singular values of the coefficient matrix; the underlying decomposition is Schmidt's [Schmidt:1907] and the linear algebra is that of Linear Algebra and Representation Theory; the Schmidt rank as the first, coarsest measure of entanglement; that the two reduced states share their spectrum, which is why bipartite entanglement entropy is unambiguous.]
Reduced states, purification and steering
[Reserved: the partial trace and the reduced density matrix as the unique object reproducing all local statistics; entanglement as the statement that the parts are mixed while the whole is pure; purification and its essential uniqueness up to a local unitary. Schrödinger's observation that the distant party can steer the local ensemble into any decomposition of the reduced state [Schroedinger:1935b], made precise as the Hughston–Jozsa–Wootters theorem [Hughston:1993]; that steering changes the ensemble description but not a single measurable local probability, which is the seed of Section 84.4.1.]
The EPR argument and the replies
Einstein, Podolsky and Rosen
[Reserved: the 1935 argument in its own terms [Einstein:1935]: the sufficient criterion of reality, the locality assumption, the position–momentum correlated two-particle state, and the conclusion that either the description is incomplete or reality depends on what is measured elsewhere; a careful statement of the premises, since almost every later dispute is about which of them a Bell test refutes; Bohr's reply of the same year, which denied the criterion of reality rather than the locality assumption; the reading of the paper Einstein himself preferred, given in his later correspondence, in which the target is separability rather than completeness.]
Verschränkung, the cat, and the amplification argument
[Reserved: Schrödinger's response in Die Naturwissenschaften [Schroedinger:1935a], which introduced the term Verschränkung, framed entanglement as the failure of the whole to be determined by its parts, and used the cat as an amplification argument — that nothing in the linear dynamics stops a microscopic superposition from being coupled to a macroscopic degree of freedom; the companion Cambridge paper with the steering theorem [Schroedinger:1935b]; the point, still worth making, that the cat is an argument about the reach of linearity and not a joke about observers, and that its modern quantitative version is the interference of massive molecules and the collapse bounds of Interpretations (Evidence-Anchored).]
Bohm's spin version
[Reserved: Bohm's replacement of the continuous EPR variables by two spin-\(\tfrac{1}{2}\) particles in the singlet [Bohm:1951], which is what made the argument experimentally tractable: a finite-dimensional Hilbert space, a rotationally invariant state, and the correlation \(-\cos\theta\) between spin components along directions at angle \(\theta\), with the spin formalism of Angular Momentum and Spin and the analyser of Experiment: Stern–Gerlach. Every subsequent inequality and every experiment uses this version.]
Two spin-\(\tfrac{1}{2}\) particles prepared in the singlet state — and the polarization-entangled photon pairs that stand in for them in the laboratory — display two properties at once. Each particle taken by itself is unpolarized: whatever direction \(\hat{\vect{a}}\) its analyser is set to, the outcomes \(\pm1\) occur with equal frequency, so a one-sided record carries no trace of the setting used on the other side. Taken in pairs the outcomes are correlated, and the correlation depends on nothing but the angle \(\theta\) between the two analyser directions,
so that parallel analysers give opposite results on every single pair while orthogonal analysers give no correlation at all. This is the Einstein–Podolsky–Rosen situation [Einstein:1935] in the two-outcome form that made it experimentally decidable, and Equation (84.1) is the curve against which every measurement of Experiment: Bell Tests is read.
Derivation. Write \(\ket{\Psi^{-}}=\bigl(\ket{\uparrow\downarrow} -\ket{\downarrow\uparrow}\bigr)/\sqrt{2}\), the unique two-spin state annihilated by the total spin \(\vect{S}=\tfrac{\hbar}{2}(\vect{\sigma}_{1}+\vect{\sigma}_{2})\). From \((\vect{\sigma}_{1}+\vect{\sigma}_{2})^{2}\ket{\Psi^{-}}=0\) and \(\vect{\sigma}^{2}=3\,\identity\) on each factor, \(6+2\,\vect{\sigma}_{1}\cdot\vect{\sigma}_{2}\) annihilates \(\ket{\Psi^{-}}\), so
The singlet carries zero total angular momentum and is therefore invariant under simultaneous rotations of both factors. Consequently the tensor \(T_{ij}=\bra{\Psi^{-}}\sigma_{1}^{i}\sigma_{2}^{j}\ket{\Psi^{-}}\) is an invariant tensor of \(\SO(3)\) of rank two, and the only such tensor is \(\delta_{ij}\) up to a factor: \(T_{ij}=\lambda\,\delta_{ij}\). Taking the trace and using Equation (84.2) gives \(3\lambda=-3\), hence \(\lambda=-1\) and
which is Equation (84.1). For the one-sided statistics, the reduced state is \(\rho_{A}=\tr_{B}\ket{\Psi^{-}}\bra{\Psi^{-}} =\tfrac{1}{2}\identity\), so \(\tr\bigl(\rho_{A}\,\vect{\sigma}\cdot\hat{\vect{a}}\bigr)=0\) for every \(\hat{\vect{a}}\): each analyser on its own sees a fair coin, which is what makes the correlation invisible until the two records are compared.
∎Bell's theorem
Local hidden variables and the 1964 inequality
[Reserved: the precise hypotheses — a shared variable \(\lambda\) with some distribution, outcome functions \(A(a,\lambda)\) and \(B(b,\lambda)\) depending only on the local setting (parameter independence) and not on the distant outcome (outcome independence), and settings statistically independent of \(\lambda\) (measurement independence) — and the derivation of Bell's original three-setting inequality on the perfect-anticorrelation singlet [Bell:1964]; the quantum prediction \(-\cos\theta\) violating it; the fact, essential to the argument's force, that no feature of the quantum formalism is used in deriving the bound.]
CHSH: the form that is actually measured
[Reserved: the four-correlator combination \(S=E(a,b)+E(a,b')+E(a',b)-E(a',b')\) with \(\abs{S}\le2\) for any local model, due to Clauser, Horne, Shimony and Holt [Clauser:1969], which drops the perfect-correlation idealization and so applies to real detectors; the Clauser–Horne form written for detection probabilities rather than correlations [Clauser:1974b]; Fine's theorem, that satisfying the full set of CHSH inequalities is equivalent to the existence of a joint distribution for all four observables [Fine:1982], which identifies exactly what a violation refutes; the quantum maximum \(2\sqrt{2}\) at the optimal analyser angles; Gisin's theorem that every entangled pure state violates some CHSH inequality [Gisin:1991].]
Let each station of a two-particle experiment choose between two analyser settings, recording an outcome \(\pm1\), and let \(E(a,b)\) be the measured correlation for the setting pair \((a,b)\). The combination
cannot exceed \(2\) in modulus in any theory in which each outcome is fixed by variables carried by the particles and is unaffected by the setting chosen at the distant station [Bell:1964]. Values of \(S\) above \(2\) are what is measured. They were first reported from atomic-cascade photon pairs in the early 1970s, sharpened with two-channel polarizers and with analysers switched while the photons were in flight in the early 1980s, and have since been reproduced with independent sources, analysers and detector technologies; the record, with its uncertainties and its residual assumptions, is Experiment: Bell Tests. Nature therefore admits no description of the assumed kind.
Derivation. The bound is the whole of the argument, and it uses no quantum mechanics whatever. Suppose the outcomes are functions \(A(a,\lambda)\), \(A(a',\lambda)\), \(B(b,\lambda)\), \(B(b',\lambda)\) taking the values \(\pm1\), each depending on the setting on its own side and on a variable \(\lambda\) distributed with some density \(p(\lambda)\geq0\), \(\int\dd\lambda\,p(\lambda)=1\), so that \(E(a,b)=\int\dd\lambda\,p(\lambda)\,A(a,\lambda)B(b,\lambda)\). For a fixed \(\lambda\) collect the four products and factor,
suppressing the arguments. Since \(B\) and \(B'\) each take only the values \(\pm1\), either \(B=B'\), in which case \(B-B'=0\) and \(\abs{B+B'}=2\), or \(B=-B'\), in which case \(B+B'=0\) and \(\abs{B-B'}=2\). In both cases exactly one bracket vanishes and the other has modulus \(2\); with \(\abs{A}=\abs{A'}=1\) this gives \(\abs{S(\lambda)}=2\) for every \(\lambda\). Averaging preserves the bound,
Quantum mechanics is not a theory of that form and is not bound by it. Insert the singlet correlation Equation (84.1) with coplanar analyser directions at angles \(a=0\), \(a'=\pi/2\), \(b=\pi/4\) and \(b'=-\pi/4\): then \(E(a,b)=E(a,b')=E(a',b)=-1/\sqrt{2}\) while \(E(a',b')=-\cos(3\pi/4)=+1/\sqrt{2}\), and Equation (84.3) gives \(S=-4/\sqrt{2}=-2\sqrt{2}\). The predicted excess over the local bound is a factor \(\sqrt{2}\), not a marginal effect — which is why the question could be settled at all with the photomultipliers of a few percent quantum efficiency available when it was first asked.
∎The quantum bound
[Reserved: Tsirelson's proof that quantum mechanics itself caps \(\abs{S}\) at \(2\sqrt{2}\) [Cirelson:1980], so that the theory is nonlocal in Bell's sense but not maximally so; the Popescu–Rohrlich box, a hypothetical device that reaches the algebraic maximum \(4\) without permitting signalling [Popescu:1994], which shows that no-signalling alone does not single out quantum correlations and turns “why \(2\sqrt{2}\)?” into a live question with several proposed information-theoretic answers; the experimental status, namely that no measured correlation has ever exceeded the Tsirelson bound.]
No experiment has produced a CHSH parameter above \(2\sqrt{2}\). This is not forced by relativity: the algebraic maximum of Equation (84.3) is \(4\), and correlation tables reaching \(4\) can be written down that leave both marginals uniform and so permit no signalling. Nor is it forced by the existence of a violation, which only requires \(\abs{S}>2\). What is observed is that the correlations of Nature stop precisely where the quantum formalism says they stop, so that the measured \(\abs{S}\) is bounded above and below by two independent theoretical statements and lies strictly between them [Hensen:2015] [Giustina:2015] [Shalm:2015].
Derivation. Represent the two alternative settings on one side by Hermitian operators \(A,A'\) and those on the other side by \(B,B'\). Outcomes \(\pm1\) mean \(A^{2}=A'^{2}=B^{2}=B'^{2}=\identity\), hence \(\norm{A}=\norm{A'}=\norm{B}=\norm{B'}=1\); operators belonging to different tensor factors commute. The CHSH observable is \(\hat{S}=A\otimes(B+B')+A'\otimes(B-B')\), and \(S=\tr(\rho\hat{S})\). Squaring, and using \((B\pm B')^{2}=2\,\identity\pm(BB'+B'B)\) together with \((B+B')(B-B')=\comm{B'}{B}\) and \((B-B')(B+B')=\comm{B}{B'}\), the anticommutators cancel between the two diagonal terms and the cross terms combine into a single commutator product,
Each commutator obeys \(\norm{\comm{A}{A'}}\leq2\norm{A}\norm{A'}=2\), so \(\norm{\hat{S}^{2}}\leq4+2\cdot2=8\) and therefore
for every state and every choice of settings. Note where the bound comes from. If the two alternative settings at a station commuted, Equation (84.5) would read \(\hat{S}^{2}=4\,\identity\) and give back \(\abs{S}\leq2\), the local bound. The whole violation, and its exact size, is therefore the non-commutativity of the two measurements a single station chooses between — not any influence travelling between the stations.
∎GHZ: contradiction without inequalities
[Reserved: the three-particle state for which local realism and quantum mechanics disagree on a single run rather than on an average, so that the refutation is algebraic rather than statistical [Greenberger:1989], in the accessible presentation of [Greenberger:1990]; the sign contradiction obtained from four perfectly correlated measurement settings; the Mermin inequality as its statistical version and its exponentially growing violation with particle number; the experimental realization belongs to Experiment: Bell Tests.]
What a violation excludes — and what it does not
[Reserved: the disciplined statement, which this treatise treats as the whole point of the chapter. A measured violation, closing the locality and detection loopholes, refutes the conjunction of the hypotheses of Section 84.3.1. It does not establish superluminal causation, since Section 84.4.1 forbids it; it does not select any interpretation of Interpretations (Evidence-Anchored); it does not refute determinism as such, only local determinism. The residual assumptions — measurement independence, or “freedom of choice” — are not removable in principle, and the cosmic-Bell experiments of Experiment: Bell Tests only push the correlation back in time. The honest summary is that local hidden-variable theories are excluded as a class, which is a statement about theories and is the strongest one the data support.]
What entanglement may not do
No-signalling
[Reserved: the theorem that the local statistics at \(A\) are independent of the setting chosen at \(B\), because the reduced state is unchanged by any local operation at \(B\) — proved directly from the partial trace and the trace-preserving property of local channels; the argument as given by Ghirardi, Rimini and Weber [Ghirardi:1980] and the careful analysis of the several distinct locality conditions by Eberhard [Eberhard:1978]; the consequence that entanglement carries no message and that relativistic causality of Minkowski Space and Its Symmetries is intact; that this is a theorem about quantum mechanics, not a postulate imposed on it.]
In every experiment on entangled pairs, the singles rate recorded at one station is independent, within statistics, of which setting the distant station chose. The check is made directly in the data of a Bell test as a control on the apparatus [Hensen:2015] [Giustina:2015] [Shalm:2015], and it has never failed. However strongly the two records are correlated, neither party can send the other a message by choosing a setting; the correlation appears only when the two records are brought together and compared, which requires a classical channel and therefore takes at least the light travel time between the stations.
Derivation. Everything one party can measure is computed from the reduced state \(\rho_{A}=\tr_{B}\rho\). Let the distant party act on their factor alone: the most general such operation is a trace-preserving map \(\mathcal{E}_{B}\), and choosing a setting and performing a measurement whose outcome is not communicated is one instance of it. The joint state becomes \((\mathrm{id}_{A}\otimes\mathcal{E}_{B})\rho\). Then for any observable \(X\) of the first factor,
because \(\tr_{B}\bigl[(\mathrm{id}_{A}\otimes\mathcal{E}_{B})\rho\bigr] =\tr_{B}\rho\) whenever \(\mathcal{E}_{B}\) preserves the trace. Hence \(\rho_{A}\), and with it every local probability, is untouched by anything done at \(B\). The argument uses only linearity and trace preservation, so it holds for every state, every setting and every separation. No-signalling is thus a theorem of the formalism, not a relativistic postulate bolted onto it — which is why Phenomenon 84.2 creates no conflict with the causal structure of Minkowski Space and Its Symmetries.
∎No-cloning
[Reserved: the impossibility of a unitary that copies an unknown state, proved in two lines from linearity, given independently by Wootters and Zurek [Wootters:1982] and by Dieks [Dieks:1982], both motivated by closing a proposed superluminal-signalling scheme; the no-broadcasting and no-deleting extensions; the constructive consequences — unconditional security of quantum key distribution [Ekert:1991] and the necessity of teleportation rather than copying in Section 84.6.1; the relation to the distinguishability of non-orthogonal states.]
No apparatus has ever been built that accepts one system in an unknown state and returns two systems in that same state. The regularity shows itself wherever a quantum signal is amplified: a laser amplifier, an optical repeater or an eavesdropper's tap always adds noise, and the minimum noise it adds is exactly the amount required to stop the output being a faithful pair of copies. It is the reason a key carried by single quanta cannot be intercepted and forwarded undetected, and the reason that the transfer of an unknown state in Section 84.6.1 must destroy the original.
Derivation. Suppose a unitary \(U\) on system plus blank register satisfied \(U\bigl(\ket{\psi}\otimes\ket{0}\bigr)=\ket{\psi}\otimes\ket{\psi}\) for every state \(\ket{\psi}\) of the system. Apply it to two states \(\ket{\psi}\) and \(\ket{\phi}\) and take the inner product of the two results. A unitary preserves inner products, and the inner product of a tensor product is the product of the inner products, so with \(z=\langle\psi|\phi\rangle\),
whence \(z\in\{0,1\}\): the only families a single unitary can copy are sets of mutually orthogonal states, which is to say classical alphabets, since a state drawn from an orthogonal set can be measured without disturbance and then rewritten at will. Nothing but linearity and unitarity was used, so the conclusion is exactly as robust as the superposition principle of The Postulates of Quantum Mechanics; in particular it cannot be evaded by a cleverer device.
∎Monogamy
[Reserved: entanglement cannot be freely shared. The Coffman–Kundu–Wootters inequality bounds the sum of the squared concurrences of \(A\) with \(B\) and with \(C\) by the squared concurrence of \(A\) with the pair, the deficit defining the three-tangle [Coffman:2000]; the extreme case, that a maximally entangled pair is uncorrelated with everything else, which is exactly what makes device-independent key distribution possible; the general \(N\)-party statement and the role of monogamy in bounding how many parties can jointly violate a Bell inequality.]
Quantifying entanglement
Separability criteria
[Reserved: the positive-partial-transpose necessary condition of Peres [Peres:1996] and the Horodecki proof that it is also sufficient in dimensions \(2\times2\) and \(2\times3\), via the theory of positive but not completely positive maps [Horodecki:1996]; bound entangled states in higher dimensions, which are entangled yet have positive partial transpose and cannot be distilled; entanglement witnesses as the operational form of the criterion, being the observables a laboratory actually measures.]
Entropy of entanglement
[Reserved: for a pure bipartite state, the von Neumann entropy of the reduced state, \(S=-\tr\rho_{A}\log\rho_{A}\), as the unique measure satisfying the natural axioms; its operational meaning as the asymptotic rate of Bell-pair distillation and of entanglement dilution [Bennett:1996]; the maximum \(\log d\) and the maximally entangled states attaining it; area laws in extended systems and the pointer to Axiomatic Quantum Field Theory for the field-theoretic case, where the same quantity is ultraviolet divergent and its regularization is a substantive physical question.]
Mixed-state measures
[Reserved: entanglement of formation and distillable entanglement and the gap between them [Bennett:1996]; Wootters' closed formula for the entanglement of formation of an arbitrary two-qubit state via the concurrence [Wootters:1998]; the negativity and logarithmic negativity, computable from the partial transpose [Vidal:2002]; the axioms an entanglement monotone must satisfy — invariance under local unitaries, non-increase under local operations and classical communication — and the fact that no single number orders mixed states completely; the review [Horodecki:2009] as the reference for the full landscape.]
Entanglement as a resource
Teleportation and entanglement swapping
[Reserved: the protocol of Bennett and co-workers [Bennett:1993], in which a shared Bell pair plus two classical bits transfer an unknown qubit exactly, destroying the original — and so respecting both Section 84.4.2 and Section 84.4.1, since without the classical message the output is the maximally mixed state; the first photonic realization [Bouwmeester:1997]; entanglement swapping, which entangles two systems that never interacted and which relies on the two-photon interference of Identical Particles; the resource accounting that makes “one ebit” a physical unit.]
A qubit in a state unknown to everyone involved can be transferred from one system to a second system that never interacted with it, using a previously shared entangled pair and two bits sent over an ordinary classical channel. What is observed is a reconstructed state whose fidelity with the input exceeds \(2/3\), the highest that any strategy of measuring the input and transmitting a classical description of the result can achieve; the original is destroyed in the process. The protocol has been realized with photon polarization [Bouwmeester:1997], with trapped ions and atoms, and with superconducting circuits, and the transfer succeeds only after the classical message arrives.
Derivation. Let the unknown state be \(\ket{\psi}=a\ket{0}+b\ket{1}\) on system \(1\), and let systems \(2\) and \(3\) share \(\ket{\Phi^{+}}=\bigl(\ket{00}+\ket{11}\bigr)/\sqrt{2}\), with \(1\) and \(2\) held by the sender and \(3\) by the receiver. Expand the product \(\ket{\psi}_{1}\otimes\ket{\Phi^{+}}_{23}\) in the Bell basis of the pair \((1,2)\), using \(\ket{00}=\bigl(\ket{\Phi^{+}}+\ket{\Phi^{-}}\bigr)/\sqrt{2}\) and its three companions. Collecting terms,
The four Bell states are orthonormal and enter with equal weight, so a measurement of \((1,2)\) in that basis yields each of the four outcomes with probability \(1/4\) — a distribution independent of \(a\) and \(b\), which is why the sender learns nothing about the state and why the outcome, taken alone, is useless. Conditioned on the outcome, system \(3\) is left in \(\identity\ket{\psi}\), \(\sigma_{z}\ket{\psi}\), \(\sigma_{x}\ket{\psi}\) or \(\sigma_{x}\sigma_{z}\ket{\psi}\) respectively, each of which the receiver undoes by applying the inverse of that operator once the two bits naming the outcome have arrived. The correction is a fixed unitary chosen from a list of four; no knowledge of \(a\) and \(b\) enters it.
Three consistency checks close the argument. Before the message arrives, the receiver's state is the average of the four branches, \(\tfrac{1}{4}\sum_{k}\sigma_{k}\ket{\psi}\bra{\psi}\sigma_{k} =\tfrac{1}{2}\identity\), independent of \(\ket{\psi}\) — so nothing has been signalled, in accordance with Phenomenon 84.4, and the classical channel really is indispensable. The input system emerges maximally entangled with the sender's half of the pair and no longer carries \(\ket{\psi}\), so nothing has been copied, in accordance with Phenomenon 84.5. And the two classical bits are not a disguised description of \(\ket{\psi}\), which would need infinitely many bits: they are a description of the sender's measurement outcome, whose distribution Equation (84.8) shows to be uniform.
∎Device-independent protocols
[Reserved: the observation that a Bell violation certifies properties of a device without any assumption about its internals — the strongest practical consequence of Section 84.3; certified randomness generation from a measured CHSH value [Pironio:2010], and device-independent key distribution descending from Ekert's protocol [Ekert:1991]; self-testing, whereby a maximal violation forces the state and measurements to be the singlet and the optimal analysers up to a local isometry; the honest caveat that the certification assumes no communication between the devices, which is a physical assumption about the laboratory and not a mathematical one.]
The experimental record
[Reserved: a short bridging section, not a duplicate of Experiment: Bell Tests, which owns the apparatus, the data tables and the uncertainties. It records only the logical sequence: the first violation with atomic cascade photons by Freedman and Clauser [Freedman:1972]; the switching experiment that first closed the locality loophole in a time-varying way [Aspect:1982b]; the loophole-free experiments of 2015, the first to close locality and detection simultaneously [Hensen:2015]; and the resulting status of local realism as an excluded class of theories. Each entry names the loophole it addressed, so that the chapter's logical claim in Section 84.3.5 can be checked against what was actually measured.]