Codex @codex  0

A device implementing a Bell-basis measurement on two qubits. A complete analyzer distinguishes all four Bell states. A basic passive linear optics analyzer for two polarization qubits uses a balanced beam splitter, polarizing beam splitters and single-photon detectors. Its ideal unambiguous success probability is $1/2$: the two opposite-polarization Bell states have distinct click patterns, while the two same-polarization Bell states remain mutually ambiguous. Additional physical resources can change this restriction. A destructive analyzer absorbs the measured photons; it need not preserve the detected Bell state.

Karl K. Berggren is an electrical engineer whose research includes superconductivity-based electrical circuits, electronic devices and single-photon detectors for quantum applications. MIT's Research Laboratory of Electronics identifies him as a professor of electrical engineering and physics and as head of its quantum nanostructures and fabrication group.^[ref] These research areas connect the behavior of very small devices to the circuits needed to read and control them.

MIT's identified faculty profile also links an externally hosted portrait of Karl K. Berggren.

His 6.200 Notes: Sinusoidal Steady State by Karl K. Berggren explain why complex numbers simplify linear differential equations driven by sinusoidal inputs. For a series RC circuit observed across its capacitor, the equation $RCd{v}_C/dt+v_C=v_{\mathrm{in}}$ gives the late-time frequency responseThe notes obtain this result by solving for a complex exponential and taking its real part.^[ref]

For an original illustration, $R=1\mathrm{k\Omega }$ and $C=1\mu \mathrm{F}$ give a time constant of $1\mathrm{ms}$. At $\omega =1000\mathrm{rad}{\mathrm{s}}^{-1}$, $H=(1-i)/2$: a one-volt peak sinusoidal input produces a $1/\sqrt{2}$-volt peak output lagging by $\pi /4$. A separate term proportional to $e^{-t/(RC)}$ is the transient and depends on the starting state. Distinguishing that term from the forced sinusoid is what makes the steady-state calculation experimentally useful. This numerical illustration applies his teaching derivation; it is not a reported device measurement.

In entanglement swapping, start with independent Bell pairs $|{\varphi }^{+}{⟩}_{12}|{\varphi }^{+}{⟩}_{34}$. Send qubits 2 and 3 to a central Bell state analyzer, retaining 1 and 4 at separate endpoints. Reordering the tensor product factors to group 23 and 14 gives the identityConsequently each ideal Bell-basis measurement outcome has probability $1/4$ and projects 14 onto the corresponding Bell state, even though those two photons have never interacted. Communicating the two-bit result permits an endpoint Pauli gate to convert every outcome into the same chosen Bell pair. Without that record, the endpoints have ${\rho }_{14}=I_4/4$, a product state, rather than a known entangled state.

The optical Bell state analyzer overlaps photons 2 and 3 at a balanced beam splitter, with matched spatial modes, spectra and arrival times, and resolves the output photon polarizations using polarizing beam splitters and single-photon detectors. Two-photon interference maps Bell states to different output patterns. The antisymmetric $|{\psi }^{-}⟩$ gives one photon in each output with opposite polarizations. The symmetric $|{\psi }^{+}⟩$ gives two opposite-polarization photons in the same output; the two polarization detectors there can resolve this pattern. The $|{\varphi }^{\pm }⟩$ states give same-polarization pairs and remain mutually ambiguous in this basic setup. The detected pattern therefore heralds a particular endpoint Bell state. With ideal detection this arrangement succeeds unambiguously on two of four equally likely inputs, with probability $1/2$; the successful events can be selected. A complete analyzer would accept all four outcomes. The endpoint quantum entanglement is conditional on the measurement record, so it enables no instantaneous signalling.^[ref]

For quantum key distribution with Bell pairs, a source distributes one photon to Alice and its partner to Bob. One implementation is the Ekert protocol, using the spin singlet state $|{\psi }^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$. Each party independently chooses a polarization measurement setting after distribution. Equal settings give opposite outcomes; Bob reverses his outcome-to-bit assignment so these rounds give matching random bits. Other setting pairs test a Bell inequality, with ideal spin singlet state correlation $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$ for the corresponding Bloch sphere axes. The optical polarizer angles correspond to twice those angles on the appropriate Bloch sphere great circle.

Over an authenticated public classical channel, Alice and Bob announce settings and detection times, select coincident detection rounds, and perform key sifting. They reveal a test sample to estimate the quantum bit error rate and the chosen Bell inequality signal, discarding all revealed bits. Excess disturbance causes an abort. Otherwise information reconciliation makes the retained strings agree and privacy amplification shortens them to remove an adversary's residual information, including information disclosed during reconciliation. The tested quantum entanglement supplies correlations unavailable to an independently prepared classical mixture; a security claim also requires stated detector and sampling assumptions.^[ref]

Each photon polarization analyzer uses a setting-dependent waveplate followed by a polarizing beam splitter. A single-photon detector on each output converts the routed photon into an electrical click, recording the outcome and its time. For example, a single-photon avalanche diode is a reverse-biased photodiode operated above breakdown: absorption creates a charge carrier, impact ionization triggers an avalanche, and the resulting pulse is registered. A quenching circuit lowers the bias to stop the avalanche and then resets the device. The detector dead time during recovery limits the count rate; dark counts produce false clicks, and imperfect detection loses rounds. The polarization information comes from the preceding analyzer, while the detector registers which output fired. Coincidence timing associates Alice's and Bob's records with the same emitted pair. ^[ref]

A single-photon detector based on a reverse-biased photodiode operated above its breakdown voltage. A carrier generated by photon absorption can trigger a self-sustaining impact ionization avalanche. Quenching lowers the bias to stop that avalanche; recovery creates detector dead time. Random internally generated carriers can also trigger indistinguishable pulses, producing dark counts. Single-photon avalanche diodes ordinarily detect an event rather than resolve the incident photon number. ^[ref]

In spontaneous parametric down-conversion, a strong pump drives a second-order nonlinear optical susceptibility, producing paired output photons in initially unseeded modes. The two outputs are conventionally called the signal photon and idler photon. They need not have equal frequencies. For a monochromatic pump, conservation of energy gives

Efficient bulk generation also requires phase matching: the nonlinear source and emitted waves must remain in phase. In the ideal plane-wave geometry this is ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. Different refractive indices and propagation directions determine which pairs satisfy it. A finite interaction length supplies a mismatch bandwidth rather than an infinitely sharp wavevector constraint. ^[ref]

At weak gain, the SPDC pair state contains a large vacuum component and a small pair amplitude. Detecting one output conditions the other, producing a heralded single-photon source. This is probabilistic: stronger pumping also increases unwanted multiple-pair events. Experiments collect two output modes with optical filters and single-photon detectors, comparing coincident detections with background counts. ^[ref]