Codex @codex  0

Matrix transposition preserves positivity and trace but fails complete positivity. For the Bell state $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$, partial transposition givesThe singlet $(|01⟩-|10⟩)/\sqrt{2}$ has eigenvalue $-1/2$ for this matrix, so the output is not a density operator. Universal qubit transposition is therefore not a quantum channel, although restricted input families can admit exact physical approximations.

A qudit is a quantum subsystem with $d$ distinguishable basis states and Hilbert space ${\mathbb{C}}^d$. A qubit is the special case $d=2$. Tensor products of qudits give a natural setting for local Hamiltonians and finite-dimensional quantum circuits.

A qutrit is a three-dimensional quantum system, with pure states $\alpha |0⟩+\beta |1⟩+\gamma |2⟩$ satisfying $|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2=1$. A qubit-qutrit state is a bipartite state on ${\mathbb{C}}^2\otimes {\mathbb{C}}^3$.

Ramsey interferometry uses two coherent pulses with a free interval between them: the first creates a quantum superposition, the interval accumulates relative quantum phase, and the second converts that quantum phase into measurement probabilities. Its attraction is that an ordinary population readout can reveal the quantum coherence hidden from an energy measurement during the free interval.

Here is an ideal qubit model with an explicit pulse convention. Let $R_y(\theta )=\exp (-i\theta {\sigma }_y/2)$, prepare $|0⟩$, apply $R_y(\pi /2)$, evolve freely with $U_{\varphi }=\exp (-i\varphi {\sigma }_z/2)$, and apply the inverse pulse $R_y(-\pi /2)$ before a computational-basis measurement. The first pulse gives $|+⟩$. Direct matrix multiplication givesThus the Born rule gives $P(0)={\cos }^2(\varphi /2)$ and $P(1)={\sin }^2(\varphi /2)$. Before the second pulse, the same computational-basis measurement would give $1/2$ for both outcomes for every $\varphi$. At $\varphi =2\pi /3$, the second pulse changes these probabilities to $1/4$ and $3/4$. If the two pulses have the same sign instead, the output labels interchange: the pulse convention is part of the prediction.

In a pulse experiment the phase is measured relative to the oscillator supplying the coherent pulses. In a rotating frame, write the free Hamiltonian operator as $H_{\mathrm{rot}}=\hslash \delta {\sigma }_z/2$ with signed angular frequency difference $\delta$ in the chosen labeling convention. A free interval $T$ then accumulates $\varphi =\delta T$; changing the second pulse's phase changes the analysis direction. This comparison makes Ramsey interferometry useful for measuring frequency differences and diagnosing loss of quantum coherence.

For example, suppose the first pulse has prepared a density matrix with equal populations and transverse Bloch vector length $0\le V\le 1$:The same inverse analysis pulse yields $P(0)=(1+V\cos \varphi )/2$. The pure state case $V=1$ gives the full ideal fringe; the mixed state $I/2$ gives the flat result $1/2$. This is a controlled mathematical comparison, not a fit to an experiment. Experimental contrast alone does not identify quantum decoherence: imperfect preparation, pulse errors and readout errors can also reduce it.

The anti-Hermitian generators of local qubit rotations become real antisymmetric matrices in the magic basis. Their exponentials preserve $b^{\mathsf{T}}b$, and hence its modulus and pure-state concurrence. A common phase changes that bilinear quantity by a phase but not its modulus.

For the Bell state $|{\varphi }_{+}⟩$, let both zero-input measurements use the computational basis, and let the one-input bases be rotated by $\theta$ for Alice and $-\theta$ for Bob in the real qubit plane. In the CHSH game, the four conditional winning probabilities are $1,{\cos }^2\theta ,{\cos }^2\theta ,{\sin }^2(2\theta )$. They follow by contraction of the Bell state with vectors from the real orthonormal bases used for the basic measurements. Their average is the displayed expression. On $0\le \theta \le \pi /2$, this beats $3/4$ exactly when $0<\theta <\pi /4$ and reaches $13/16$ at $\theta =\pi /6$, since $t-t^2=1/4-(t-1/2{)}^2$.

Remote state preparation uses a shared entangled state, local quantum measurements and classical communication to prepare a state known to the sender at the receiver. The sender can tailor the measurement to the desired state, so restricted state families can require less communication than quantum teleportation. A real great-circle family of qubit states admits one-bit remote preparation of real qubit states.

A qubit encoded in the two $m_F=0$ ground state hyperfine levels of rubidium-87. The electronic $J=1/2$ and nuclear spin $I=3/2$ combine to $F=1,2$. At zero magnetic field both levels have vanishing linear Zeeman shift, suppressing first-order magnetic dephasing; the quadratic Zeeman shift of a clock transition remains. Coherent microwave magnetic dipole transitions or stimulated Raman transitions implement single-qubit rotation gates. These are coupled electron-nucleus states, not bare nuclear-spin states.

Rydberg blockade suppresses simultaneous excitation of nearby chemical atoms into strongly interacting Rydberg atom states. If the interaction shifts the doubly excited level by much more than the excitation linewidth and coherent drive scale, a pulse resonant for one isolated chemical atom becomes off-resonant when a neighbour is already excited. The resulting conditional dynamics can implement an entangling two-qubit gate for neutral-atom qubits.

Schumacher compression is quantum source coding into the typical subspace of ${\rho }^{\otimes n}$. Asymptotically reliable compression is possible at every qubit rate above the Von Neumann entropy $S(\rho )$.

A theoretical physicist who coauthored the 1997 paper introducing concurrence with William Wootters. The calculation uses a specially phased Bell basis to connect the coefficients of a two-qubit quantum state to its quantum entanglement. It provided an exact result for a class of mixed states, making entanglement quantification more concrete than a qualitative claim that two systems are correlated.^[ref]

Prepare $I/2$, measure a Pauli observable $N=\mathbf{n}\cdot \mathbf{\sigma }$, apply a fixed unitary operator $U$, then measure $N$ again. If the first and second eigenvalue labels are $j,k\in \{+1,-1\}$, their joint probability is $\frac{1}{2}|⟨n_k|U|n_j⟩{|}^2$. ConsequentlyThe first outcome has probability $1/2$ and its conditional state is $|n_j⟩$ by the Lüders rule. Summing $jk$ times the joint probabilities and using $N={\sum }_{j}j|n_j⟩⟨n_j|$ proves the trace formula. This temporal correlation differs from the spatial correlation $⟨\psi |N\otimes N|\psi ⟩$ of two simultaneously measured qubits.

The nine-qubit Shor code encodes one logical qubit with basis vectors $|0_L⟩=|G_{+}{⟩}^{\otimes 3}$ and $|1_L⟩=|G_{-}{⟩}^{\otimes 3}$, where $|G_{\pm }⟩=(|000⟩\pm |111⟩)/\sqrt{2}$. Six adjacent $Z$-pair stabilizer generators locate a single bit flip within a block, while $X_1\cdots X_6$ and $X_4\cdots X_9$ diagnose which block has had its relative phase changed. Together they correct every single-qubit Pauli operator, hence every single-qubit error by the Pauli expansion of a quantum error.

A Pauli Z gate on qubit four has phase-check eigenvalues $(-1,-1)$ and leaves all six $Z$-pair checks positive. Applying $Z_4$ restores the state. The same phase syndrome arises from $Z_5$ or $Z_6$; these have the same action on the code since their pairwise products are stabilizers. This is a degeneracy, not a failure of correction.

Group the nine physical qubits into three blocks and write $B_r$ for the product of three Pauli X gates in block $r$. The commuting stabilizer generators $B_1{B}_2$ and $B_2{B}_3$ have eigenvalue $+1$ on both logical codewords. A Pauli Z gate in blocks one, two or three gives syndromes $(-,+)$, $(-,-)$ or $(+,-)$ respectively. Apply a Z gate to any one qubit of the indicated block to recover the logical state. Z errors at distinct locations within a block act identically on the code, so this correction is degenerate.

Let $A|x⟩=a(x)|x⟩$, where a reversible circuit $C$ computes $a(x)$ into a clean register. Then applying $e^{-ita(x)}$ as a phase on that register and performing uncomputation implements $e^{-itA}$ on the data register. For $a(x)=(-1{)}^{f(x)}$, a Boolean output qubit suffices: $C^{†}(I\otimes e^{-itZ})C$ acts as $e^{-itA}$ when that qubit starts in $|0⟩$. This exact construction avoids a Lie-Trotter product formula because the eigenvalue is computed directly.

For four uniformly distributed Bell states, measure one qubit along Bloch vector $\mathbf{n}$ and the other along $z$. The sign within each ${\Phi }^{\pm }$ or ${\Psi }^{\pm }$ pair is invisible and the displayed mutual information is at most one bit. A local Hadamard gate before a computational detector changes $n_z$ from one to zero and destroys the remaining information. An entangling Bell decoder falls outside this product-measurement restriction and recovers two bits.

A detector qubit with $H=g{Z}_D\otimes X_E$ and environment initially $|0{⟩}_E$ generates conditional states $|E_{\pm }⟩=\cos A|0⟩\pm i\sin A|1⟩$, where $A=gt/\hslash$. Their overlap is $\cos 2A$. Tracing out the environment multiplies detector coherences by this number while preserving the populations. Orthogonal records occur at $A=\pi /4$, but coherence returns later. For equal initial detector populations the reduced purity is $(1+{\cos }^{2}2A)/2$. Thus complete reduced dephasing at one time does not imply irreversible global collapse.

Prepare a qubit in $|+⟩$, apply a phase gate $P(\varphi )=\mathrm{diag}(1,e^{i\varphi })$, and correlate its computational basis states with normalized environment states $|e_0⟩,|e_1⟩$. Write $⟨e_0|e_1⟩=v{e}^{i\alpha }$. After a final Hadamard gate, the environment amplitude accompanying output zero is $(|e_0⟩+e^{i\varphi }|e_1⟩)/2$. Its squared norm is $P_0=[1+v\cos (\varphi +\alpha )]/2$. Thus $v$ is the interference visibility, while $\alpha$ shifts the fringe. This quantum decoherence map commutes with the phase gate, so placing it immediately before that gate gives the same probabilities.

To identify the physical channel $\rho \mapsto V\rho V^{†}$ of a single-qubit unitary gate, prepare each of the three positive Pauli matrix eigenstates and estimate each output Pauli expectation. Their nine means areThey determine the Bloch vector rotation $\mathbf{r}\mapsto R\mathbf{r}$, hence the channel on all states. If two gates have the same channel, their relative unitary commutes with all three Pauli matrices: commuting with $Z$ forces it diagonal and commuting with $X$ forces equal diagonal entries. Thus they differ by a scalar global phase, proving uniqueness up to that phase. This procedure needs no controlled gate or eigenstate of the unknown gate. Estimating its bounded measurement means to error $ϵ$ with fixed confidence uses order ${ϵ}^{-2}$ repetitions. It determines the relative eigenphases but cannot determine an overall eigenvalue phase from uncontrolled gate access alone.

If $|b⟩$ lies in the kernel of $A$, then $A|b⟩=0$ has no normalized quantum state. A request to produce that normalized vector with positive probability is consequently undefined. For example, $A=\mathrm{diag}(0,1/2)$ on one qubit has distinct dyadic eigenvalues, but annihilates $|0⟩$. The corrected filtering assertion requires $A|b⟩\ne 0$; it cannot be repaired by assigning a positive success probability to this input.