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Entanglement dilution uses local operations and classical communication to prepare partially entangled states from maximally entangled states, such as Bell pairs. For many identical bipartite pure targets, the asymptotic cost in Bell pairs per target, allowing a vanishing approximation error, is their entanglement entropy in bits. The quantum typical subspace of the target's Schmidt coefficients has almost all its weight and dimension at most $2^{m(S+\delta )}$. A maximally entangled state of this dimension can be converted to the normalized truncated target by Nielsen's pure-state conversion theorem. The resulting state approaches the full target as the retained weight tends to one. Average monotonicity of pure-state entanglement entropy and entropy continuity give the converse lower bound. This is the reverse asymptotic task to entanglement concentration.

For $0<p<1$, every bipartition of ${\Psi }_p^{\otimes n}$ has Schmidt rank $2^n$, while $m$ copies of a GHZ state have Schmidt rank $2^m$. Schmidt-rank contraction under product operators therefore forbids any exact successful LOCC branch when $m<n$. Approximate entanglement dilution avoids this obstruction by retaining only a high-weight quantum typical subspace, whose dimension has logarithm $n{h}_2(p)+o(n)$. Exact finite-block conversion and asymptotically faithful conversion are different tasks.

The Schumacher compression theorem concerns a memoryless quantum information source with average density matrix $\rho$ on a finite-dimensional Hilbert space. For $n$ uses the average state is ${\rho }_n={\rho }^{\otimes n}$. Compression consists of an encoding quantum channel into a space of dimension $D_n$ and a decoding channel back to the original block space, with asymptotic rate $n^{-1}{\log }_2{D}_n$ qubits per source use. The optimal reliable rate is $S(\rho )$: every larger rate is achievable, and no smaller rate is reliable. This holds for average pure-source fidelity; the direct construction also preserves a purification reference, and the converse for that stronger entanglement fidelity criterion holds with fidelity tending to zero below the threshold. The encoding and decoding are deterministic channels; no free classical transmission of the source label is allowed.

Diagonalize $\rho ={\sum }_j{\lambda }_j|j⟩⟨j|$ and omit zero-probability eigenvalues when sampling. Define the quantum typical subspace projector $P_{n,\delta }$ by retaining product eigenvectors withThe sampled random variable $-{\log }_2{\lambda }_j$ has expectation $S=S(\rho )$. The weak law of large numbers gives $w_n=\mathrm{Tr}({\rho }_n{P}_{n,\delta })\to 1$. Every retained eigenvalue is at least $2^{-n(S+\delta )}$, and the total probability is at most one, soThis proves the two typical subspace theorem estimates used in the direct part.

Write $P=P_{n,\delta }$. Encode its range isometrically into a compressed space and reserve one orthogonal failure flag. Decode the range back isometrically and the failure flag to a fixed pure state $|\phi ⟩$. This typical-subspace compression with a failure flag is a physical channel whose composite action isIndeed its Kraus operators are $P$ and $|\phi ⟩⟨e_a|$, where the $e_a$ form an orthonormal basis of $\ker P$, and their adjoint products sum to $I$. The Kraus formula for entanglement fidelity givesFor any pure-state ensemble of ${\rho }_n$, the average squared quantum fidelity is at least this number: apply convexity of $|z{|}^2$ separately to the amplitudes $⟨{\psi }_x|K_{\alpha }|{\psi }_x⟩$ with their source probabilities. Thus average fidelity also tends to one. The compressed dimension is at most $2^{n(S+\delta )}+1$, and its rate is at most $S+\delta +o(1)$. Choose $\delta <R-S$ for any prescribed $R>S$. One may also take ${\delta }_n=n^{-1/4}$: the finite spectral information variance and Chebyshev's inequality still give $w_n\to 1$, with rate tending to $S$. A pure source, $S=0$, can simply be re-prepared exactly without sending any qubits.

For the coherent converse, we prove the finite-dimensional quantum compression converse. If the compressed space has dimension $D$, the state of reference plus compressed system has a pure-state decomposition with Schmidt rank at most $D$. Applying decoder Kraus operators locally cannot increase the Schmidt rank of those vectors, so the final reference–output state has Schmidt number at most $D$. Let $|{\Psi }_{{\rho }_n}⟩$ purify the source, with squared Schmidt coefficients equal to the decreasing eigenvalues ${\mu }_1\ge {\mu }_2\ge \cdots$ of ${\rho }_n$. For any normalized vector $|v⟩$ of Schmidt rank at most $D$, its reference support lies in a rank-at-most-$D$ projector $Q$. HenceThe last bound follows by maximizing ${\sum }_j{\mu }_j⟨j|Q|j⟩$ subject to $0\le ⟨j|Q|j⟩\le 1$ and their sum at most $D$. Average over a Schmidt-rank-bounded decomposition of the final state to obtain $F_e\le {\sum }_{j=1}^D{\mu }_j$.

At rate $R<S$, choose $0<\delta <S-R$. The total source probability outside the quantum typical subspace tends to zero, while every eigenvalue inside it is at most $2^{-n(S-\delta )}$. Therefore for $D_n\le 2^{nR}$,This argument allows arbitrary encoding and decoding channels, not only subspace projection.

For completeness, average pure-source reliability also forces the same rate, even though it is a weaker criterion than entanglement fidelity. Here is a separate converse for that formulation. Let the emitted block pure states be ${\psi }_x$, with probabilities $p_x$, and let ${\sigma }_x$ be the decoded states. If their mean squared fidelity tends to one, then their mean trace distance ${\bar{\delta }}_n$ tends to zero, since $\frac{1}{2}\Vert {\sigma }_x-{\psi }_x{\Vert }_1\le \sqrt{1-⟨{\psi }_x|{\sigma }_x|{\psi }_x⟩}$. This inequality follows by choosing a purification with that overlap and using contraction of trace distance under a partial trace.

We need only an elementary entropy continuity from Jordan decomposition bound, derivable from Question 4. If $\delta =\frac{1}{2}\Vert \sigma -\tau {\Vert }_1$, write the positive and negative parts of $\sigma -\tau$ as $\delta {\omega }_{+},\delta {\omega }_{-}$. The common density matrixand the two mixture bounds giveHere $g$ is increasing and concave, $g(0)=0$. Apply this in block dimension $d^n$, both to the average states and to individual outputs, using convexity of trace distance and concavity of $g$. Since the original ensemble is pure, its Holevo quantity is $S({\rho }_n)=nS$. The decoded ensemble hasThe Holevo quantity under a quantum channel cannot increase. To specify the entropy ingredient, Strong subadditivity of Von Neumann entropy states $S(XB)+S(BE)\ge S(B)+S(XBE)$. Attach the classical source label $X$, dilate the decoder isometrically to $BE$, and trace out $E$. The displayed inequality says $I(X:B)\le I(X:BE)$; the isometry preserves the input mutual information. Since $I(X:B)$ of a labelled ensemble is its Holevo quantity, this proves the needed decoder monotonicity. Before decoding, the system has dimension $D_n$, so its Holevo quantity is at most ${\log }_2{D}_n$. Thus ${\log }_2{D}_n\ge nS-o(n)$ and the rate cannot be below $S$. Both operational fidelity formulations therefore have the claimed noiseless coding threshold.

For fixed $\epsilon >0$, let the quantum typical subspace ${\mathcal{T}}_{\epsilon }^{(n)}$ be spanned by the product eigenvectors with nonzero eigenvalues satisfyingEquivalently their spectral information $-n^{-1}{\log }_2{\lambda }_{\mathbf{j}}$ lies within $\epsilon$ of $S(\pi )$. Denote its orthogonal projection by $P_{\epsilon }^{(n)}$.

The typical subspace theorem states that for every $\delta >0$ there is $n_0$ such that for $n\ge n_0$,Also the operator bounds on its range areTo see the concentration statement, sample $j_s$ independently with probabilities ${\lambda }_j$. The mean of $-{\log }_2{\lambda }_{j_s}$ is $S(\pi )$, so the weak law of large numbers gives typical probability tending to one. Zero-eigenvalue letters have probability zero and are excluded. The dimension bounds follow by summing the upper and lower typical eigenvalue bounds and using the typical mass between $1-\delta$ and $1$.

Use base-two logarithms throughout, so classical rates are measured in bits and quantum compression rates in qubits. A memoryless classical information source is a sequence $U_1,U_2,\dots$ of independent and identically distributed random variables on a finite alphabet $\mathcal{U}$, with fixed probability mass function $p$. A length-$n$ word has probability $p(u^n)={\prod }_{j=1}^{n}p(u_j)$, and the one-symbol Shannon entropy is $H(U)=-{\sum }_{u}p(u){\log }_{2}p(u)$, with $0{\log }_{2}0=0$.

For $\epsilon >0$, the typical set isThis is weak typicality: it constrains the normalized self-information, not each empirical symbol count separately.

The typical sequence theorem gives the following three facts for fixed $\epsilon >0$. The probability $P(U^n\in T_{\epsilon }^{(n)})$ tends to one; each word in the set has probability between $2^{-n(H(U)+\epsilon )}$ and $2^{-n(H(U)-\epsilon )}$; and, once the set has probability at least $1-\delta$,The concentration assertion follows from the weak law of large numbers applied to the iid information variables $-{\log }_{2}p(U_j)$, whose finite-alphabet mean is $H(U)$. The probability bounds are the definition of typicality exponentiated. Summing the lower bound over the typical words and using total probability at most one proves the cardinality upper bound; summing the upper bound and using total typical probability at least $1-\delta$ proves the lower bound.

Fix $R>H(U)$ and choose $0<\epsilon <R-H(U)$. Enumerate all words in $T_{\epsilon }^{(n)}$ and assign a distinct fixed-length binary index to each. Reserve one more index as a failure symbol for every atypical input. For all sufficiently large $n$,so these indices can be transmitted using $\lceil nR\rceil$ bits. The decoder inverts the typical-word enumeration; on the failure index it returns any fixed word. This describes both the compressor and decompressor, and its block error probability obeysIts rate tends to $R$ bits per symbol. Thus every $R>H(U)$ permits reliable fixed-rate compression, with no claim that atypical words are reconstructed correctly.

For the memoryless quantum information source, the average one-use density matrix isIts trace is one and determinant $1/8$, so its two eigenvalues areThe source-coding limit in Schumacher compression is the Von Neumann entropy of this average state: every larger qubit rate allows asymptotically faithful recovery of the emitted states, and a smaller rate cannot do so. Applying that limit giveswhere $h_2(t)=-t{\log }_{2}t-(1-t){\log }_2(1-t)$ is the binary entropy. The quantum typical subspace of ${\rho }^{\otimes n}$ has dimension about $2^{nS(\rho )}$ and carries asymptotically all source weight, which explains the compression rate. The entropy of the classical emission label is one bit; it is not the quantum rate because the two emitted pure states are nonorthogonal.

Use the usual asymptotic convention: the output approaches the desired $m(n)$-copy pure target in trace distance, and any recorded failure probability tends to zero. Define the binary entropyThe entanglement entropies per source and target are $E_{\psi }=h_2(s)$ and $E_{\varphi }=h_2(t)$. The asymptotic pure-state entanglement conversion rate isBoth achievability and the upper bound matter here.

For entanglement concentration, expand $n$ sources in their correlated computational basis. Every pair of strings of Hamming weight $k$ has the same amplitude $(1-s{)}^{(n-k)/2}{s}^{k/2}$. Alice measures only this weight, without resolving the individual string. Bob's string has the same weight automatically. Conditional on $k$, the normalized state is maximally entangled on $D_k=\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ correlated strings. Its outcome probability is $\left(\genfrac{}{}{0ex}{}{n}{k}\right)s^k(1-s{)}^{n-k}$, a binomial distribution. By the weak law of large numbers, $k/n\to s$ in probability; Stirling's formula then gives $n^{-1}{\log }_2{D}_k\to h_2(s)$. A uniform Schmidt vector on $D_k$ entries is majorized by a uniform vector on $2^{\lfloor {\log }_2{D}_k\rfloor }$ entries padded with zeros. Thus Nielsen's pure-state conversion theorem converts the conditional state into that many Bell pairs exactly. For any fixed $\delta >0$, one obtains at least $n(E_{\psi }-\delta )$ Bell pairs with probability tending to one.

For entanglement dilution, consider the Schmidt strings of $m$ targets with probabilities $q_x$. The quantum typical subspace retains strings satisfyingThe retained probability $Q_m$ tends to one by the weak law applied to the independent variables $-{\log }_2{q}_{x_j}$. The number of retained strings is at most $2^{m(E_{\varphi }+\delta )}$, because their probabilities sum to at most one. Normalize the target after truncation to those strings. Its squared overlap with the full target is $Q_m\to 1$. A uniform Schmidt vector of dimension $2^L$, with $L=\lceil m(E_{\varphi }+\delta )\rceil$, is majorized by the truncated target's Schmidt vector padded with zeros: the sum of its largest $k$ entries is at least $k/2^L$. Hence $L$ Bell pairs prepare the truncated target exactly by LOCC. Its trace distance from the true target is $\sqrt{1-Q_m}\to 0$. Combining concentration and dilution achieves any rate $R<E_{\psi }/E_{\varphi }$. Choosing margins tending sufficiently slowly to zero attains the ratio as a limiting rate.

For the converse, average monotonicity of pure-state entanglement entropy follows from Concavity of Von Neumann entropy: a measurement by one party leaves the other party's reduction unchanged on average, so its average conditional entropy cannot increase. Refine local measurement records, including local discarded systems, to obtain pure output branches $\{p_r,|{\chi }_r⟩\}$ on the $m$ output qubits of each party. Iterating the concavity argument givesLet $|{\Phi }_m⟩=|\varphi {⟩}^{\otimes m}$ and let the squared-overlap error of the average output ${\rho }_{\mathrm{out}}$ be $\epsilon =1-⟨{\Phi }_m|{\rho }_{\mathrm{out}}|{\Phi }_m⟩\to 0$. Define $d_r=\sqrt{1-|⟨{\Phi }_m|{\chi }_r⟩{|}^2}$, the pure-state trace distance. Concavity of the square root gives ${\sum }_r{p}_r{d}_r\le \sqrt{\epsilon }$. Partial trace cannot increase trace distance, so the reduced states are also within $d_r$. The continuity bound for quantum conditional entropy, with trivial conditioning and local dimension $2^m$, impliesConsequentlywhich excludes a limiting rate exceeding $E_{\psi }/E_{\varphi }$.

The vanishing-error convention is essential. If exact deterministic conversion were required for every finite block, majorization would also require the largest squared Schmidt coefficient to satisfy $(1-s{)}^n\le (1-t{)}^m$, henceThis can be strictly smaller than the entropy ratio: at $s=0.1,t=0.2$, the two bounds are approximately $0.4722$ and $0.6496$. Thus the boxed rate answers the standard asymptotic task, rather than imposing unmentioned exact zero-error conditions on every finite block.

Diagonalize the source average $\pi ={\sum }_j{\lambda }_j|j⟩⟨j|$, using its positive eigenvalues. For fixed $\delta >0$, let ${\Pi }_{n,\delta }$ project onto product eigenvectors whose eigenvalues obeyThe typical subspace theorem states that, for every $ϵ>0$ and all sufficiently large $n$,The probability statement is the weak law of large numbers applied to $-{\log }_2{\lambda }_j$; the dimension bounds follow by summing the typical eigenvalue bounds. This explains why the quantum typical subspace retains almost all probability using about $nS(\pi )$ qubits.

For $R>S(\pi )$ choose $0<\delta <R-S(\pi )$. Measure $\{{\Pi }_{n,\delta },I-{\Pi }_{n,\delta }\}$. On success, encode the projected state isometrically into a space of dimension $\dim {\mathcal{T}}_{n,\delta }$; on failure, output a separate fixed flag. Decode the successful sector by the inverse isometry, and map the flag to a fixed state $|{\phi }_0⟩$. The compressed dimension is at most $2^{n(S(\pi )+\delta )}+1$, hence fits within rate $R$ for all sufficiently large $n$. Both maps are trace-preserving quantum channels, not merely successful postselected operations.

Writing $\Pi ={\Pi }_{n,\delta }$, the composite channel isFor a pure source signal $|{\Psi }_k⟩$, set $a_k=⟨{\Psi }_k|\Pi |{\Psi }_k⟩$. Its squared quantum fidelity after the composite channel is at least $a_k^2$. Since the source is memoryless, its average state on $n$ uses is ${\pi }^{\otimes n}$. Convexity of the square givesThe Kraus formula for entanglement fidelity gives the same lower bound for $F_e({\pi }^{\otimes n},{\mathcal{N}}_n)$, because one Kraus operator is $\Pi$ and the other terms are nonnegative. Taking $ϵ\to 0$ proves reliable compression at every $R>S(\pi )$, in both average pure-signal fidelity and the stronger entanglement-fidelity sense. This is typical-subspace compression with a failure flag.

Memorylessness means that the ensemble's average density operator is ${\rho }^{(n)}={\pi }^{\otimes n}$. Its spectral decomposition isThe product eigenvalues giveZero eigenvalues contribute zero to the Von Neumann entropy and are excluded from logarithmic typicality tests.

The quantum typical subspace is the span of eigenvectors whose eigenvalues satisfyIts orthogonal projection $P_{\epsilon }^{(n)}$ selects precisely the classical typical set for the eigenvalue distribution $q$. The typical-set cardinality bounds and the weak law of large numbers therefore givefor any fixed $\delta >0$ and all sufficiently large $n$.

A compression-decompression scheme consists of quantum channels ${\mathcal{E}}_n:\mathcal{D}({\mathcal{H}}^{\otimes n})\to \mathcal{D}({\mathcal{K}}_n)$ and ${\mathcal{D}}_n:\mathcal{D}({\mathcal{K}}_n)\to \mathcal{D}({\mathcal{H}}^{\otimes n})$, where ${\mathrm{limsup}}_n{n}^{-1}{\log }_2\dim {\mathcal{K}}_n\le R$. For the emitted pure-state ensemble, reliability means that the mean squared quantum fidelity tends to one:The average is taken over the actual source distribution, rather than the worst possible input vector.

A standard stronger formulation of reliable quantum source compression requireswhere entanglement fidelity tests preservation of a purification and its reference-system correlations. It implies the mean-fidelity condition for every pure-state ensemble of ${\pi }^{\otimes n}$. Schumacher compression achieves this at any rate $R>S(\pi )$: choose $0<\epsilon <R-S(\pi )$, encode the quantum typical subspace, and map atypical outcomes to a fallback state. The composed channel has a Kraus term $P_{\epsilon }^{(n)}$, so its entanglement fidelity is at least $[\mathrm{Tr}({\pi }^{\otimes n}{P}_{\epsilon }^{(n)}){]}^2\to 1$.

A quantum source-compression scheme has encoding and decoding quantum channels with compressed space ${\mathcal{K}}_n$ and rate ${\mathrm{limsup}}_n{n}^{-1}{\log }_2\dim {\mathcal{K}}_n$. For a pure-state source ensemble, average reliability means that the mean squared quantum fidelity after decoding tends to one. A stronger coherent criterion is $F_e({\pi }^{\otimes n},{\mathcal{D}}_n\circ {\mathcal{E}}_n)\to 1$, where entanglement fidelity also tests a purification reference. Encoding the quantum typical subspace achieves both criteria above $S(\pi )$. Schumacher's original quantum coding paper establishes quantum source coding.

Measure the quantum typical subspace projector $\Pi$. Encode its successful sector isometrically and reserve one orthogonal compressed flag for failure; decode the flag to a fixed state. The composite channel is ${\mathcal{N}}_n(\tau )=\Pi \tau \Pi +\mathrm{Tr}[(I-\Pi )\tau ]|{\phi }_0⟩⟨{\phi }_0|$. It is trace preserving. The Kraus formula for entanglement fidelity gives the displayed lower bound, while convexity of the square gives the same bound for average pure-source fidelity. One flag adds only a vanishing asymptotic rate overhead.

For each fixed $\delta >0$, the quantum typical subspace projector selects eigenvalues of ${\pi }^{\otimes n}$ between $2^{-n(S(\pi )+\delta )}$ and $2^{-n(S(\pi )-\delta )}$. Its probability tends to one by the weak law of large numbers for the spectral information variable. If its probability is at least $1-ϵ$, its dimension is between $(1-ϵ)2^{n(S(\pi )-\delta )}$ and $2^{n(S(\pi )+\delta )}$. These statements give the size and concentration estimates behind Schumacher compression.