Schmidt decomposition

The Schmidt decomposition, named after the German mathematician Erhard Schmidt who introduced it in his 1906–1907 work on integral equations, is a canonical form for representing vectors in the tensor product of two Hilbert spaces, analogous to the singular value decomposition for matrices.^[1] In the context of quantum mechanics, it provides a unique expansion for any pure state |\psi\rangle of a bipartite system \mathcal{H}_A \otimes \mathcal{H}_B as |\psi\rangle = \sum_i \sqrt{\lambda_i} |u_i\rangle_A |v_i\rangle_B, where \{|u_i\rangle_A\} and \{|v_i\rangle_B\} are orthonormal bases for the subsystems \mathcal{H}_A and \mathcal{H}_B, respectively, and the \lambda_i are non-negative real numbers (Schmidt coefficients) that sum to 1 and are ordered non-increasingly. This decomposition is fundamental in quantum information theory because it directly quantifies entanglement in pure bipartite states: the Schmidt rank, defined as the number of non-zero \lambda_i, measures the effective dimensionality of the entanglement, while the von Neumann entropy of the reduced density operator S(\rho_A) = -\sum_i \lambda_i \log_2 \lambda_i serves as the entanglement entropy, providing a precise metric for the state's inseparability. The theorem guarantees the existence and uniqueness of this form (up to irrelevant phase factors in the bases), making it a cornerstone for analyzing quantum correlations, state preparation, and protocols like quantum teleportation and dense coding. Unlike for multipartite systems, where no universal Schmidt-like decomposition exists, the bipartite case fully captures the entanglement structure through local bases and coefficients, facilitating efficient numerical computations via singular value decomposition algorithms. Applications extend to continuous-variable systems, where it aids in modal analysis of Gaussian states, and to error correction, where Schmidt coefficients inform the fidelity of noisy entangled resources.

Bipartite Pure States

Formal Statement

In quantum mechanics, a bipartite pure state is represented by a normalized vector |\psi\rangle in the tensor product of two complex Hilbert spaces \mathcal{H}_A \otimes \mathcal{H}_B, where \mathcal{H}_A and \mathcal{H}_B describe the subsystems A and B, respectively. The Schmidt decomposition theorem asserts that any such pure bipartite state admits a canonical form

|\psi\rangle = \sum_{i=1}^r \lambda_i |u_i\rangle_A \otimes |v_i\rangle_B,

where r \leq \min(\dim \mathcal{H}_A, \dim \mathcal{H}_B) is the Schmidt rank, the coefficients \{\lambda_i\}_{i=1}^r are positive real numbers (the Schmidt coefficients) satisfying \sum_{i=1}^r \lambda_i^2 = 1, and \{|u_i\rangle_A\}_{i=1}^r, \{|v_i\rangle_B\}_{i=1}^r are orthonormal sets spanning the relevant subspaces of \mathcal{H}_A and \mathcal{H}_B. The Schmidt coefficients \{\lambda_i\} are uniquely determined by the state |\psi\rangle, while the orthonormal bases \{|u_i\rangle_A\} and \{|v_i\rangle_B\} are unique up to individual phase factors e^{i\theta_i} for each i, such that the overall decomposition remains invariant under these transformations. This quantum decomposition is analogous to the singular value decomposition of matrices in linear algebra.

Proof Sketch

The Schmidt decomposition theorem can be proved constructively using techniques from linear algebra, specifically the singular value decomposition (SVD). Consider a bipartite pure state |\psi\rangle_{AB} in the tensor product Hilbert space \mathcal{H}_A \otimes \mathcal{H}_B, expressed in a product basis as

|\psi\rangle_{AB} = \sum_{j=1}^{d_A} \sum_{k=1}^{d_B} c_{jk} |j\rangle_A |k\rangle_B,

where \{|j\rangle_A\} and \{|k\rangle_B\} are orthonormal bases for \mathcal{H}_A and \mathcal{H}_B, respectively, with dimensions d_A = \dim \mathcal{H}_A and d_B = \dim \mathcal{H}_B, and the coefficients satisfy \sum_{j,k} |c_{jk}|^2 = 1. Form a complex d_A \times d_B matrix C whose elements are the coefficients, i.e., C_{jk} = c_{jk}. Apply the SVD to this matrix:

C = U \Sigma V^\dagger,

where U is a d_A \times d_A unitary matrix, V is a d_B \times d_B unitary matrix, and \Sigma is a d_A \times d_B rectangular diagonal matrix containing the singular values \lambda_i (non-negative real numbers) on its main diagonal, with at most r = \min(d_A, d_B) non-zero entries ordered such that \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_r > 0. The normalization of |\psi\rangle ensures \sum_i \lambda_i^2 = 1. The columns of U and V provide the required orthonormal bases for the decomposition. Define

|u_i\rangle_A = \sum_{j=1}^{d_A} U_{ji} |j\rangle_A, \quad |v_i\rangle_B = \sum_{k=1}^{d_B} V_{ki} |k\rangle_B

for i = 1, \dots, r. These states form orthonormal sets in \mathcal{H}_A and \mathcal{H}_B, respectively, because U and V are unitary, ensuring \langle u_i | u_{i'}\rangle_A = \delta_{ii'} and \langle v_i | v_{i'}\rangle_B = \delta_{ii'}. Substituting the SVD into the expansion of |\psi\rangle yields

|\psi\rangle_{AB} = \sum_{i=1}^r \lambda_i |u_i\rangle_A |v_i\rangle_B,

completing the decomposition, with the Schmidt rank r bounding the number of terms.

Key Properties

Reduced Density Operators

In the context of a bipartite pure state |\psi\rangle_{AB} = \sum_{i=1}^r \sqrt{\lambda_i} |u_i\rangle_A |v_i\rangle_B, where the Schmidt coefficients \lambda_i are non-negative real numbers satisfying \sum_{i=1}^r \lambda_i = 1 and \{|u_i\rangle_A\}, \{|v_i\rangle_B\} are orthonormal bases for subsystems A and B, respectively, the reduced density operator for subsystem A is obtained by performing the partial trace over B:

\rho_A = \operatorname{Tr}_B(|\psi\rangle\langle\psi|) = \sum_{i=1}^r \lambda_i |u_i\rangle\langle u_i|_A.

The partial trace operation effectively sums over the degrees of freedom in B, yielding a diagonal form in the Schmidt basis for A.^[2]^[3] Similarly, the reduced density operator for B is

\rho_B = \operatorname{Tr}_A(|\psi\rangle\langle\psi|) = \sum_{i=1}^r \lambda_i |v_i\rangle\langle v_i|_B,

which is diagonal in the corresponding Schmidt basis for B.^[2]^[3] To see that \rho_A and \rho_B share the same non-zero eigenvalues, note that the Schmidt decomposition arises from the singular value decomposition of the coefficient matrix representing |\psi\rangle, where the singular values are precisely the \sqrt{\lambda_i}.^[2] The eigenvalues of \rho_A are thus \{\lambda_i\}_{i=1}^r (with additional zero eigenvalues if the local dimensions exceed r), and the same spectrum holds for \rho_B due to the symmetry in the decomposition.^[3] This shared spectrum reflects the intrinsic correlation between the subsystems, as the partial trace preserves the singular value structure. The Schmidt bases \{|u_i\rangle_A\} and \{|v_i\rangle_B\} therefore diagonalize \rho_A and \rho_B, respectively, providing a canonical representation that captures the local mixedness induced by entanglement in the global pure state.^[2]^[3] A key property of these reduced operators is their purity, quantified by \operatorname{Tr}(\rho_A^2) = \sum_{i=1}^r \lambda_i^2. Since \sum_{i=1}^r \lambda_i = 1 and the \lambda_i are non-negative, it follows that \sum_{i=1}^r \lambda_i^2 \leq 1, because (\sum_{i=1}^r \lambda_i)^2 = \sum_{i=1}^r \lambda_i^2 + 2 \sum_{i<j} \lambda_i \lambda_j = 1 \geq \sum_{i=1}^r \lambda_i^2, with equality if and only if r=1 (i.e., the state is a product state).^[2] The same holds for \operatorname{Tr}(\rho_B^2). This measure of purity thus encodes the extent to which the local description deviates from a pure state, directly tied to the distribution of the Schmidt coefficients.^[3]

Schmidt Coefficients and Rank

In the Schmidt decomposition of a bipartite pure state |\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B, the Schmidt coefficients \lambda_i are the non-negative real numbers satisfying \sum_i \lambda_i = 1, appearing as eigenvalues in the expansion |\psi\rangle = \sum_i \sqrt{\lambda_i} |u_i\rangle_A |v_i\rangle_B, where the states |u_i\rangle_A and |v_i\rangle_B form orthonormal bases for their respective subspaces, and only terms with \lambda_i > 0 contribute. The Schmidt rank r, also known as the Schmidt number, is defined as the number of non-zero Schmidt coefficients \lambda_i. This rank equals the dimension of the support of the decomposition and corresponds directly to the rank of the reduced density operator \rho_A = \mathrm{Tr}_B(|\psi\rangle\langle\psi|) or \rho_B = \mathrm{Tr}_A(|\psi\rangle\langle\psi|), both of which share the same non-zero eigenvalues \lambda_i. The Schmidt rank provides a measure of the effective dimensionality of entanglement in the bipartite system. A state with r = 1 is separable, meaning it can be expressed as a product state |\psi\rangle = |\phi\rangle_A \otimes |\chi\rangle_B with no entanglement, as the decomposition reduces to a single term. In contrast, r > 1 indicates an entangled state, where the non-zero coefficients reflect correlations that cannot be factored into independent local descriptions. The maximum possible Schmidt rank is bounded by r \leq \min(\dim \mathcal{H}_A, \dim \mathcal{H}_B), determined by the smaller Hilbert space dimension, which limits the number of independent basis states available for the expansion. This rank quantifies the "effective dimension" of the entanglement support, revealing how many orthonormal product terms are required to span the state without redundancy. Notably, while the full density operator |\psi\rangle\langle\psi| of a pure state always has rank 1, the Schmidt rank focuses on the entanglement structure across the bipartition and can be much higher, highlighting the multipartite nature inherent in entangled systems.

Entanglement Entropy

The entanglement entropy provides a quantitative measure of entanglement for a bipartite pure state |\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B, derived from the von Neumann entropy of the reduced density operator \rho_A = \Tr_B(|\psi\rangle\langle\psi|), whose eigenvalues are the Schmidt coefficients \lambda_i. This measure, denoted E(|\psi\rangle), captures the degree of quantum correlations between the subsystems and serves as a fundamental resource quantifier in quantum information theory. The entanglement entropy is formally defined as

S(\rho_A) = -\Tr(\rho_A \log_2 \rho_A) = -\sum_{i=1}^r \lambda_i \log_2 \lambda_i,

where r is the Schmidt rank and \lambda_i are the Schmidt coefficients with \sum_i \lambda_i = 1. For separable states, where r=1 and \rho_A is a pure state projector, E(|\psi\rangle) = 0. In contrast, it reaches its maximum value of \log_2 \min(\dim \mathcal{H}_A, \dim \mathcal{H}_B) for maximally entangled states, such as the uniform superposition over a balanced basis, reflecting complete delocalization of information across subsystems. This entropy exhibits additivity for tensor product states, satisfying E(|\psi\rangle \otimes |\phi\rangle) = E(|\psi\rangle) + E(|\phi\rangle), due to the independence of the reduced operators. It is also monotonic non-increasing under local operations and classical communication (LOCC), ensuring that entanglement cannot be created locally but can only be diluted or preserved. Introduced in quantum information theory during the 1990s, the entanglement entropy emerged as a cornerstone for assessing entanglement as a resource in protocols like distillation and teleportation.

Multipartite Extensions

Schmidt Rank Vector

In multipartite quantum systems, the Schmidt rank vector extends the notion of the Schmidt rank from bipartite pure states to capture the entanglement structure across multiple bipartitions. For a tripartite pure state |\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C, the Schmidt rank vector is defined as (r_{AB}, r_{AC}, r_{BC}), where r_{AB} denotes the Schmidt rank across the A|BC cut (equivalently, the rank of the reduced density operator \rho_A = \mathrm{Tr}_{BC}(|\psi\rangle\langle\psi|)), r_{AC} is the rank across the B|AC cut (rank of \rho_B), and r_{BC} is the rank across the C|AB cut (rank of \rho_C).^[4]^[5] This vector provides a compact characterization of the entanglement dimensionality, with each component reflecting the effective dimension of the entangled subspace for the corresponding bipartition.^[4] The components of the Schmidt rank vector can be constructed directly from the ranks of the partial traces over the appropriate subsystems, as each r_{XY} corresponds to the dimension of the support of the reduced density matrix on the single subsystem side of the cut.^[5] Alternatively, it can be obtained through successive bipartite Schmidt decompositions: for example, first performing the decomposition across one cut (e.g., A|BC) yields |\psi\rangle = \sum_k \sqrt{\lambda_k} |\phi_k\rangle_A |\chi_k\rangle_{BC}, after which the states |\chi_k\rangle_{BC} can be further decomposed across B|C to determine the relevant ranks.^[4] In the bipartite case, the Schmidt rank vector collapses to a single scalar r, corresponding to the unique bipartition.^[5] The components of the Schmidt rank vector satisfy nontrivial inequalities that encode constraints from the multipartite entanglement structure. In particular, r_{AB} \leq \min(r_{AC} r_{BC}, \dim \mathcal{H}_A), with analogous bounds for the other components; these arise from the submultiplicativity of matrix ranks under partial traces, ensuring that the support of one reduced density matrix is contained within the tensor product of supports from finer bipartitions.^[5] Such relations highlight how multipartite systems impose tighter bounds than independent bipartite entanglements, influencing classifications of entanglement types.^[4]

Applications to Multipartite Entanglement

The Schmidt rank vector plays a crucial role in detecting genuine multipartite entanglement (GME) in pure quantum states. For an N-partite system, the vector consists of the Schmidt ranks across all possible bipartitions, where a rank greater than 1 for every entry indicates that the state cannot be separated into a product across any cut, confirming GME rather than biseparability.^[6] This property extends the bipartite Schmidt rank to multipartite settings, providing a necessary condition for full inseparability and enabling the identification of states useful for quantum information protocols requiring collective entanglement, such as multipartite quantum secret sharing. In the context of stochastic local operations and classical communication (SLOCC), the Schmidt rank vector serves as an invariant that classifies entanglement types by distinguishing inequivalent orbits under local transformations. States with distinct vectors, such as (4,2,2) and (3,3,2) in a tripartite qudit system, belong to different SLOCC classes and are incomparable under LOCC, highlighting structural differences in their entanglement.^[6] However, the vector does not resolve all distinctions; for instance, the GHZ state |\text{GHZ}\rangle = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle) and the W state |\text{W}\rangle = \frac{1}{\sqrt{3}} (|001\rangle + |010\rangle + |100\rangle) in three qubits both possess the vector (2,2,2), yet they represent fundamentally different entanglement classes under SLOCC. Despite its utility, the Schmidt rank vector has limitations in fully capturing multipartite entanglement structures, as it primarily detects dimensionality without quantifying the degree or type of correlations. It fails to differentiate states within the same vector class, necessitating complementary measures such as entanglement entropy computed across bipartitions to assess the effective entanglement strength—for example, the von Neumann entropy S(\rho_A) = 1 for GHZ versus approximately 0.918 for W across a single-qubit cut.^[6] This combination is essential for comprehensive analysis in tasks like entanglement certification and quantum network design. The concept underlying the Schmidt rank vector emerged in the early 2000s amid efforts to extend bipartite entanglement tools to multipartite quantum information processing, with foundational work on multipartite Schmidt ranks appearing around 2001 and formal vectorization developed by 2013 to address classification challenges in higher-dimensional systems.^[7]

Illustrative Examples

Two-Qubit Bipartite Case

A quintessential example of the Schmidt decomposition in the two-qubit bipartite setting is the Bell state |\Phi^+\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle + |11\rangle \right). This state is already expressed in its canonical Schmidt form, featuring two equal Schmidt coefficients \lambda_1 = \lambda_2 = \frac{1}{2} and orthonormal bases \{ |0\rangle, |1\rangle \} for both qubits.^[8] Tracing out the second qubit yields the reduced density operator \rho_A = \mathrm{Tr}_B \left( |\Phi^+\rangle\langle\Phi^+| \right) = \frac{1}{2} I, where I denotes the $2 \times 2 identity matrix; this operator has full rank r = 2 and eigenvalues \frac{1}{2}, \frac{1}{2}. The associated entanglement entropy E = -\sum_i \lambda_i \log_2 \lambda_i = 1 bit, marking maximal entanglement for a two-qubit system.^[8] By contrast, the separable state |00\rangle admits a trivial Schmidt decomposition with a single non-zero coefficient \lambda_1 = 1, local bases \{ |0\rangle \} (effectively rank-1), and reduced density operator \rho_A = |0\rangle\langle 0| of rank r = 1 and entanglement entropy E = 0.^[8] This Schmidt form of the Bell state visually underscores its maximal entanglement, as the equal coefficients fully span the two-dimensional local Hilbert spaces, distinguishing it sharply from product states.^[8]

Three-Qubit Multipartite Case

The three-qubit Greenberger-Horne-Zeilinger (GHZ) state serves as a canonical example for illustrating the Schmidt rank vector in the multipartite setting. This state is defined as

|\mathrm{GHZ}\rangle = \frac{1}{\sqrt{2}} \left( |000\rangle + |111\rangle \right),

where the kets denote the standard computational basis for each qubit. The Schmidt rank vector for this state, which captures the Schmidt ranks across the three possible bipartitions (one qubit versus the other two), is (2, 2, 2). This indicates that the state exhibits maximal Schmidt rank of 2 for every cut, reflecting balanced tripartite entanglement symmetric under permutation of parties.^[9] To compute these ranks, consider the partial trace over one qubit, say the third, yielding the reduced density operator for the first two qubits: \rho_{AB} = \mathrm{Tr}_C (|\mathrm{GHZ}\rangle\langle\mathrm{GHZ}|) = \frac{1}{2} (|00\rangle\langle00| + |11\rangle\langle11|). This operator has rank 2, as it is supported on two orthogonal product states in the AB subspace. Analogous calculations for the other bipartitions confirm the rank-2 reduced states, underscoring the state's full-rank entanglement structure across all partitions.^[9] In contrast, the three-qubit W state,

|W\rangle = \frac{1}{\sqrt{3}} \left( |001\rangle + |010\rangle + |100\rangle \right),

also possesses a Schmidt rank vector of (2, 2, 2). For instance, tracing over the third qubit gives \rho_{AB} = \frac{2}{3} |\Psi^+\rangle\langle\Psi^+| + \frac{1}{3} |00\rangle\langle00|, where |\Psi^+\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle), which again has rank 2 despite the reduced state being entangled. The identical rank vectors for GHZ and W states highlight a key multipartite distinction: while local unitaries preserve these ranks, the two states belong to inequivalent classes under stochastic local operations and classical communication (SLOCC), with GHZ exhibiting a more fragile global phase-sensitive entanglement compared to the permutation-symmetric W state.^[9] The non-product structure of these states is further illuminated through successive Schmidt decomposition, where one first decomposes across a bipartition (e.g., A|BC for GHZ, yielding two terms with correlated bases) and then applies the procedure to the effective subspace on BC. For the GHZ state, this reveals perfectly aligned product bases (|0\rangle|0\rangle and |1\rangle|1\rangle) in the correlated terms, demonstrating the irreducible tripartite coherence that cannot be expressed as a bipartite product even after partial decomposition. The generalized Schmidt form for three qubits, expanding the state in a canonical basis of up to five orthogonal products, similarly requires multiple terms for both GHZ (two terms) and W (three terms), emphasizing their multipartite inseparability beyond bipartite ranks.^[10]^[9]