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research28 min studyLesson 566 in Physics

Matrix product state canonical forms

Relate tensor gauges to orthonormal block bases and identify the Schmidt spectrum without mistaking representation for approximation.

Before you begin
On this page

The question for today

What you’ll understand

  • Write an open-chain MPS with explicit boundary dimensions.
  • Distinguish gauge freedom from physical changes of state.
  • Read Schmidt weights and entanglement entropy from a mixed canonical form.

A generic many-body state lists exponentially many amplitudes. A matrix product state factors those amplitudes into local tensors linked by auxiliary indices. This factorization can represent any finite chain state if bond dimensions are large enough; practical compression comes from restricting those dimensions. Canonical forms then organize the representation so that local tensor coefficients refer to orthonormal left and right blocks. They make normalization, expectation values, and truncation interpretable without turning a finite bond dimension into a universal accuracy guarantee.

A tensor factorization with controlled index sizes

For an open chain, the amplitude of a product configuration is the product of matrices selected by its local physical labels. The first and last bond dimensions are one, making the final product a scalar. With local dimension d and maximum internal dimension D, storage grows roughly as LdD2LdD^2, apart from edge savings and symmetry structure. This estimate is polynomial only when D stays manageable; an exact generic state can require D exponential in chain length.

Bond indices are bookkeeping spaces, not extra physical particles. The Schmidt rank across a cut cannot exceed the dimension of its connecting bond. Conversely, successive singular-value decompositions give an exact MPS whose minimal necessary dimensions track the Schmidt ranks at each cut. A product state needs dimension one everywhere; a state with small local magnetization can still require a large bond dimension if it has broad bipartite entanglement. Local expectation values alone therefore do not determine compressibility.

ψ(s1,…,sL)=A1s1A2s2⋯ALsL,Ajsj∈CDj−1×Dj,D0=DL=1\psi(s_1,\ldots,s_L)=A_1^{s_1}A_2^{s_2}\cdots A_L^{s_L},\qquad A_j^{s_j}\in\mathbb C^{D_{j-1}\times D_j},\qquad D_0=D_L=1
The physical index s_j has d values and the coefficients are dimensionless amplitudes. Open boundary conditions are assumed; a periodic-chain representation instead closes virtual indices with a trace.↔ Scroll wide equations horizontally.

Gauge choices establish orthonormal environments

Inserting an invertible matrix X and its inverse on neighboring virtual legs leaves every physical amplitude unchanged. This gauge freedom means individual tensor entries are not observables. Left-canonical tensors obey a column-isometry condition, making the accumulated left-block states orthonormal. Right-canonical tensors obey the complementary row-isometry condition. QR or singular-value factorizations move the non-isometric center through the chain while preserving the represented state when no nonzero singular values are discarded.

In mixed canonical form, choose a bond between orthonormal left and right blocks. The connecting diagonal matrix contains nonnegative Schmidt coefficients. Their squares are eigenvalues of either reduced density matrix, and they sum to one for a normalized pure state. Normalizing every local tensor independently does not in general normalize the whole MPS, because contractions link their entries. Canonical identities give the appropriate global norm through the remaining center object instead of arbitrary tensor-by-tensor rescaling.

∑s(Ajs)†Ajs=I,∑sBjs(Bjs)†=I,∣ψ⟩=∑αλα∣Lα⟩∣Rα⟩\sum_s(A_j^s)^\dagger A_j^s=I,\qquad \sum_sB_j^s(B_j^s)^\dagger=I,\qquad |\psi\rangle=\sum_\alpha\lambda_\alpha|L_\alpha\rangle|R_\alpha\rangle
A denotes left-canonical tensors and B right-canonical tensors, which need not occur at the same site simultaneously. The Schmidt block states are orthonormal, lambda is nonnegative, and sum lambda squared equals one.↔ Scroll wide equations horizontally.

Canonical spectra expose the representational limit

The entropy of a pure-state bipartition is the Shannon entropy of its Schmidt weights. With at most D nonzero weights, it cannot exceed the natural logarithm of D. This is a rank bound, not a claim that every dimension-D state is maximally entangled. A sharply decaying spectrum may be accurately compressed with modest D; a flat spectrum uses its full rank. Truncating tensors outside a suitable canonical setting can lose the direct relationship between local singular values and physical Schmidt weights.

For one-dimensional gapped ground states with local interactions, limited entanglement often makes MPS methods effective. Critical states, long-range interactions, highly excited states, and real-time entanglement growth can require much larger dimensions. An area-law statement alone should not be read as an automatic numerical certificate for a chosen D and observable. The Bloch instrument supplies two coefficients that can be stipulated as Schmidt coefficients of a paired-state embedding, but it does not contain a chain of tensors or compute a reduced density matrix.

wα=λα2,S=−∑αwαln⁡wα≤ln⁡Dw_\alpha=\lambda_\alpha^2,\qquad S=-\sum_\alpha w_\alpha\ln w_\alpha\leq\ln D
Weights and entropy are dimensionless, with entropy measured in nats. The bound concerns one pure-state cut of an open-chain MPS whose connecting bond dimension is D.↔ Scroll wide equations horizontally.

Work it through

Read a two-weight Schmidt spectrum

Across one cut, a normalized state is 0.8∣L1R1⟩+0.2∣L2R2⟩\sqrt{0.8}|L_1R_1\rangle+\sqrt{0.2}|L_2R_2\rangle with orthonormal block states. Find its Schmidt rank, minimal exact bond dimension, and entropy.

  1. Both coefficients are nonzero, so the Schmidt rank is two. An exact MPS across this cut needs a bond dimension at least two, independent of which gauge represents its local tensors.

  2. Square the coefficients to obtain weights 0.8 and 0.2. Their sum is one, confirming the normalization assumed in the entropy formula.

  3. Evaluate S=−0.8ln⁡0.8−0.2ln⁡0.2≃0.5004S=-0.8\ln0.8-0.2\ln0.2\simeq0.5004 nats. This is below ln⁡2≃0.6931\ln2\simeq0.6931, because the two weights are unequal rather than maximally mixed.

The result

The cut needs exact bond dimension two and has entropy about 0.5004 nats; bond dimension alone does not fix the entropy.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Two stipulated Schmidt weights

Use the displayed basis probabilities as Schmidt weights of a separately stipulated paired-state embedding. Reset and set θ=60∘\theta=60^\circ, ϕ=0∘\phi=0^\circ: predict weights 0.75 and 0.25. Repeat from Reset with both controls 90∘90^\circ, 0∘0^\circ: predict equal weights and the larger entropy ln 2. This pure two-state lab does not create an MPS or measure bipartite entropy.

Two stipulated Schmidt weightsA normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Current readings: Probability |0⟩: 71.1%; Probability |1⟩: 28.9%; Bloch (x, y, z): 0.78, 0.45, 0.42. All parameters are adjustable with the labeled controls below.xyz|0⟩|1⟩Basis probabilitiesP(0)71.1%P(1)28.9%
Bloch vector / P(0)P(1)Phase latitude
Probability |0⟩
71.1%
Probability |1⟩
28.9%
Bloch (x, y, z)
0.78, 0.45, 0.42
t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA Schmidt spectrum concerns a specified bipartition; the lab only provides coefficients for the stated mathematical comparison.

Pause & reason

Check your understanding

What does inserting X and X inverse on neighboring internal MPS legs do when X is invertible and no truncation occurs?

Your turn

Try it yourself

A maximally entangled pair has Schmidt weights one half and one half. Can an open-chain bond of dimension one represent it exactly? Find its entropy and the largest squared overlap of any normalized rank-one approximation.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

A dimension-one bond permits only a product state across the cut; use the largest Schmidt weight for the best rank-one squared overlap.

Compare with the solution

The pair has Schmidt rank two and entropy ln⁡2\ln2, so a dimension-one bond cannot represent it exactly. The best normalized rank-one approximation keeps one Schmidt component and has squared overlap one half with the original state. Its missing Schmidt weight is also one half. A gauge change cannot overcome this rank restriction because it leaves the physical state unchanged.

Keep the idea

Take it with you

  • An exact MPS factorization and a restricted-bond approximation are different statements.
  • Canonical gauges turn block bases into orthonormal bases and expose Schmidt coefficients.
  • Bond dimension bounds rank and entropy but does not certify a chosen observable’s accuracy.

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