Gaussian quantum marginal problem
Jan Vlach
June 7, 2011
Quantum marginal problem
quantum ellectromagnetic oscillator with N modes
ith mode is characterised by ˆxi (position) and ˆpi (momentum)
reduction
we measure only subset of all modes
2 reduced systems S1 and S2
solution: Characterize possible states of reduced systems if
global state is given.
Quantum marginal problem
Simplifications
Gaussian states
characterise possible states of only one of the reduced
subsystems
Mathematical formulation
vector space V of dimension 2n
V = span(ˆx1, ˆp1, ˆx2, ˆp2, . . . , ˆxn, ˆpn)
symplectic form ω . . . matrix Ω
symmetric positive definite form γ . . . covariance matrix K
F = K−1Ω . . . operator on V
eigenvalues of iF . . . symplectic eigenvalues
symplectic eigenvalues determine state of the system
State of the art
Eisert, J., Tyc, T., Rudolph, T., Sanders, B.C.: Gaussian
Quantum Marginal Problem. Commun. Math. Phys. 280,
263-280 (2008)
global system reduced to N 1-mode systems
cj . . . symplectic eigenvalues of the ith mode
dk . . . kth global symplectic eigenvalue
State of the art
necessary and sufficient condition . . . n + 1 inequalities
k
j=1
cj ≥
k
j=1
dj , k = 1, . . . , N
and
cn −
N−1
j=1
cj ≤ dn −
N−1
j=1
dj
Williamson theorem
there exists a basis such that
K = diag(
1
µ1
,
1
µ2
, . . . ,
1
µn
,
1
µ1
,
1
µ2
, . . .
1
µn
)
and
Ω =
N
j=1
0 1
−1 0
where µ1, µ2, . . . , µn are symplectic eigenvalues
Restrictions on subsystems of M = N − 1 modes
M = N − 1:
µ1 ≥ µ2 ≥ · · · ≥ µN . . . global symplectic eigenvalues
ν1 ≥ ν2 ≥ · · · ≥ νN−1 . . . local symplectic eigenvalues
solution:
µ1 ≥ ν1 ≥ µ2 ≥ · · · ≥ νN−1 ≥ µN
Restrictions on subsystems of M < N modes
M < N:
µ1 ≥ µ2 ≥ · · · ≥ µN . . . global symplectic eigenvalues
ν1 ≥ ν2 ≥ · · · ≥ νM . . . local symplectic eigenvalues
solution:
µj ≥ νj ≥ µN−M+j , j < M + 1
Future work
find restrictions for reduced and complementary subsystem
together
consider arbitrary number of subsystems of arbitrary size
consider non-Gaussian states